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New fractional identities, associated novel fractional inequalities with applications to means and error estimations for quadrature formulas

Abstract

In this paper, the authors derive some new generalizations of fractional trapezium-like inequalities using the class of harmonic convex functions. Moreover, three new fractional integral identities are given, and on using them as auxiliary results some interesting integral inequalities are found. Finally, in order to show the efficiency of our main results, some applications to special means for different positive real numbers and error estimations for quadrature formulas are obtained.

Introduction and preliminaries

Computational and Fractional Analysis are nowadays more and more at the center of mathematics and of other related sciences, either by themselves because of their rapid development, which is based on very old foundations, or because they cover a great variety of applications in the real world. In recent years, fractional calculus \((FC)\) is applied in many phenomena in applied sciences, fluid mechanics, physics and also biology can be described as very effective using the mathematical tools of FC. The fractional derivatives have occurred in many applied sciences equations such as reaction and diffusion processes, system identification, velocity signal analysis, relaxation of damping behavior fabrics and creeping of polymer composites [1, 2, 26, 32]. A set \(S\in \mathbb{R}\) is said to be convex, if

$$\begin{aligned} (1-{\tau })b_{1}+{\tau } {b_{2}}\in S,\quad \forall b_{1},{b_{2}}\in S,{ \tau }\in [0,1]. \end{aligned}$$

Similarly, a function \({\Upsilon }:S\to \mathbb{R}\) is said to be convex, if

$$\begin{aligned} {\Upsilon }\bigl((1-{\tau })b_{1}+{\tau } {b_{2}}\bigr) \leq (1-{\tau }){\Upsilon }(b_{1})+{ \tau } {\Upsilon }({b_{2}}),\quad b_{1},{b_{2}}\in S,{\tau } \in [0,1]. \end{aligned}$$

Recently, İşcan [3] introduced the class of harmonic convex functions as:

A function \({\Upsilon }:I\subset (0,+\infty )\to \mathbb{R}\) is said to be harmonic convex, if

$$\begin{aligned} {\Upsilon } \biggl( \frac{{b_{1}{b_{2}}}}{{\tau b_{1}}+(1-{\tau }){{b_{2}}}} \biggr)\leq (1-{ \tau }){\Upsilon }({b_{1}})+{\tau } {\Upsilon }({{b_{2}}}),\quad \forall {b_{1}},{{b_{2}}}\in I,{\tau }\in [0,1]. \end{aligned}$$

The harmonic property has played a significant role in different fields of pure and applied sciences. In [4] the authors discussed the important role of the harmonic mean in Asian options of stock. Interestingly, harmonic means have applications in electric circuit theory. To be more precise, the total resistance of a set of parallel resistors is just half of the total resistor’s harmonic means. For example, if \(\mathcal{R}_{1}\) and \(\mathcal{R}_{2}\) are the resistances of two parallel resistors, then the total resistance is computed by the formula:

$$\begin{aligned} \mathcal{R}_{T}= \frac{\mathcal{R}_{1}\mathcal{R}_{2}}{\mathcal{R}_{1}+\mathcal{R}_{2}}= \frac{1}{2}H( \mathcal{R}_{1},\mathcal{R}_{2}), \end{aligned}$$

which is half of the harmonic mean.

Noor [5] showed that the harmonic mean also played a crucial role in developing parallel algorithms for solving nonlinear problems. The author used the harmonic means and harmonic convex functions to suggest some iterative methods for solving linear and nonlinear equations.

The theory of convexity also has a wide range of applications in other areas of pure and applied sciences. It also has a great impact on the development of the theory of inequalities. Several inequalities are consequences of the applications of convex functions. Many generalizations, variants and extensions for the convexity have attracted the attention of many researchers. An interesting result pertaining to convex functions is the trapezium inequality (Hermite–Hadamard inequality) that provides an integral average of a continuous convex function on a compact interval. This result reads as:

Let \({\Upsilon }:I=[{b_{1}},{{b_{2}}}]\subset \mathbb{R}\to \mathbb{R}\) be a convex function, then

$$\begin{aligned} {\Upsilon } \biggl(\frac{{b_{1}}+{{b_{2}}}}{2} \biggr)\leq \frac{1}{{{b_{2}}}-{b_{1}}} \int _{{b_{1}}}^{{{b_{2}}}}{ \Upsilon }(x) \,\mathrm{d}x\leq \frac{{\Upsilon }({b_{1}})+{\Upsilon }({{b_{2}}})}{2}. \end{aligned}$$

Over the years, a variety of new generalizations of this classical result have been obtained in the literature. For example, İşcan [3] obtained a new refinement of the trapezium inequality using the class of harmonic convex functions. He derived the following version of the trapezium inequality.

Let \({\Upsilon }:I=[{b_{1}},{{b_{2}}}]\subset (0,+\infty )\to \mathbb{R}\) be an harmonic convex function, then

$$\begin{aligned} {\Upsilon } \biggl(\frac{2{b_{1}{b_{2}}}}{{b_{1}}+{{b_{2}}}} \biggr) \leq \frac{{b_{1}{b_{2}}}}{{{b_{2}}}-{b_{1}}} \int _{{b_{1}}}^{{{b_{2}}}} \frac{{\Upsilon }(x)}{x^{2}} \,\mathrm{d}x \leq \frac{{\Upsilon }({b_{1}})+{\Upsilon }({{b_{2}}})}{2}. \end{aligned}$$

We now recall some useful definitions. For brevity, the set of integrable functions on the interval \([b_{1},b_{2}]\) is denoted by \(L_{1}[{b_{1}},{{b_{2}}}]\).

Definition 1.1

Let \({\Upsilon }\in L_{1}[{b_{1}},{{b_{2}}}]\). The left- and right-sided Riemann–Liouville fractional integrals \(J_{{b_{1}}^{+}}^{\alpha }{\Upsilon }\) and \(J_{{{b_{2}}}^{-}}^{\alpha }{\Upsilon }\) of order \(\alpha >0\) with \({b_{1}}\geq 0 \) are defined by

$$ J_{{b_{1}}^{+}}^{\alpha }{\Upsilon }(x)=\frac{1}{\Gamma (\alpha )} \int _{b_{1}}^{x} ( x-{\tau } ) ^{\alpha -1}{ \Upsilon }({ \tau }) \,\mathrm{d} {\tau },\quad x>b_{1} $$

and

$$ J_{{{b_{2}}}^{-}}^{\alpha }{\Upsilon }(x)=\frac{1}{\Gamma (\alpha )} \int _{x}^{{b_{2}}} ( {\tau }-x ) ^{\alpha -1}{ \Upsilon }({\tau }) \,\mathrm{d} {\tau },\quad x< {b_{2}}, $$

respectively, and \(\Gamma (\alpha )\) is Gamma function. Also, we define \(J_{{b_{1}}^{+}}^{0}{\Upsilon }(x)=J_{{{b_{2}}}^{-}}^{0}{\Upsilon }(x)={ \Upsilon }(x)\).

Definition 1.2

Let \({\Upsilon }\in L_{1}[{b_{1}},{{b_{2}}}]\). The k–Riemann–Liouville fractional integrals \(_{k}J_{{b_{1}}^{+}}^{\alpha }{\Upsilon }\) and \({}_{k}J_{{{b_{2}}}^{-}}^{\alpha }{\Upsilon }\) of order \(\alpha , k >0\) with \({b_{1}}\geq 0 \) are given as follows:

$$ _{k}J_{{b_{1}}^{+}}^{\alpha }{\Upsilon }(x)= \frac{1}{k\Gamma _{k} (\alpha )} \int _{b_{1}}^{x} ( x-{ \tau } ) ^{\frac{\alpha }{k} -1}{ \Upsilon }({\tau }) \,\mathrm{d} {\tau }, \quad x>b_{1} $$

and

$$ _{k}J_{{{b_{2}}}^{-}}^{\alpha }{\Upsilon }(x)= \frac{1}{k\Gamma _{k} (\alpha )} \int _{x}^{{b_{2}}} ( { \tau }-x ) ^{\frac{\alpha }{k} -1}{ \Upsilon }({\tau }) \,\mathrm{d} {\tau }, \quad x< {b_{2}}, $$

respectively.

Definition 1.3

A hypergeometric function \({_{2}F_{1}} (b_{1},{b_{2}},{b_{3}}, z )\) has the following integral representation

$$ {_{2}F_{1}} (b_{1},{b_{2}},{b_{3}}, z )= \frac{1}{\beta ({b_{2}}, {b_{3}}-{b_{2}} )} \int _{0}^{1}x^{{b_{2}}-1}(1-x)^{{b_{3}}-{b_{2}}-1}(1-zx)^{-b_{1}} \,\mathrm{d}x, \quad {b_{3}}>{b_{2}}>0, $$

where \(\beta (\cdot )\) is the Beta function and \(|z|<1\).

Sarikaya et al. [6] opened up a new direction of research in the field of inequalities involving convex functions. They derived a fractional version of the trapezium inequality. This result reads as:

Let \({\Upsilon }:[{b_{1}},{{b_{2}}}]\to \mathbb{R}\) be a positive function with \(0\leq {b_{1}}<{{b_{2}}}\) and \({\Upsilon }\in L_{1}[{b_{1}},{{b_{2}}}]\). If ϒ is a convex function on \([{b_{1}},{{b_{2}}}]\), then

$$\begin{aligned} {\Upsilon } \biggl(\frac{{b_{1}}+{{b_{2}}}}{2} \biggr)\leq \frac{\Gamma (\alpha +1)}{2({{b_{2}}}-{b_{1}})^{\alpha }} \bigl[J_{{b_{1}}^{+}}^{ \alpha }{\Upsilon }({{b_{2}}})+J_{{{b_{2}}}^{-}}^{\alpha }{ \Upsilon }({b_{1}})\bigr] \leq \frac{{\Upsilon }({b_{1}})+{\Upsilon }({{b_{2}}})}{2}, \end{aligned}$$

with \(\alpha >0\).

Using this idea, İşcan and Wu [7] obtained the fractional trapezium inequality using the class of harmonic convex functions. Their result is stated as follows:

Let \({\Upsilon }:[{b_{1}},{{b_{2}}}]\subset (0,+\infty )\to \mathbb{R}\) be a function with \({\Upsilon }\in L_{1}[{b_{1}},{{b_{2}}}]\). If ϒ is an harmonic convex function, then

$$\begin{aligned} {\Upsilon } \biggl(\frac{2{b_{1}{b_{2}}}}{{b_{1}}+{{b_{2}}}} \biggr)& \leq \frac{\Gamma (\alpha +1)}{2} \biggl( \frac{{b_{1}{b_{2}}}}{{{b_{2}}}-{b_{1}}} \biggr)^{\alpha } \biggl[J_{ \frac{1}{{b_{1}}}^{-}}^{\alpha }{ \Upsilon }\circ {\Psi } \biggl( \frac{1}{{{b_{2}}}} \biggr)+J_{\frac{1}{{{b_{2}}}}^{+}}^{\alpha }{ \Upsilon }\circ \Psi \biggl(\frac{1}{{b_{1}}} \biggr) \biggr]\\ &\leq \frac{{\Upsilon }({b_{1}})+{\Upsilon }({{b_{2}}})}{2}, \end{aligned}$$

where \(\alpha >0\) and \(\Psi (x):=\frac{1}{x}\).

For more details on the trapezium inequality, its generalizations and applications, see [8, 9, 1725, 2731].

The aim of this paper is to derive some new generalizations of fractional trapezium-like inequalities using the class of harmonic convex functions. In order to establish some of our main results, we derive three new fractional integral identities. These identities will be used as auxiliary results. Moreover, in order to show the efficiency of our main results, some applications to special means for positive different real numbers and error estimations for quadrature formulas will also be obtained. We also discuss special cases that show that our results represent significant generalizations and under suitable conditions one can obtain many other new and known results.

Before moving to the main results, let us recall some previously known concepts and results that will help us to obtain our main results. Let \(\Phi :[0,+\infty )\rightarrow [0,+\infty )\) be a function satisfying the following conditions:

  1. 1.

    \({\int _{0}^{1}\frac{\Phi ({\tau })}{{\tau }} \,\mathrm{d}{ \tau }}<+\infty \),

  2. 2.

    \(\frac{1}{M_{1}}\leq \frac{\Phi (s)}{\Phi (r)}\leq M_{1} \) for \(\frac{1}{2}\leq \frac{s}{r}\leq 2\),

  3. 3.

    \(\frac{\Phi (r)}{r^{2}}\leq M_{2}\frac{\Phi (s)}{s^{2}}\) for \(s\leq r\),

  4. 4.

    \(\vert \frac{\Phi (r)}{r^{2}}-\frac{\Phi (s)}{s^{2}} \vert \leq M_{3} \vert r-s \vert \frac{\Phi (r)}{r^{2}}\) for \(\frac{1}{2}\leq \frac{s}{r}\leq 2\),

where \(M_{1}\), \(M_{2}\) and \(M_{3}\) are independent of \(r,s>0\). Under the assumptions of Φ, the left- and right-sided generalized fractional integrals are

$$\begin{aligned}& {_{{b_{1}}^{+}}I_{\Phi }} {\Upsilon }(x)= \int _{b_{1}}^{x} \frac{\Phi (x-{\tau })}{x-{\tau }}{\Upsilon }( \tau ) \,\mathrm{d} {\tau }, \quad x>{b_{1}}, \end{aligned}$$
(1.1)
$$\begin{aligned}& {_{{{b_{2}}}^{-}}I_{\Phi }} {\Upsilon }(x)= \int _{x}^{{b_{2}}} \frac{\Phi ({\tau }-x)}{{\tau }-x}{\Upsilon }( \tau ) \,\mathrm{d} {\tau }, \quad x< {{b_{2}}}. \end{aligned}$$
(1.2)

Actually, these fractional integrals are the generalization of some well-known fractional integrals like the Riemann–Liouville fractional integrals [10], the k–Riemann–Liouville fractional integrals [11], the Katugampola fractional integrals [12], conformable fractional integrals [13], etc.

  1. 1.

    If we take \(\Phi ({\tau })={\tau }\) in operators (1.1) and (1.2), we have the classical Riemann integrals.

  2. 2.

    If we choose \(\Phi ({\tau })=\frac{{\tau }^{\alpha }}{\Gamma (\alpha )}\) in operators (1.1) and (1.2), we obtain the Riemann–Liouville fractional integrals, see [10].

  3. 3.

    If we substitute \(\Phi ({\tau })= \frac{{\tau }^{\frac{\alpha }{k}}}{k\Gamma _{k}(\alpha )}\) in operators (1.1) and (1.2), we obtain the k–Riemann–Liouville fractional integrals, see [11].

  4. 4.

    If we take \(\Phi ({\tau })={\tau }(x-{\tau })^{\alpha -1}\) in operators (1.1) and (1.2), we have conformable fractional integrals that are defined by Khalil et al. [14].

  5. 5.

    If we choose \(\Phi ({\tau })=\frac{{\tau }}{\alpha }\exp (- \frac{1-\alpha }{\alpha }{\tau } )\) for \(\alpha \in (0,1]\), in operators (1.1) and (1.2), we get left-sided and right-sided fractional integrals with an exponential kernel that were defined in [15, 16].

Main results

In this section, before we discuss our main results, let us denote, respectively

$$\begin{aligned}& \Delta ({\tau }):= \int _{0}^{\tau } \frac{\Phi (\frac{{{b_{2}}}-{b_{1}}}{{b_{1}{b_{2}}}}\mu )}{\mu } \,\mathrm{d} \mu \quad \text{and}\quad \delta ({\tau }):= \int _{\tau }^{1} \frac{\Phi (\frac{{{b_{2}}}-{b_{1}}}{{b_{1}{b_{2}}}}\mu )}{\mu } \,\mathrm{d} \mu , \\& \eta ({\tau }):= \int _{0}^{\tau } \frac{\Phi (\frac{{{b_{2}}}-{b_{1}}}{{b_{1}{b_{2}}}(m+1)}\mu )}{\mu } \,\mathrm{d} \mu \quad \text{and}\quad \Omega ({\tau }):= \int _{0}^{\tau } \frac{\Phi (\frac{{{b_{2}}}-{b_{1}}}{{b_{1}{b_{2}}}(\lambda +\mu )}\mu )}{\mu } \,\mathrm{d} \mu . \end{aligned}$$

Generalized trapezium inequality

We now derive a new generalized fractional trapezium-type integral inequality using the class of harmonic convex functions. For brevity, we denote in the following \({\Psi }(\tau ):={\frac{1}{\tau }}\).

Theorem 2.1

Let \({\Upsilon }:[{b_{1}},{{b_{2}}}]\rightarrow \mathbb{R}\) be an harmonic convex function, then

$$\begin{aligned} {\Upsilon } \biggl(\frac{2{b_{1}{b_{2}}}}{{b_{1}}+{{b_{2}}}} \biggr)& \leq \frac{1}{2\eta (1)} \biggl[{_{ (\frac{1}{{{b_{2}}}} )^{+}}I_{ \Phi }} {\Upsilon \circ \Psi } \biggl( \frac{m{b_{1}}+{{b_{2}}}}{(m+1){b_{1}{b_{2}}}} \biggr) +{_{ ( \frac{1}{{b_{1}}} )^{-}}I_{\Phi }} {\Upsilon \circ \Psi } \biggl( \frac{{b_{1}}+m{{b_{2}}}}{(m+1){b_{1}{b_{2}}}} \biggr) \biggr] \\ &\leq \frac{[{\Upsilon }({b_{1}})+{\Upsilon }({{b_{2}}})]}{2}, \end{aligned}$$

where \(m\in \mathbb{N}\).

Proof

Since ϒ is an harmonic convex function, then

$$\begin{aligned} {\Upsilon } \biggl(\frac{2xy}{x+y} \biggr)\leq \frac{1}{2} \bigl[{\Upsilon }(x)+{ \Upsilon }(y)\bigr]. \end{aligned}$$

This implies

$$\begin{aligned} &2{\Upsilon } \biggl(\frac{2{b_{1}{b_{2}}}}{{b_{1}}+{{b_{2}}}} \biggr) \leq {\Upsilon } \biggl( \frac{(m+1){b_{1}{b_{2}}}}{(1-{\tau }){b_{1}}+(m+{\tau }){{b_{2}}}} \biggr)+{\Upsilon } \biggl( \frac{(m+1){b_{1}{b_{2}}}}{(m+{\tau }){b_{1}}+(1-{\tau }){{b_{2}}}} \biggr). \end{aligned}$$

Multiplying both sides by \(\frac{\Phi (\frac{{{b_{2}}}-{b_{1}}}{(m+1){b_{1}{b_{2}}}}{\tau } )}{{\tau }}\) and integrating with respect to τ on \([0,1]\), we have

$$\begin{aligned} &2{\Upsilon } \biggl(\frac{2{b_{1}{b_{2}}}}{{b_{1}}+{{b_{2}}}} \biggr) \int _{0}^{1} \frac{\Phi (\frac{{{b_{2}}}-{b_{1}}}{(m+1){b_{1}{b_{2}}}}{\tau } )}{{\tau }} \,\mathrm{d} {\tau } \\ &\quad \leq \biggl[ \int _{0}^{1} \frac{\Phi (\frac{{{b_{2}}}-{b_{1}}}{(m+1){b_{1}{b_{2}}}}{\tau } )}{{\tau }}{ \Upsilon } \biggl( \frac{(m+1){b_{1}{b_{2}}}}{(1-{\tau }){b_{1}}+(m+{\tau }){{b_{2}}}} \biggr) \,\mathrm{d} {\tau } \\ &\qquad{} + \int _{0}^{1} \frac{\Phi (\frac{{{b_{2}}}-{b_{1}}}{(m+1){b_{1}{b_{2}}}}{\tau } )}{{\tau }}{ \Upsilon } \biggl( \frac{(m+1){b_{1}{b_{2}}}}{(m+{\tau }){b_{1}}+(1-{\tau }){{b_{2}}}} \biggr) \,\mathrm{d} {\tau } \biggr]. \end{aligned}$$

This implies

$$\begin{aligned} &{2\eta (1)} {\Upsilon } \biggl( \frac{2{b_{1}{b_{2}}}}{{b_{1}}+{{b_{2}}}} \biggr) \\ &\quad \leq \int _{0}^{1} \frac{\Phi (\frac{{{b_{2}}}-{b_{1}}}{(m+1){b_{1}{b_{2}}}}{\tau } )}{{\tau }}{ \Upsilon } \biggl( \frac{(m+1){b_{1}{b_{2}}}}{(1-{\tau }){b_{1}}+(m+{\tau }){{b_{2}}}} \biggr) \,\mathrm{d} {\tau } \\ &\qquad{} + \int _{0}^{1} \frac{\Phi (\frac{{{b_{2}}}-{b_{1}}}{(m+1){b_{1}{b_{2}}}}{\tau } )}{{\tau }}{ \Upsilon } \biggl( \frac{(m+1){b_{1}{b_{2}}}}{(m+{\tau }){b_{1}}+(1-{\tau }){{b_{2}}}} \biggr) \,\mathrm{d} {\tau } \\ &\quad = \int _{\frac{1}{{{b_{2}}}}}^{ \frac{m{b_{1}}+{{b_{2}}}}{(m+1){b_{1}{b_{2}}}}} \frac{\Phi (\frac{m{b_{1}}+{{b_{2}}}}{(m+1){b_{1}{b_{2}}}}-x )}{ (\frac{m{b_{1}}+{{b_{2}}}}{(m+1){b_{1}{b_{2}}}}-x )}{ \Upsilon \circ \Psi }(x) \,\mathrm{d}x+ \int _{ \frac{{b_{1}}+m{{b_{2}}}}{(m+1){b_{1}{b_{2}}}}}^{\frac{1}{{b_{1}}}} \frac{\Phi (x-\frac{{b_{1}}+m{{b_{2}}}}{(m+1){b_{1}{b_{2}}}} )}{ (x-\frac{{b_{1}}+m{{b_{2}}}}{(m+1){b_{1}{b_{2}}}} )}{ \Upsilon \circ \Psi }(x) \,\mathrm{d}x \\ &\quad ={{}_{ (\frac{1}{{{b_{2}}}} )^{+}}I_{\Phi }} {\Upsilon \circ \Psi } \biggl( \frac{m{b_{1}}+{{b_{2}}}}{(m+1){b_{1}{b_{2}}}} \biggr)+{_{ (\frac{1}{{b_{1}}} )^{-}}I_{\Phi }} {\Upsilon \circ \Psi } \biggl(\frac{{b_{1}}+m{{b_{2}}}}{(m+1){b_{1}{b_{2}}}} \biggr). \end{aligned}$$

Now, we prove the second inequality, for this we have

$$\begin{aligned} &{\Upsilon } \biggl( \frac{(m+1){b_{1}{b_{2}}}}{(1-{\tau }){b_{1}}+(m+{\tau }){{b_{2}}}} \biggr)\leq \frac{m+{\tau }}{m+1}{ \Upsilon }({b_{1}})+ \frac{1-{\tau }}{m+1}{\Upsilon }({{b_{2}}}). \end{aligned}$$
(2.1)
$$\begin{aligned} &{\Upsilon } \biggl( \frac{(m+1){b_{1}{b_{2}}}}{(m+{\tau }){b_{1}}+(1-{\tau }){{b_{2}}}} \biggr)\leq \frac{m+{\tau }}{m+1}{ \Upsilon }({{b_{2}}})+ \frac{1-{\tau }}{m+1}{\Upsilon }({b_{1}}). \end{aligned}$$
(2.2)

Adding (2.1) and (2.2) and multiplying both sides by \(\frac{\Phi (\frac{{{b_{2}}}-{b_{1}}}{(m+1){b_{1}{b_{2}}}}{\tau } )}{{\tau }}\) and integrating with respect to τ on \([0,1]\), we have

$$\begin{aligned} & \int _{0}^{1} \frac{\Phi (\frac{{{b_{2}}}-{b_{1}}}{(m+1){b_{1}{b_{2}}}}{\tau } )}{{\tau }}{ \Upsilon } \biggl( \frac{(m+1){b_{1}{b_{2}}}}{(1-{\tau }){b_{1}}+(m+{\tau }){{b_{2}}}} \biggr) \,\mathrm{d} {\tau }\\ &\qquad {}+ \int _{0}^{1} \frac{\Phi (\frac{{{b_{2}}}-{b_{1}}}{(m+1){b_{1}{b_{2}}}}{\tau } )}{{\tau }}{ \Upsilon } \biggl( \frac{(m+1){b_{1}{b_{2}}}}{(m+{\tau }){b_{1}}+(1-{\tau }){{b_{2}}}} \biggr) \,\mathrm{d} {\tau } \\ &\quad \leq {\bigl[{\Upsilon }({b_{1}})+{\Upsilon }({{b_{2}}}) \bigr]} \int _{0}^{1} \frac{\Phi (\frac{{{b_{2}}}-{b_{1}}}{(m+1){b_{1}{b_{2}}}}{\tau } )}{{\tau }} \,\mathrm{d} {\tau }. \end{aligned}$$

Using generalized fractional integrals, we obtain our second inequality. This completes the proof. □

Corollary 2.1

If we choose \(\Phi ({\tau })={\tau }\) and \(m=1\) in Theorem 2.1, we have

$$\begin{aligned} {\Upsilon } \biggl(\frac{2{b_{1}{b_{2}}}}{{b_{1}}+{{b_{2}}}} \biggr)& \leq \frac{{b_{1}{b_{2}}}}{{{b_{2}}}-{b_{1}}} \int _{ \frac{1}{{{b_{2}}}}}^{\frac{1}{{b_{1}}}}{\Upsilon \circ \Psi }(x) \,\mathrm{d}x\leq \frac{[{\Upsilon }({b_{1}})+{\Upsilon }({{b_{2}}})]}{2}. \end{aligned}$$

Corollary 2.2

If we choose \(\Phi ({\tau })=\frac{{\tau }^{{\alpha }}}{\Gamma (\alpha )}\) in Theorem 2.1, we obtain

$$\begin{aligned} {\Upsilon } \biggl(\frac{2{b_{1}{b_{2}}}}{{b_{1}}+{{b_{2}}}} \biggr)\leq{}& \frac{({b_{1}{b_{2}}}(m+1))^{\alpha }\Gamma (\alpha +1)}{2({{b_{2}}}-{b_{1}})^{\alpha }}\\ &{}\times \biggl[{J_{ (\frac{1}{{{b_{2}}}} )^{+}}^{\alpha }} {\Upsilon \circ \Psi } \biggl( \frac{m{b_{1}}+{{b_{2}}}}{(m+1){b_{1}{b_{2}}}} \biggr) +{J_{ (\frac{1}{{b_{1}}} )^{-}}} {\Upsilon \circ \Psi } \biggl( \frac{{b_{1}}+m{{b_{2}}}}{(m+1){b_{1}{b_{2}}}} \biggr) \biggr] \\ \leq {}&\frac{[{\Upsilon }({b_{1}})+{\Upsilon }({{b_{2}}})]}{2}. \end{aligned}$$

For \(m=1\), we obtain

$$\begin{aligned} {\Upsilon } \biggl(\frac{2{b_{1}{b_{2}}}}{{b_{1}}+{{b_{2}}}} \biggr)\leq {}& \frac{2^{\alpha -1}({b_{1}{b_{2}}})^{\alpha }\Gamma (\alpha +1)}{({{b_{2}}}-{b_{1}})^{\alpha }}\\ &{}\times \biggl[{J_{ (\frac{1}{{{b_{2}}}} )^{+}}^{\alpha }} {\Upsilon \circ \Psi } \biggl( \frac{{b_{1}}+{{b_{2}}}}{2{b_{1}{b_{2}}}} \biggr) +{J_{ (\frac{1}{{b_{1}}} )^{-}}} {\Upsilon \circ \Psi } \biggl( \frac{{b_{1}}+{{b_{2}}}}{2{b_{1}{b_{2}}}} \biggr) \biggr] \\ \leq {}&\frac{[{\Upsilon }({b_{1}})+{\Upsilon }({{b_{2}}})]}{2}. \end{aligned}$$

Corollary 2.3

If we choose \(\Phi ({\tau })= \frac{{\tau }^{\frac{\alpha }{k}}}{k\Gamma _{k}(\alpha )}\) in Theorem 2.1, we have

$$\begin{aligned} {\Upsilon } \biggl(\frac{2{b_{1}{b_{2}}}}{{b_{1}}+{{b_{2}}}} \biggr) \leq {}&\frac{((m+1){b_{1}{b_{2}}})^{\frac{\alpha }{k}}\Gamma _{k}(\alpha +k)}{2({{b_{2}}}-{b_{1}})^{\frac{\alpha }{k}}} \\ &{}\times\biggl[{_{k}J_{ (\frac{1}{{{b_{2}}}} )^{+}}^{\alpha }} { \Upsilon \circ \Psi } \biggl( \frac{m{b_{1}}+{{b_{2}}}}{(m+1){b_{1}{b_{2}}}} \biggr) +{_{k}J_{ (\frac{1}{{b_{1}}} )^{-}}^{\alpha }} {\Upsilon \circ \Psi } \biggl( \frac{{b_{1}}+m{{b_{2}}}}{(m+1){b_{1}{b_{2}}}} \biggr) \biggr] \\ \leq {}&\frac{[{\Upsilon }({b_{1}})+{\Upsilon }({{b_{2}}})]}{2}. \end{aligned}$$

For \(m=1\), we obtain

$$\begin{aligned} {\Upsilon } \biggl(\frac{2{b_{1}{b_{2}}}}{{b_{1}}+{{b_{2}}}} \biggr) \leq {}&\frac{2^{\frac{\alpha }{k}-1}({b_{1}{b_{2}}})^{\frac{\alpha }{k}}\Gamma _{k}(\alpha +k)}{({{b_{2}}}-{b_{1}})^{\frac{\alpha }{k}}} \\ &{}\times \biggl[{_{k}J_{ (\frac{1}{{{b_{2}}}} )^{+}}^{\alpha }} { \Upsilon \circ \Psi } \biggl(\frac{{b_{1}}+{{b_{2}}}}{2{b_{1}{b_{2}}}} \biggr) +{_{k}J_{ (\frac{1}{{b_{1}}} )^{-}}^{\alpha }} { \Upsilon \circ \Psi } \biggl(\frac{{b_{1}}+{{b_{2}}}}{2{b_{1}{b_{2}}}} \biggr) \biggr] \\ \leq{}& \frac{[{\Upsilon }({b_{1}})+{\Upsilon }({{b_{2}}})]}{2}. \end{aligned}$$

Auxiliary results

In this subsection, we derive three new fractional integral identities that will be used in the following.

Lemma 2.2

Let \({\Upsilon }:[{b_{1}},{{b_{2}}}]\rightarrow \mathbb{R}\) be a differentiable function on \(({b_{1}},{{b_{2}}})\) with \({b_{1}}<{{b_{2}}}\) and \(m\in \mathbb{N}\), then

$$\begin{aligned} &\frac{{\Upsilon }({b_{1}})+{\Upsilon }({{b_{2}}})}{m+1}\\ &\qquad {}- \frac{1}{(m+1)\eta (1)} \biggl[{_{ (\frac{1}{{{b_{2}}}} )^{+}}I_{ \Phi }} {\Upsilon \circ \Psi } \biggl( \frac{m{b_{1}}+{{b_{2}}}}{(m+1){b_{1}{b_{2}}}} \biggr) +{_{ ( \frac{1}{{b_{1}}} )^{-}}I_{\Phi }} {\Upsilon \circ \Psi } \biggl( \frac{{b_{1}}+m{{b_{2}}}}{(m+1){b_{1}{b_{2}}}} \biggr) \biggr] \\ &\quad =\frac{{b_{1}{b_{2}}}({{b_{2}}}-{b_{1}})}{\eta (1)} \biggl[ \int _{0}^{1} \frac{\eta ({\tau })}{((m+{\tau }){b_{1}}+(1-{\tau }){{b_{2}}})^{2}}{ \Upsilon }' \biggl( \frac{(m+1){b_{1}{b_{2}}}}{(m+{\tau }){b_{1}}+(1-{\tau }){{b_{2}}}} \biggr) \,\mathrm{d} {\tau } \\ &\qquad {} - \int _{0}^{1} \frac{\eta ({\tau })}{((1-{\tau }){b_{1}}+(m+{\tau }){{b_{2}}})^{2}}{ \Upsilon }' \biggl( \frac{(m+1){b_{1}{b_{2}}}}{(1-{\tau }){b_{1}}+(m+{\tau }){{b_{2}}}} \biggr) \,\mathrm{d} {\tau } \biggr]. \end{aligned}$$

Proof

Consider the right-hand side

$$\begin{aligned} I:={}&\frac{{b_{1}{b_{2}}}({{b_{2}}}-{b_{1}})}{\eta (1)} \biggl[ \int _{0}^{1} \frac{\eta ({\tau })}{((m+{\tau }){b_{1}}+(1-{\tau }){{b_{2}}})^{2}}{ \Upsilon }' \biggl( \frac{(m+1){b_{1}{b_{2}}}}{(m+{\tau }){b_{1}}+(1-{\tau }){{b_{2}}}} \biggr) \,\mathrm{d} {\tau } \\ &{} - \int _{0}^{1} \frac{\eta ({\tau })}{((1-{\tau }){b_{1}}+(m+{\tau }){{b_{2}}})^{2}}{ \Upsilon }' \biggl( \frac{(m+1){b_{1}{b_{2}}}}{(1-{\tau }){b_{1}}+(m+{\tau }){{b_{2}}}} \biggr) \,\mathrm{d} {\tau } \biggr] \\ ={}&\frac{{b_{1}{b_{2}}}({{b_{2}}}-{b_{1}})}{\eta (1)}[I_{1}-I_{2}], \end{aligned}$$

where

$$\begin{aligned} I_{1}:={}& \int _{0}^{1} \frac{\eta ({\tau })}{((m+{\tau }){b_{1}}+(1-{\tau }){{b_{2}}})^{2}}{ \Upsilon }' \biggl( \frac{(m+1){b_{1}{b_{2}}}}{(m+{\tau }){b_{1}}+(1-{\tau }){{b_{2}}}} \biggr) \,\mathrm{d} {\tau } \\ ={}& \frac{\eta (1){\Upsilon }({{b_{2}}})}{(m+1){b_{1}{b_{2}}}({{b_{2}}}-{b_{1}})}\\ &{}- \frac{1}{{b_{1}{b_{2}}}({{b_{2}}}-{b_{1}})(m+1)} \int _{ \frac{1}{{{b_{2}}}}}^{\frac{m{b_{1}}+{{b_{2}}}}{(m+1){b_{1}{b_{2}}}}} \frac{\Phi (\frac{m{b_{1}}+{{b_{2}}}}{(m+1){b_{1}{b_{2}}}}-x )}{\frac{m{b_{1}}+{{b_{2}}}}{(m+1){b_{1}{b_{2}}}}-x}{ \Upsilon \circ \Psi }(x) \,\mathrm{d}x \\ ={}& \frac{\eta (1){\Upsilon }({{b_{2}}})}{(m+1){b_{1}{b_{2}}}({{b_{2}}}-{b_{1}})}- \frac{1}{{b_{1}{b_{2}}}({{b_{2}}}-{b_{1}})(m+1)}{_{ ( \frac{1}{{{b_{2}}}} )^{+}}I_{\Phi }} {\Upsilon \circ \Psi } \biggl( \frac{m{b_{1}}+{{b_{2}}}}{(m+1){b_{1}{b_{2}}}} \biggr). \end{aligned}$$

Similarly,

$$\begin{aligned} I_{2}:={}& \int _{0}^{1} \frac{\eta ({\tau })}{((1-{\tau }){b_{1}}+(m+{\tau }){{b_{2}}})^{2}}{ \Upsilon }' \biggl( \frac{(m+1){b_{1}{b_{2}}}}{(1-{\tau }){b_{1}}+(m+{\tau }){{b_{2}}}} \biggr) \,\mathrm{d} {\tau } \\ ={}&{-} \frac{\eta (1){\Upsilon }({b_{1}})}{(m+1){b_{1}{b_{2}}}({{b_{2}}}-{b_{1}})}\\ &{}+ \frac{1}{{b_{1}{b_{2}}}({{b_{2}}}-{b_{1}})(m+1)} \int _{ \frac{{b_{1}}+m{{b_{2}}}}{(m+1){b_{1}{b_{2}}}}}^{\frac{1}{{b_{1}}}} \frac{\Phi (x-\frac{{b_{1}}+m{{b_{2}}}}{(m+1){b_{1}{b_{2}}}} )}{x-\frac{{b_{1}}+m{{b_{2}}}}{(m+1){b_{1}{b_{2}}}}}{ \Upsilon \circ \Psi }(x) \,\mathrm{d}x \\ ={}&{-} \frac{\eta (1){\Upsilon }({b_{1}})}{(m+1){b_{1}{b_{2}}}({{b_{2}}}-{b_{1}})}+ \frac{1}{{b_{1}{b_{2}}}({{b_{2}}}-{b_{1}})(m+1)}{_{ ( \frac{1}{{b_{1}}} )^{-}}I_{\Phi }} {\Upsilon \circ \Psi } \biggl( \frac{{b_{1}}+m{{b_{2}}}}{(m+1){b_{1}{b_{2}}}} \biggr). \end{aligned}$$

Substituting the values of \(I_{1}\) and \(I_{2}\) in I, we obtain our required result. □

Remark 2.1

If we choose \(m=1\) and \(\Phi ({\tau })={\tau }\), we have

$$\begin{aligned} &\frac{{\Upsilon }({b_{1}})+{\Upsilon }({{b_{2}}})}{2}- \frac{{b_{1}{b_{2}}}}{({{b_{2}}}-{b_{1}})} \int _{\frac{1}{{{b_{2}}}}}^{ \frac{1}{{b_{1}}}}{\Upsilon \circ \Psi }(x) \,\mathrm{d}x \\ &\quad ={{b_{1}{b_{2}}}({{b_{2}}}-{b_{1}})} \biggl[ \int _{0}^{1} \frac{{\tau }}{((1+{\tau }){b_{1}}+(1-{\tau }){{b_{2}}})^{2}}{\Upsilon }' \biggl(\frac{2{b_{1}{b_{2}}}}{(1+{\tau }){b_{1}}+(1-{\tau }){{b_{2}}}} \biggr) \,\mathrm{d} {\tau } \\ & \qquad {}- \int _{0}^{1} \frac{{\tau }}{((1-{\tau }){b_{1}}+(1+{\tau }){{b_{2}}})^{2}}{\Upsilon }' \biggl(\frac{2{b_{1}{b_{2}}}}{(1-{\tau }){b_{1}}+(1+{\tau }){{b_{2}}}} \biggr) \,\mathrm{d} {\tau } \biggr]. \end{aligned}$$

Corollary 2.4

If we take \(m=1\) and \(\Phi (\tau )=\frac{{\tau }^{\alpha }}{\Gamma (\alpha )}\) in Lemma 2.2, we obtain

$$\begin{aligned} &\frac{{\Upsilon }({b_{1}})+{\Upsilon }({{b_{2}}})}{2}\\ &\qquad {}- \frac{2^{\alpha -1}({b_{1}{b_{2}}})^{\alpha }\Gamma (\alpha +1)}{({{b_{2}}}-{b_{1}})^{\alpha }} \biggl[{J_{ (\frac{1}{{{b_{2}}}} )^{+}}^{\alpha }} {\Upsilon \circ \Psi } \biggl(\frac{{b_{1}}+{{b_{2}}}}{2{b_{1}{b_{2}}}} \biggr) +J_{ (\frac{1}{{b_{1}}} )^{-}}^{\alpha }{ \Upsilon \circ \Psi } \biggl(\frac{{b_{1}}+{{b_{2}}}}{2{b_{1}{b_{2}}}} \biggr) \biggr] \\ &\quad ={{b_{1}{b_{2}}}({{b_{2}}}-{b_{1}})} \biggl[ \int _{0}^{1} \frac{{\tau }^{\alpha }}{((1+{\tau }){b_{1}}+(1-{\tau }){{b_{2}}})^{2}}{ \Upsilon }' \biggl( \frac{2{b_{1}{b_{2}}}}{(1+{\tau }){b_{1}}+(1-{\tau }){{b_{2}}}} \biggr) \,\mathrm{d} {\tau } \\ &\qquad {} - \int _{0}^{1} \frac{{\tau }^{\alpha }}{((1-{\tau }){b_{1}}+(1+{\tau }){{b_{2}}})^{2}}{ \Upsilon }' \biggl( \frac{2{b_{1}{b_{2}}}}{(1-{\tau }){b_{1}}+(1-{\tau }){{b_{2}}}} \biggr) \,\mathrm{d} {\tau } \biggr]. \end{aligned}$$

Corollary 2.5

If we choose \(m=1\) and \(\Phi (\tau )=\frac{{\tau }^{\frac{\alpha }{k}}}{k\Gamma _{k}(\alpha )}\) in Lemma 2.2, we obtain

$$\begin{aligned} &\frac{{\Upsilon }({b_{1}})+{\Upsilon }({{b_{2}}})}{2}\\ &\qquad {}- \frac{2^{\frac{\alpha }{k}-1}({b_{1}{b_{2}}})^{\frac{\alpha }{k}}\Gamma _{k}(\alpha +k)}{({{b_{2}}}-{b_{1}})^{\frac{\alpha }{k}}} \biggl[{_{k}J_{ (\frac{1}{{{b_{2}}}} )^{+}}^{\alpha }} { \Upsilon \circ \Psi } \biggl(\frac{{b_{1}}+{{b_{2}}}}{2{b_{1}{b_{2}}}} \biggr) +{_{k}J_{ (\frac{1}{{b_{1}}} )^{-}}^{\alpha }} { \Upsilon \circ \Psi } \biggl(\frac{{b_{1}}+{{b_{2}}}}{2{b_{1}{b_{2}}}} \biggr) \biggr] \\ &\quad ={{b_{1}{b_{2}}}({{b_{2}}}-{b_{1}})} \biggl[ \int _{0}^{1} \frac{{\tau }^{\frac{\alpha }{k} }}{((1+{\tau }){b_{1}}+(1-{\tau }){{b_{2}}})^{2}}{ \Upsilon }' \biggl( \frac{2{b_{1}{b_{2}}}}{(1+{\tau }){b_{1}}+(1-{\tau }){{b_{2}}}} \biggr) \,\mathrm{d} {\tau } \\ &\qquad {} - \int _{0}^{1} \frac{{\tau }^{\frac{\alpha }{k}}}{((1-{\tau }){b_{1}}+(1+{\tau }){{b_{2}}})^{2}}{ \Upsilon }' \biggl( \frac{2{b_{1}{b_{2}}}}{(1-{\tau }){b_{1}}+(1-{\tau }){{b_{2}}}} \biggr) \,\mathrm{d} {\tau } \biggr]. \end{aligned}$$

Lemma 2.3

Let \(\Upsilon :[{b_{1}},{{b_{2}}}]\rightarrow \mathbb{R}\) be a differentiable function on \(({b_{1}},{{b_{2}}})\) with \({b_{1}}<{{b_{2}}}\) and \(\lambda ,\mu \in [0,\infty )\) with \(\lambda +\mu \neq0\), then

$$\begin{aligned} &\frac{\Omega (\lambda )\Upsilon ({{b_{2}}})+\Omega (\mu )\Upsilon ({b_{1}})}{\lambda +\mu }\\ &\qquad {}- \frac{1}{\lambda +\mu } \biggl[{_{ (\frac{1}{{{b_{2}}}} )^{+}}I_{ \Phi }{ \Upsilon \circ \Psi } \biggl( \frac{\lambda {{b_{2}}}+\mu {b_{1}}}{{b_{1}{b_{2}}}(\lambda +\mu )} \biggr)} +{_{ (\frac{1}{{b_{1}}} )^{-}}I_{\Phi }{ \Upsilon \circ \Psi } \biggl( \frac{\lambda {{b_{2}}}+\mu {b_{1}}}{{b_{1}{b_{2}}}(\lambda +\mu )} \biggr)} \biggr] \\ &\quad ={b_{1}{b_{2}}}({{b_{2}}}-{b_{1}}) \biggl[ \int _{0}^{\lambda } \frac{\Omega ({\tau })}{((\lambda -{\tau }){{b_{2}}}+(\mu +{\tau }){b_{1}})^{2}} \Upsilon ' \biggl( \frac{{b_{1}{b_{2}}}(\lambda +\mu )}{(\lambda -{\tau }){{b_{2}}}+(\mu +{\tau }){b_{1}}} \biggr) \,\mathrm{d} {\tau } \\ &\qquad {}- \int _{0}^{\mu } \frac{\Omega ({\tau })}{((\lambda +{\tau }){{b_{2}}}+(\mu -{\tau }){b_{1}})^{2}} \Upsilon ' \biggl( \frac{{b_{1}{b_{2}}}(\lambda +\mu )}{(\lambda +{\tau }){{b_{2}}}+(\mu -{\tau }){b_{1}}} \biggr) \,\mathrm{d} {\tau } \biggr]. \end{aligned}$$

Proof

Consider the right-hand side

$$\begin{aligned} I:={}&{b_{1}{b_{2}}}({{b_{2}}}-{b_{1}}) \biggl[ \int _{0}^{\lambda } \frac{\Omega ({\tau })}{((\lambda -{\tau }){{b_{2}}}+(\mu +{\tau }){b_{1}})^{2}} \Upsilon ' \biggl( \frac{{b_{1}{b_{2}}}(\lambda +\mu )}{(\lambda -{\tau }){{b_{2}}}+(\mu +{\tau }){b_{1}}} \biggr) \,\mathrm{d} {\tau } \\ &{}- \int _{0}^{\mu } \frac{\Omega ({\tau })}{((\lambda +{\tau }){{b_{2}}}+(\mu -{\tau }){b_{1}})^{2}} \Upsilon ' \biggl( \frac{{b_{1}{b_{2}}}(\lambda +\mu )}{(\lambda +{\tau }){{b_{2}}}+(\mu -{\tau }){b_{1}}} \biggr) \,\mathrm{d} {\tau } \biggr] \\ ={}&{b_{1}{b_{2}}}({{b_{2}}}-{b_{1}})[I_{3}-I_{4}], \end{aligned}$$
(2.3)

where

$$\begin{aligned} I_{3}:={}& \int _{0}^{\lambda } \frac{\Omega ({\tau })}{((\lambda -{\tau }){{b_{2}}}+(\mu +{\tau }){b_{1}})^{2}} \Upsilon ' \biggl( \frac{{b_{1}{b_{2}}}(\lambda +\mu )}{(\lambda -{\tau }){{b_{2}}}+(\mu +{\tau }){b_{1}}} \biggr) \,\mathrm{d} {\tau } \\ ={}& \frac{\Omega (\lambda )\Upsilon ({{b_{2}}})}{{b_{1}{b_{2}}}({{b_{2}}}-{b_{1}})(\lambda +\mu )}\\ &{}- \frac{1}{{b_{1}{b_{2}}}({{b_{2}}}-{b_{1}})(\lambda +\mu )} \int _{ \frac{1}{{{b_{2}}}}}^{ \frac{\lambda {{b_{2}}}+\mu {b_{1}}}{{b_{1}{b_{2}}}(\lambda +\mu )}} \frac{\Phi (\frac{\lambda {{b_{2}}}+\mu {b_{1}}}{{b_{1}{b_{2}}}(\lambda +\mu )}-x )}{ (\frac{\lambda {{b_{2}}}+\mu {b_{1}}}{{b_{1}{b_{2}}}(\lambda +\mu )}-x )}{ \Upsilon \circ \Psi }(x) \,\mathrm{d}x \\ ={}& \frac{\Omega (\lambda )\Upsilon ({{b_{2}}})}{{b_{1}{b_{2}}}({{b_{2}}}-{b_{1}})(\lambda +\mu )}- \frac{1}{{b_{1}{b_{2}}}({{b_{2}}}-{b_{1}})(\lambda +\mu )}{_{ ( \frac{1}{{{b_{2}}}} )^{+}}I_{\Phi }{ \Upsilon \circ \Psi } \biggl( \frac{\lambda {{b_{2}}}+\mu {b_{1}}}{{b_{1}{b_{2}}}(\lambda +\mu )} \biggr)}. \end{aligned}$$

Similarly,

$$\begin{aligned} I_{4}:={}& \int _{0}^{\mu } \frac{\Omega ({\tau })}{((\lambda +{\tau }){{b_{2}}}+(\mu -{\tau }){b_{1}})^{2}} \Upsilon ' \biggl( \frac{{b_{1}{b_{2}}}(\lambda +\mu )}{(\lambda +{\tau }){{b_{2}}}+(\mu -{\tau }){b_{1}}} \biggr) \,\mathrm{d} {\tau } \\ ={}&{-} \frac{\Omega (\mu )\Upsilon ({b_{1}})}{{b_{1}{b_{2}}}({{b_{2}}}-{b_{1}})(\lambda +\mu )}\\ &{}+ \frac{1}{{b_{1}{b_{2}}}({{b_{2}}}-{b_{1}})(\lambda +\mu )} \int _{ \frac{\lambda {{b_{2}}}+\mu {b_{1}}}{{b_{1}{b_{2}}}(\lambda +\mu )}}^{ \frac{1}{{b_{1}}}} \frac{\Phi (x-\frac{\lambda {{b_{2}}}+\mu {b_{1}}}{{b_{1}{b_{2}}}(\lambda +\mu )} )}{ (x-\frac{\lambda {{b_{2}}}+\mu {b_{1}}}{{b_{1}{b_{2}}}(\lambda +\mu )} )}{ \Upsilon \circ \Psi }(x) \,\mathrm{d}x \\ ={}&{-} \frac{\Omega (\mu )\Upsilon ({{b_{2}}})}{{b_{1}{b_{2}}}({{b_{2}}}-{b_{1}})(\lambda +\mu )}- \frac{1}{{b_{1}{b_{2}}}({{b_{2}}}-{b_{1}})(\lambda +\mu )}{_{ ( \frac{1}{{b_{1}}} )^{-}}I_{\Phi }{ \Upsilon \circ \Psi } \biggl( \frac{\lambda {{b_{2}}}+\mu {b_{1}}}{{b_{1}{b_{2}}}(\lambda +\mu )} \biggr)}. \end{aligned}$$

Substituting the values of \(I_{3}\) and \(I_{4}\) in (2.3), we obtain our required result. □

Corollary 2.6

Choosing \(\Phi ({\tau })={\tau }\) in Lemma 2.3, we have

$$\begin{aligned} &\frac{\lambda \Upsilon ({{b_{2}}})+\mu \Upsilon ({b_{1}})}{\lambda +\mu }- \frac{{b_{1}{b_{2}}}}{{{b_{2}}}-{b_{1}}} \int _{\frac{1}{{{b_{2}}}}}^{ \frac{1}{{b_{1}}}}{\Upsilon \circ \Psi }(x) \,\mathrm{d}x \\ &\quad ={b_{1}{b_{2}}}({{b_{2}}}-{b_{1}}) \biggl[ \int _{0}^{\lambda } \frac{{\tau }}{((\lambda -{\tau }){{b_{2}}}+(\mu +{\tau }){b_{1}})^{2}} \Upsilon ' \biggl( \frac{{b_{1}{b_{2}}}(\lambda +\mu )}{(\lambda -{\tau }){{b_{2}}}+(\mu +{\tau }){b_{1}}} \biggr) \,\mathrm{d} {\tau } \\ &\qquad {}- \int _{0}^{\mu } \frac{{\tau }}{((\lambda +{\tau }){{b_{2}}}+(\mu -{\tau }){b_{1}})^{2}} \Upsilon ' \biggl( \frac{{b_{1}{b_{2}}}(\lambda +\mu )}{(\lambda +{\tau }){{b_{2}}}+(\mu -{\tau }){b_{1}}} \biggr) \,\mathrm{d} {\tau } \biggr]. \end{aligned}$$

Corollary 2.7

Taking \(\Phi ({\tau })=\frac{{\tau }^{\alpha }}{\Gamma (\alpha )}\) in Lemma 2.3, we obtain

$$\begin{aligned} &\frac{\lambda ^{\alpha }\Upsilon ({{b_{2}}})+\mu ^{\alpha }\Upsilon ({b_{1}})}{\lambda +\mu }- \frac{({b_{1}{b_{2}}})^{\alpha }(\lambda +\mu )^{\alpha -1}\Gamma (\alpha +1)}{({{b_{2}}}-{b_{1}})^{\alpha }} \biggl[{J_{ (\frac{1}{{{b_{2}}}} )^{+}}^{\alpha }{ \Upsilon \circ \Psi } \biggl( \frac{\lambda {{b_{2}}}+\mu {b_{1}}}{{b_{1}{b_{2}}}(\lambda +\mu )} \biggr)} \\ &\qquad {}+{J_{ (\frac{1}{{b_{1}}} )^{-}}^{\alpha }{\Upsilon \circ \Psi } \biggl( \frac{\lambda {{b_{2}}}+\mu {b_{1}}}{{b_{1}{b_{2}}}(\lambda +\mu )} \biggr)} \biggr] \\ &\quad ={b_{1}{b_{2}}}({{b_{2}}}-{b_{1}}) \biggl[ \int _{0}^{\lambda } \frac{{\tau }^{\alpha }}{((\lambda -{\tau }){{b_{2}}}+(\mu +{\tau }){b_{1}})^{2}} \Upsilon ' \biggl( \frac{{b_{1}{b_{2}}}(\lambda +\mu )}{(\lambda -{\tau }){{b_{2}}}+(\mu +{\tau }){b_{1}}} \biggr) \,\mathrm{d} {\tau } \\ &\qquad {}- \int _{0}^{\mu } \frac{{\tau }^{\alpha }}{((\lambda +{\tau }){{b_{2}}}+(\mu -{\tau }){b_{1}})^{2}} \Upsilon ' \biggl( \frac{{b_{1}{b_{2}}}(\lambda +\mu )}{(\lambda +{\tau }){{b_{2}}}+(\mu -{\tau }){b_{1}}} \biggr) \,\mathrm{d} {\tau } \biggr]. \end{aligned}$$

Corollary 2.8

Choosing \(\Phi ({\tau })= \frac{{\tau }^{\frac{\alpha }{k}}}{k\Gamma _{k}(\alpha )}\) in Lemma 2.3, we obtain

$$\begin{aligned} &\frac{\lambda ^{\frac{\alpha }{k}}\Upsilon ({{b_{2}}})+\mu ^{\frac{\alpha }{k}}\Upsilon ({b_{1}})}{\lambda +\mu }- \frac{k({b_{1}{b_{2}}})^{\frac{\alpha }{k}}(\lambda +\mu )^{\frac{\alpha }{k}-1}\Gamma _{k}(\alpha +k)}{({{b_{2}}}-{b_{1}})^{\frac{\alpha }{k}}} \biggl[{_{k}{J_{ (\frac{1}{{{b_{2}}}} )^{+}}^{\alpha }} { \Upsilon \circ \Psi } \biggl( \frac{\lambda {{b_{2}}}+\mu {b_{1}}}{{b_{1}{b_{2}}}(\lambda +\mu )} \biggr)} \\ &\qquad {}+{{_{k}J_{ (\frac{1}{{b_{1}}} )^{-}}^{\alpha }} { \Upsilon \circ \Psi } \biggl( \frac{\lambda {{b_{2}}}+\mu {b_{1}}}{{b_{1}{b_{2}}}(\lambda +\mu )} \biggr)} \biggr] \\ &\quad ={b_{1}{b_{2}}}({{b_{2}}}-{b_{1}}) \biggl[ \int _{0}^{\lambda } \frac{{\tau }^{\frac{\alpha }{k}}}{((\lambda -{\tau }){{b_{2}}}+(\mu +{\tau }){b_{1}})^{2}} \Upsilon ' \biggl( \frac{{b_{1}{b_{2}}}(\lambda +\mu )}{(\lambda -{\tau }){{b_{2}}}+(\mu +{\tau }){b_{1}}} \biggr) \,\mathrm{d} {\tau } \\ &\qquad {}- \int _{0}^{\mu } \frac{{\tau }^{\frac{\alpha }{k}}}{((\lambda +{\tau }){{b_{2}}}+(\mu -{\tau }){b_{1}})^{2}} \Upsilon ' \biggl( \frac{{b_{1}{b_{2}}}(\lambda +\mu )}{(\lambda +{\tau }){{b_{2}}}+(\mu -{\tau }){b_{1}}} \biggr) \,\mathrm{d} {\tau } \biggr]. \end{aligned}$$

Lemma 2.4

Let \({\Upsilon }:[{b_{1}},{{b_{2}}}]\subset (0,+\infty )\rightarrow \mathbb{R}\) be a differentiable mapping on \(({b_{1}},{{b_{2}}})\) with \({b_{1}}<{{b_{2}}}\), then

$$\begin{aligned} &{\Upsilon } \biggl(\frac{2{b_{1}{b_{2}}}}{{b_{1}}+{{b_{2}}}} \biggr)- \frac{1}{2\Delta (1)} \biggl[_{\frac{1}{{{b_{2}}}}^{+}}I_{\Phi }{{ \Upsilon }\circ {\Psi }} \biggl( \frac{1}{{b_{1}}} \biggr)+ _{ \frac{1}{{b_{1}}}^{-}}I_{\Phi }{{\Upsilon } \circ {\Psi }} \biggl( \frac{1}{{{b_{2}}}} \biggr) \biggr]\\ &\quad = \frac{{b_{1}{b_{2}}}({{b_{2}}}-{b_{1}})}{2\Delta (1)}\sum_{j=1}^{4}M_{j}, \end{aligned}$$

where

$$\begin{aligned}& M_{1}:= \int _{0}^{\frac{1}{2}} \frac{\Delta ({\tau })}{({\tau b_{1}}+(1-{\tau }){{b_{2}}})^{2}}{ \Upsilon }' \biggl( \frac{{b_{1}{b_{2}}}}{{\tau b_{1}}+(1-{\tau }){{b_{2}}}} \biggr) \,\mathrm{d} {\tau }, \\& M_{2}:= \int _{0}^{\frac{1}{2}} \frac{ (-\Delta ({\tau }) )}{((1-{\tau }){b_{1}}+{\tau {b_{2}}})^{2}}{ \Upsilon }' \biggl( \frac{{b_{1}{b_{2}}}}{(1-{\tau }){b_{1}}+{\tau {b_{2}}}} \biggr) \,\mathrm{d} {\tau }, \\& M_{3}:= \int _{\frac{1}{2}}^{1} \frac{ (-\delta ({\tau }) )}{({\tau b_{1}}+(1-{\tau }){{b_{2}}})^{2}}{ \Upsilon }' \biggl( \frac{{b_{1}{b_{2}}}}{{\tau b_{1}}+(1-{\tau }){{b_{2}}}} \biggr) \,\mathrm{d} {\tau }, \\& M_{4}:= \int _{\frac{1}{2}}^{1} \frac{\delta ({\tau })}{((1-{\tau }){b_{1}}+{\tau {b_{2}}})^{2}}{ \Upsilon }' \biggl( \frac{{b_{1}{b_{2}}}}{(1-{\tau }){b_{1}}+{\tau {b_{2}}}} \biggr) \,\mathrm{d} {\tau }. \end{aligned}$$

Proof

Integrating by parts \(M_{i}\) for \(i=1,2,3,4\), and changing the variables, we have

$$\begin{aligned}& \begin{aligned} M_{1}={}&\frac{1}{{b_{1}{b_{2}}}({{b_{2}}}-{b_{1}})}{\Upsilon } \biggl( \frac{2{b_{1}{b_{2}}}}{{b_{1}}+{{b_{2}}}} \biggr) \int _{0}^{ \frac{1}{2}} \frac{\Phi (\frac{{{b_{2}}}-{b_{1}}}{{b_{1}{b_{2}}}}\mu )}{\mu } \,\mathrm{d} \mu \\ &{}-\frac{1}{{b_{1}{b_{2}}}({{b_{2}}}-{b_{1}})} \int _{0}^{ \frac{1}{2}} \frac{\Phi (\frac{{{b_{2}}}-{b_{1}}}{{b_{1}{b_{2}}}}{\tau } )}{{\tau }}{ \Upsilon } \biggl( \frac{{b_{1}{b_{2}}}}{{\tau b_{1}}+(1-{\tau }){{b_{2}}}} \biggr) \,\mathrm{d} {\tau }, \end{aligned} \\& \begin{aligned} M_{2}={}&\frac{1}{{b_{1}{b_{2}}}({{b_{2}}}-{b_{1}})}{\Upsilon } \biggl( \frac{2{b_{1}{b_{2}}}}{{b_{1}}+{{b_{2}}}} \biggr) \int _{0}^{ \frac{1}{2}} \frac{\Phi (\frac{{{b_{2}}}-{b_{1}}}{{b_{1}{b_{2}}}}\mu )}{\mu } \,\mathrm{d} \mu \\ &{}-\frac{1}{{b_{1}{b_{2}}}({{b_{2}}}-{b_{1}})} \int _{0}^{ \frac{1}{2}} \frac{\Phi (\frac{{{b_{2}}}-{b_{1}}}{{b_{1}{b_{2}}}}{\tau } )}{{\tau }}{ \Upsilon } \biggl( \frac{{b_{1}{b_{2}}}}{(1-{\tau }){b_{1}}+{\tau {b_{2}}}} \biggr) \,\mathrm{d} {\tau }, \end{aligned} \\& \begin{aligned} M_{3}={}&\frac{1}{{b_{1}{b_{2}}}({{b_{2}}}-{b_{1}})}{\Upsilon } \biggl( \frac{2{b_{1}{b_{2}}}}{{b_{1}}+{{b_{2}}}} \biggr) \int _{\frac{1}{2}}^{1} \frac{\Phi (\frac{{{b_{2}}}-{b_{1}}}{{b_{1}{b_{2}}}}\mu )}{\mu } \,\mathrm{d} \mu \\ &{}-\frac{1}{{b_{1}{b_{2}}}({{b_{2}}}-{b_{1}})} \int _{ \frac{1}{2}}^{1} \frac{\Phi (\frac{{{b_{2}}}-{b_{1}}}{{b_{1}{b_{2}}}}{\tau } )}{{\tau }}{ \Upsilon } \biggl( \frac{{b_{1}{b_{2}}}}{{\tau b_{1}}+(1-{\tau }){{b_{2}}}} \biggr) \,\mathrm{d} {\tau }, \end{aligned} \\& \begin{aligned} M_{4}={}&\frac{1}{{b_{1}{b_{2}}}({{b_{2}}}-{b_{1}})}{\Upsilon } \biggl( \frac{2{b_{1}{b_{2}}}}{{b_{1}}+{{b_{2}}}} \biggr) \int _{\frac{1}{2}}^{1} \frac{\Phi (\frac{{{b_{2}}}-{b_{1}}}{{b_{1}{b_{2}}}}\mu )}{\mu } \,\mathrm{d} \mu \\ &{}-\frac{1}{{b_{1}{b_{2}}}({{b_{2}}}-{b_{1}})} \int _{ \frac{1}{2}}^{1} \frac{\Phi (\frac{{{b_{2}}}-{b_{1}}}{{b_{1}{b_{2}}}}{\tau } )}{{\tau }}{ \Upsilon } \biggl( \frac{{b_{1}{b_{2}}}}{(1-{\tau }){b_{1}}+{\tau {b_{2}}}} \biggr) \,\mathrm{d} {\tau }. \end{aligned} \end{aligned}$$

Adding \(M_{1}\), \(M_{2}\), \(M_{3}\) and \(M_{4}\) and multiplying by the factor \(\frac{{b_{1}{b_{2}}}({{b_{2}}}-{b_{1}})}{2\Delta (1)}\), we obtain our required result. □

Corollary 2.9

Taking \(\Phi ({\tau })={\tau }\) in Lemma 2.4, then

$$\begin{aligned} &{\Upsilon } \biggl(\frac{2{b_{1}{b_{2}}}}{{b_{1}}+{{b_{2}}}} \biggr)- \frac{{b_{1}{b_{2}}}}{{{b_{2}}}-{b_{1}}} \int _{b_{1}}^{{b_{2}}} \frac{{\Upsilon }(x)}{x^{2}} \,\mathrm{d}x= \frac{{b_{1}{b_{2}}}({{b_{2}}}-{b_{1}})}{2}\sum_{j=1}^{4}L_{j}, \end{aligned}$$

where

$$\begin{aligned}& L_{1}:= \int _{0}^{\frac{1}{2}} \frac{{\tau }}{({\tau b_{1}}+(1-{\tau }){{b_{2}}})^{2}}{\Upsilon }' \biggl(\frac{{b_{1}{b_{2}}}}{{\tau b_{1}}+(1-{\tau }){{b_{2}}}} \biggr) \,\mathrm{d} {\tau }, \\& L_{2}:= \int _{0}^{\frac{1}{2}} \frac{{-\tau }}{((1-{\tau }){b_{1}}+{\tau {b_{2}}})^{2}}{\Upsilon }' \biggl(\frac{{b_{1}{b_{2}}}}{{\tau b_{1}}+(1-{\tau }){{b_{2}}}} \biggr) \,\mathrm{d} {\tau }, \\& L_{3}:= \int _{\frac{1}{2}}^{1} \frac{-{\tau }}{({\tau b_{1}}+(1-{\tau }){{b_{2}}})^{2}}{\Upsilon }' \biggl(\frac{{b_{1}{b_{2}}}}{{\tau b_{1}}+(1-{\tau }){{b_{2}}}} \biggr) \,\mathrm{d} {\tau }, \\& L_{4}:= \int _{\frac{1}{2}}^{1} \frac{{\tau }}{((1-{\tau }){b_{1}}+{\tau {b_{2}}})^{2}}{\Upsilon }' \biggl( \frac{{b_{1}{b_{2}}}}{(1-{\tau }){b_{1}}+{\tau {b_{2}}}} \biggr) \,\mathrm{d} {\tau }. \end{aligned}$$

Corollary 2.10

Choosing \(\Phi ({\tau })=\frac{{\tau }^{\alpha }}{\Gamma (\alpha )}\) in Lemma 2.4, then

$$\begin{aligned} &{\Upsilon } \biggl(\frac{2{b_{1}{b_{2}}}}{{b_{1}}+{{b_{2}}}} \biggr)- \frac{\Gamma (\alpha +1)}{2} \biggl( \frac{{b_{1}{b_{2}}}}{{{b_{2}}}-{b_{1}}} \biggr)^{\alpha } \biggl[J_{ \frac{1}{{{b_{2}}}}^{+}}^{\alpha }{{ \Upsilon }\circ {\Psi }} \biggl( \frac{1}{{b_{1}}} \biggr) +J_{\frac{1}{{b_{1}}}^{-}}^{\alpha }{{ \Upsilon }\circ {\Psi }} \biggl( \frac{1}{{{b_{2}}}} \biggr) \biggr] \\ &\quad =\frac{{b_{1}{b_{2}}}({{b_{2}}}-{b_{1}})}{2}\sum_{j=5}^{8}L_{j}, \end{aligned}$$

where

$$\begin{aligned}& L_{5}:= \int _{0}^{\frac{1}{2}} \frac{{\tau }^{\alpha }}{({\tau b_{1}}+(1-{\tau }){{b_{2}}})^{2}}{\Upsilon }' \biggl(\frac{{b_{1}{b_{2}}}}{{\tau b_{1}}+(1-{\tau }){{b_{2}}}} \biggr) \,\mathrm{d} {\tau }, \\& L_{6}:= \int _{0}^{\frac{1}{2}} \frac{(-{\tau })^{\alpha }}{((1-{\tau }){b_{1}}+{\tau {b_{2}}})^{2}}{ \Upsilon }' \biggl( \frac{{b_{1}{b_{2}}}}{(1-{\tau }){b_{1}}+{\tau {b_{2}}}} \biggr) \,\mathrm{d} {\tau }, \\& L_{7}:= \int _{\frac{1}{2}}^{1} \frac{(-{\tau })^{\alpha }}{({\tau b_{1}}+(1-{\tau }){{b_{2}}})^{2}}{ \Upsilon }' \biggl( \frac{{b_{1}{b_{2}}}}{{\tau b_{1}}+(1-{\tau }){{b_{2}}}} \biggr) \,\mathrm{d} {\tau }, \\& L_{8}:= \int _{\frac{1}{2}}^{1} \frac{{\tau }^{\alpha }}{((1-{\tau }){b_{1}}+{\tau {b_{2}}})^{2}}{\Upsilon }' \biggl(\frac{{b_{1}{b_{2}}}}{(1-{\tau }){b_{1}}+{\tau {b_{2}}}} \biggr) \,\mathrm{d} {\tau }. \end{aligned}$$

Corollary 2.11

Taking \(\Phi ({\tau })= \frac{{\tau }^{\frac{\alpha }{k}}}{k\Gamma _{k}(\alpha )}\) in Lemma 2.4, then

$$\begin{aligned} &{\Upsilon } \biggl(\frac{2{b_{1}{b_{2}}}}{{b_{1}}+{{b_{2}}}} \biggr)- \frac{\Gamma _{k}(\alpha +k)}{2} \biggl( \frac{{b_{1}{b_{2}}}}{{{b_{2}}}-{b_{1}}} \biggr)^{\frac{\alpha }{k}} \biggl[ _{k}J_{\frac{1}{{{b_{2}}}}^{+}}^{\alpha }{{ \Upsilon }\circ { \Psi }} \biggl(\frac{1}{{b_{1}}} \biggr) + _{k}J_{\frac{1}{{b_{1}}}^{-}}^{ \alpha }{{\Upsilon }\circ {\Psi }} \biggl(\frac{1}{{{b_{2}}}} \biggr) \biggr] \\ &\quad =\frac{{b_{1}{b_{2}}}({{b_{2}}}-{b_{1}})}{2}\sum_{j=9}^{12}L_{j}, \end{aligned}$$

where

$$\begin{aligned}& L_{9}:= \int _{0}^{\frac{1}{2}} \frac{{\tau }^{\frac{\alpha }{k}}}{({\tau b_{1}}+(1-{\tau }){{b_{2}}})^{2}}{ \Upsilon }' \biggl( \frac{{b_{1}{b_{2}}}}{{\tau b_{1}}+(1-{\tau }){{b_{2}}}} \biggr) \,\mathrm{d} {\tau }, \\& L_{10}:= \int _{0}^{\frac{1}{2}} \frac{(-{\tau })^{\frac{\alpha }{k}}}{((1-{\tau }){b_{1}}+{\tau {b_{2}}})^{2}}{ \Upsilon }' \biggl( \frac{{b_{1}{b_{2}}}}{(1-{\tau }){b_{1}}+{\tau {b_{2}}}} \biggr) \,\mathrm{d} {\tau }, \\& L_{11}:= \int _{\frac{1}{2}}^{1} \frac{(-{\tau })^{\frac{\alpha }{k}}}{({\tau b_{1}}+(1-{\tau }){{b_{2}}})^{2}}{ \Upsilon }' \biggl( \frac{{b_{1}{b_{2}}}}{{\tau b_{1}}+(1-{\tau }){{b_{2}}}} \biggr) \,\mathrm{d} {\tau }, \\& L_{12}:= \int _{\frac{1}{2}}^{1} \frac{{\tau }^{\frac{\alpha }{k}}}{((1-{\tau }){b_{1}}+{\tau {b_{2}}})^{2}}{ \Upsilon }' \biggl( \frac{{b_{1}{b_{2}}}}{(1-{\tau }){b_{1}}+{\tau {b_{2}}}} \biggr) \,\mathrm{d} {\tau }. \end{aligned}$$

Corollary 2.12

Choosing \(\Phi ({\tau })=\frac{{\tau }}{\alpha }\exp (-A{\tau } )\) in Lemma 2.4with \(A=\frac{1-\alpha }{\alpha }\) and \(\alpha \in (0,1]\), then

$$\begin{aligned} &{\Upsilon } \biggl(\frac{2{b_{1}{b_{2}}}}{{b_{1}}+{{b_{2}}}} \biggr)- \frac{1-\alpha }{2(1-\exp (-A))} \biggl( \frac{{b_{1}{b_{2}}}}{{{b_{2}}}-{b_{1}}} \biggr)^{\alpha } \biggl[I_{ \frac{1}{{{b_{2}}}}^{+}}^{\alpha }{{ \Upsilon }\circ {\Psi }} \biggl( \frac{1}{{b_{1}}} \biggr) +I_{\frac{1}{{b_{1}}}^{-}}^{\alpha }{{ \Upsilon }\circ {\Psi }} \biggl( \frac{1}{{{b_{2}}}} \biggr) \biggr] \\ &\quad =\frac{{b_{1}{b_{2}}}({{b_{2}}}-{b_{1}})}{2(1-\exp (-A))}\sum_{j=13}^{16}L_{j}, \end{aligned}$$

where

$$\begin{aligned}& L_{13}:= \int _{0}^{\frac{1}{2}} \frac{[\exp (-A{\tau })-1]}{({\tau b_{1}}+(1-{\tau }){{b_{2}}})^{2}}{ \Upsilon }' \biggl( \frac{{b_{1}{b_{2}}}}{{\tau b_{1}}+(1-{\tau }){{b_{2}}}} \biggr) \,\mathrm{d} {\tau }, \\& L_{14}:= \int _{0}^{\frac{1}{2}} \frac{[1-\exp (-A{\tau })]}{((1-{\tau }){b_{1}}+{\tau {b_{2}}})^{2}}{ \Upsilon }' \biggl( \frac{{b_{1}{b_{2}}}}{(1-{\tau }){b_{1}}+{\tau {b_{2}}}} \biggr) \,\mathrm{d} {\tau }, \\& L_{15}:= \int _{\frac{1}{2}}^{1} \frac{[\exp (-A(1-{\tau }))-\exp (-A{\tau })]}{({\tau b_{1}}+(1-{\tau }){{b_{2}}})^{2}}{ \Upsilon }' \biggl( \frac{{b_{1}{b_{2}}}}{{\tau b_{1}}+(1-{\tau }){{b_{2}}}} \biggr) \,\mathrm{d} {\tau }, \\& L_{16}:= \int _{\frac{1}{2}}^{1} \frac{[\exp (-A{\tau })-\exp (-A(1-{\tau }))]}{((1-{\tau }){b_{1}}+{\tau {b_{2}}})^{2}}{ \Upsilon }' \biggl( \frac{{b_{1}{b_{2}}}}{(1-{\tau }){b_{1}}+{\tau {b_{2}}}} \biggr) \,\mathrm{d} {\tau }. \end{aligned}$$

Further results

Now, utilizing auxiliary results obtained in the previous subsection, we derive some further generalized fractional trapezium-like inequalities using the class of harmonic convex functions.

Theorem 2.5

Let \({\Upsilon }:[{b_{1}},{{b_{2}}}]\rightarrow \mathbb{R}\) be a continuous function on \(({b_{1}},{{b_{2}}})\) with \({b_{1}}<{{b_{2}}}\) and \(|{\Upsilon }'|^{q}\) be an harmonic convex function with \(\frac{1}{p}+\frac{1}{q}=1\), then

$$\begin{aligned} & \biggl\vert \frac{{\Upsilon }({b_{1}})+{\Upsilon }({{b_{2}}})}{m+1}\\ &\qquad {}- \frac{1}{(m+1)\eta (1)} \biggl[{_{ (\frac{1}{{{b_{2}}}} )^{+}}I_{ \Phi }} {\Upsilon \circ \Psi } \biggl( \frac{m{b_{1}}+{{b_{2}}}}{(m+1){b_{1}{b_{2}}}} \biggr) +{_{ ( \frac{1}{{b_{1}}} )^{-}}I_{\Phi }} {\Upsilon \circ \Psi } \biggl( \frac{{b_{1}}+m{{b_{2}}}}{(m+1){b_{1}{b_{2}}}} \biggr) \biggr] \biggr\vert \\ &\quad \leq \frac{{b_{1}{b_{2}}}({{b_{2}}}-{b_{1}})}{\eta (1)} \biggl[\pi _{1}^{ \frac{1}{p}} \biggl( \int _{0}^{1}\eta ^{q}({\tau }) \biggl( \frac{1-{\tau }}{m+1} \bigl\vert {\Upsilon }'({b_{1}}) \bigr\vert ^{q}+\frac{m+{\tau }}{m+1} \bigl\vert { \Upsilon }'({{b_{2}}}) \bigr\vert ^{q} \biggr) \,\mathrm{d} {\tau } \biggr)^{ \frac{1}{q}} \\ &\qquad {}+ \pi _{2}^{\frac{1}{p}} \biggl( \int _{0}^{1}\eta ^{q}({\tau }) \biggl(\frac{m+{\tau }}{m+1} \bigl\vert {\Upsilon }'({b_{1}}) \bigr\vert ^{q}+ \frac{1-{\tau }}{m+1} \bigl\vert {\Upsilon }'({{b_{2}}}) \bigr\vert ^{q} \biggr) \,\mathrm{d} { \tau } \biggr)^{\frac{1}{q}} \biggr], \end{aligned}$$

where

$$\begin{aligned} &\pi _{1} := \frac{(m{b_{1}}+{{b_{2}}})^{1-2p}}{({{b_{2}}}-{b_{1}})(1-2p)} \biggl[1- \biggl( \frac{(m+1){b_{1}}}{m{b_{1}}+{{b_{2}}}} \biggr)^{1-2p} \biggr], \\ &\pi _{2} := \frac{({b_{1}}+m{{b_{2}}})^{1-2p}}{({{b_{2}}}-{b_{1}})(1-2p)} \biggl[ \biggl( \frac{(m+1){{b_{2}}}}{{b_{1}}+m{{b_{2}}}} \biggr)^{1-2p}-1 \biggr]. \end{aligned}$$

Proof

Using Lemma 2.2, the modulus property, Hölder’s inequality and the harmonic convexity of \(|{\Upsilon }'|^{q}\), we have

$$\begin{aligned} & \biggl\vert \frac{{\Upsilon }({b_{1}})+{\Upsilon }({{b_{2}}})}{m+1}\\ &\qquad {}- \frac{1}{(m+1)\eta (1)} \biggl[{_{ (\frac{1}{{{b_{2}}}} )^{+}}I_{ \Phi }} {\Upsilon \circ \Psi } \biggl( \frac{m{b_{1}}+{{b_{2}}}}{(m+1){b_{1}{b_{2}}}} \biggr) +{_{ ( \frac{1}{{b_{1}}} )^{-}}I_{\Phi }} {\Upsilon \circ \Psi } \biggl( \frac{{b_{1}}+m{{b_{2}}}}{(m+1){b_{1}{b_{2}}}} \biggr) \biggr] \biggr\vert \\ &\quad \leq \frac{{b_{1}{b_{2}}}({{b_{2}}}-{b_{1}})}{\eta (1)} \biggl[ \int _{0}^{1} \frac{\eta ({\tau })}{((m+{\tau }){b_{1}}+(1-{\tau }){{b_{2}}})^{2}} \biggl\vert {\Upsilon }' \biggl( \frac{(m+1){b_{1}{b_{2}}}}{(m+{\tau }){b_{1}}+(1-{\tau }){{b_{2}}}} \biggr) \biggr\vert \,\mathrm{d} {\tau } \\ &\qquad {} + \int _{0}^{1} \frac{\eta ({\tau })}{((1-{\tau }){b_{1}}+(m+{\tau }){{b_{2}}})^{2}} \biggl\vert {\Upsilon }' \biggl( \frac{(m+1){b_{1}{b_{2}}}}{(1-{\tau }){b_{1}}+(m+{\tau }){{b_{2}}}} \biggr) \biggr\vert \,\mathrm{d} {\tau } \biggr] \\ &\quad \leq \frac{{b_{1}{b_{2}}}({{b_{2}}}-{b_{1}})}{\eta (1)} \biggl[ \biggl( \int _{0}^{1}\bigl((m+{\tau }){b_{1}}+(1-{\tau }){{b_{2}}}\bigr)^{-2p} \,\mathrm{d} { \tau } \biggr)^{\frac{1}{p}}\\ &\qquad {}\times \biggl( \int _{0}^{1}{\eta ^{p}({\tau })} \biggl\vert {\Upsilon }' \biggl( \frac{(m+1){b_{1}{b_{2}}}}{(m+{\tau }){b_{1}}+(1-{\tau }){{b_{2}}}} \biggr) \biggr\vert ^{q} \,\mathrm{d} {\tau } \biggr)^{\frac{1}{q}} \\ & \qquad {}+ \biggl( \int _{0}^{1}\bigl((1-{\tau }){b_{1}}+(m+{\tau }){{b_{2}}}\bigr)^{-2p} \,\mathrm{d} {\tau } \biggr)^{\frac{1}{p}}\\ &\qquad {}\times \biggl( \int _{0}^{1}{\eta ^{q}({ \tau })} \biggl\vert {\Upsilon }' \biggl( \frac{(m+1){b_{1}{b_{2}}}}{(1-{\tau }){b_{1}}+(m+{\tau }){{b_{2}}}} \biggr) \biggr\vert ^{q} \,\mathrm{d} {\tau } \biggr)^{\frac{1}{q}} \biggr] \\ &\quad \leq \frac{{b_{1}{b_{2}}}({{b_{2}}}-{b_{1}})}{\eta (1)} \biggl[ \biggl( \int _{0}^{1}\bigl((m+{\tau }){b_{1}}+(1-{\tau }){{b_{2}}}\bigr)^{-2p} \,\mathrm{d} { \tau } \biggr)^{\frac{1}{p}} \\ &\qquad {}\times\biggl( \int _{0}^{1}\eta ^{q}({\tau }) \biggl(\frac{1-{\tau }}{m+1} \bigl\vert {\Upsilon }'({b_{1}}) \bigr\vert ^{q}+ \frac{m+{\tau }}{m+1} \bigl\vert {\Upsilon }'({{b_{2}}}) \bigr\vert ^{q} \biggr) \,\mathrm{d} { \tau } \biggr)^{\frac{1}{q}} \\ &\qquad {} + \biggl( \int _{0}^{1}\bigl((1-{\tau }){b_{1}}+(m+{\tau }){{b_{2}}}\bigr)^{-2p} \,\mathrm{d} {\tau } \biggr)^{\frac{1}{p}}\\ &\qquad {}\times \biggl( \int _{0}^{1}\eta ^{q}({ \tau }) \biggl(\frac{m+{\tau }}{m+1} \bigl\vert {\Upsilon }'({b_{1}}) \bigr\vert ^{q}+ \frac{1-{\tau }}{m+1} \bigl\vert {\Upsilon }'({{b_{2}}}) \bigr\vert ^{q} \biggr) \,\mathrm{d} { \tau } \biggr)^{\frac{1}{q}} \biggr]. \end{aligned}$$

After simple calculations, we obtain our required result. □

Corollary 2.13

Choosing \(\Phi ({\tau })={\tau }\) in Theorem 2.5, we have

$$\begin{aligned} & \biggl\vert \frac{{\Upsilon }({b_{1}})+{\Upsilon }({{b_{2}}})}{m+1}- \frac{{b_{1}{b_{2}}}}{{{b_{2}}}-{b_{1}}} \int _{\frac{1}{{{b_{2}}}}}^{ \frac{1}{{b_{1}}}}{\Upsilon \circ \Psi }(x) \,\mathrm{d}x \biggr\vert \\ &\quad \leq {{b_{1}{b_{2}}}({{b_{2}}}-{b_{1}})}\\ &\qquad {}\times \biggl[\pi _{1}^{\frac{1}{p}} \biggl(\frac{1}{(m+1)(q+1)(q+2)} \bigl\vert {\Upsilon }'({b_{1}}) \bigr\vert ^{q}+ \frac{m(q+2)+(q+1)}{(m+1)(q+1)(q+2)} \bigl\vert {\Upsilon }'({{b_{2}}}) \bigr\vert ^{q} \biggr)^{\frac{1}{q}} \\ &\qquad {}+ \pi _{2}^{\frac{1}{p}} \biggl(\frac{1}{(m+1)(q+1)(q+2)} \bigl\vert { \Upsilon }'({{b_{2}}}) \bigr\vert ^{q}+\frac{m(q+2)+(q+1)}{(m+1)(q+1)(q+2)} \bigl\vert { \Upsilon }'({b_{1}}) \bigr\vert ^{q} \biggr)^{\frac{1}{q}} \biggr], \end{aligned}$$

Corollary 2.14

Taking \(\Phi ({\tau })=\frac{{\tau }^{\alpha }}{\Gamma (\alpha )}\) in Theorem 2.5, we obtain

$$\begin{aligned} &\biggl\vert \frac{{\Upsilon }({b_{1}})+{\Upsilon }({{b_{2}}})}{m+1}- \frac{(m+1)^{\alpha -1}({b_{1}{b_{2}}})^{\alpha }\Gamma (\alpha +1)}{({{b_{2}}}-{b_{1}})^{\alpha }} \biggl[{J_{ (\frac{1}{{{b_{2}}}} )^{+}}^{\alpha }} {\Upsilon \circ \Psi } \biggl( \frac{m{b_{1}}+{{b_{2}}}}{{b_{1}{b_{2}}}(m+1)} \biggr) \\ &\qquad {}+J_{ (\frac{1}{{b_{1}}} )^{-}}^{\alpha }{ \Upsilon \circ \Psi } \biggl( \frac{{b_{1}}+m{{b_{2}}}}{(m+1){b_{1}{b_{2}}}} \biggr) \biggr]\biggr\vert \\ &\quad \leq {{b_{1}{b_{2}}}({{b_{2}}}-{b_{1}})} \biggl[\pi _{1}^{\frac{1}{p}} \biggl(\frac{1}{(m+1)({\alpha }q+1)({\alpha }q+2)} \bigl\vert {\Upsilon }'({b_{1}}) \bigr\vert ^{q}\\ &\qquad {}+ \frac{m({\alpha }q+2)+({\alpha }q+1)}{(m+1)({\alpha }q+1)({\alpha }q+2)} \bigl\vert {\Upsilon }'({{b_{2}}}) \bigr\vert ^{q} \biggr)^{ \frac{1}{q}} \\ &\qquad {}+ \pi _{2}^{\frac{1}{p}} \biggl( \frac{1}{(m+1)({\alpha } q+1)({\alpha }q+2)} \bigl\vert {\Upsilon }'({{b_{2}}}) \bigr\vert ^{q}\\ &\qquad {}+ \frac{m({\alpha }q+2)+({\alpha }q+1)}{(m+1)({\alpha }q+1)({\alpha }q+2)} \bigl\vert { \Upsilon }'({b_{1}}) \bigr\vert ^{q} \biggr)^{\frac{1}{q}} \biggr], \end{aligned}$$

where \(\pi _{1}\) and \(\pi _{2}\) are already defined.

Corollary 2.15

Choosing \(\Phi ({\tau })= \frac{{\tau }^{\frac{\alpha }{k}}}{k\Gamma _{k}(\alpha )}\) in Theorem 2.5, we obtain

$$\begin{aligned} &\biggl\vert \frac{{\Upsilon }({b_{1}})+{\Upsilon }({{b_{2}}})}{m+1}- \frac{(m+1)^{\frac{\alpha }{k}-1}({b_{1}{b_{2}}})^{\frac{\alpha }{k}}\Gamma _{k}(\alpha +k)}{({{b_{2}}}-{b_{1}})^{\frac{\alpha }{k}}} \biggl[{_{k}J_{ (\frac{1}{{{b_{2}}}} )^{+}}^{\alpha }} { \Upsilon \circ \Psi } \biggl( \frac{m{b_{1}}+{{b_{2}}}}{{b_{1}{b_{2}}}(m+1)} \biggr) \\ &\qquad {} +{_{k}J_{ (\frac{1}{{b_{1}}} )^{-}}^{\alpha }} {\Upsilon \circ \Psi } \biggl( \frac{{b_{1}}+m{{b_{2}}}}{(m+1){b_{1}{b_{2}}}} \biggr) \biggr]\biggr\vert \\ &\quad \leq {{b_{1}{b_{2}}}({{b_{2}}}-{b_{1}})} \biggl[\pi _{1}^{\frac{1}{p}} \biggl(\frac{k}{(m+1)({\alpha }q+k)({\alpha }q+2k)} \bigl\vert {\Upsilon }'({b_{1}}) \bigr\vert ^{q}\\ &\qquad {}+ \frac{km({\alpha }q+2k)+k({\alpha }q+k)}{(m+1)({\alpha }q+k)({\alpha }q+2k)} \bigl\vert { \Upsilon }'({{b_{2}}}) \bigr\vert ^{q} \biggr)^{\frac{1}{q}} \\ &\qquad {}+ \pi _{2}^{\frac{1}{p}} \biggl( \frac{k}{(m+1)({\alpha } q+k)({\alpha }q+2k)} \bigl\vert {\Upsilon }'({{b_{2}}}) \bigr\vert ^{q}\\ &\qquad {}+ \frac{mk({\alpha }q+2k)+(k{\alpha }q+k)}{(m+1)({\alpha }q+k)({\alpha }q+2k)} \bigl\vert { \Upsilon }'({b_{1}}) \bigr\vert ^{q} \biggr)^{\frac{1}{q}} \biggr]. \end{aligned}$$

Theorem 2.6

Let \({\Upsilon }:[{b_{1}},{{b_{2}}}]\rightarrow \mathbb{R}\) be a continuous function on \(({b_{1}},{{b_{2}}})\) with \({b_{1}}<{{b_{2}}}\) and \(|{\Upsilon }'|^{q}\) be an harmonic convex function with \(q\geq 1\), then

$$\begin{aligned} & \biggl\vert \frac{{\Upsilon }({b_{1}})+{\Upsilon }({{b_{2}}})}{m+1}\\ &\qquad {}- \frac{1}{(m+1)\eta (1)} \biggl[{_{ (\frac{1}{{{b_{2}}}} )^{+}}I_{ \Phi }} {\Upsilon \circ \Psi } \biggl( \frac{m{b_{1}}+{{b_{2}}}}{(m+1){b_{1}{b_{2}}}} \biggr) +{_{ ( \frac{1}{{b_{1}}} )^{-}}I_{\Phi }} {\Upsilon \circ \Psi } \biggl( \frac{{b_{1}}+m{{b_{2}}}}{(m+1){b_{1}{b_{2}}}} \biggr) \biggr] \biggr\vert \\ &\quad \leq \frac{{b_{1}{b_{2}}}({{b_{2}}}-{b_{1}})}{\eta (1)}\biggl[ \biggl( \int _{0}^{1}\eta ({\tau }) \bigl((m+{\tau }){b_{1}}+(1-{\tau }){{b_{2}}}\bigr)^{-2} \,\mathrm{d} {\tau } \biggr)^{1-\frac{1}{q}} \\ &\qquad {}\times \biggl( \int _{0}^{1}{\eta ({\tau })}\bigl((m+{\tau }){b_{1}}+(1-{ \tau }){{b_{2}}}\bigr)^{-2} \biggl(\frac{1-{\tau }}{m+1} \bigl\vert {\Upsilon }'({b_{1}}) \bigr\vert ^{q}+ \frac{m+{\tau }}{m+1} \bigl\vert {\Upsilon }'({{b_{2}}}) \bigr\vert ^{q} \biggr) \,\mathrm{d} { \tau } \biggr)^{\frac{1}{q}} \\ &\qquad {}+ \biggl( \int _{0}^{1}\eta ({\tau }) \bigl((1-{\tau }){b_{1}}+(m+{\tau }){{b_{2}}}\bigr)^{-2} \,\mathrm{d} {\tau } \biggr)^{1-\frac{1}{q}} \\ &\qquad {}\times \biggl( \int _{0}^{1}{\eta ({\tau })}\bigl((1-{\tau }){b_{1}}+(m+{ \tau }){{b_{2}}}\bigr)^{-2}\\ &\qquad {}\times \biggl(\frac{m+{\tau }}{m+1} \bigl\vert {\Upsilon }'({b_{1}}) \bigr\vert ^{q}+ \frac{1-{\tau }}{m+1} \bigl\vert {\Upsilon }'({{b_{2}}}) \bigr\vert ^{q} \biggr) \,\mathrm{d} { \tau } \biggr)^{\frac{1}{q}}\biggr]. \end{aligned}$$

Proof

Using Lemma 2.2, the modulus property, the power mean inequality and the convexity of \(|{\Upsilon }'|^{q}\), we have

$$\begin{aligned} & \biggl\vert \frac{{\Upsilon }({b_{1}})+{\Upsilon }({{b_{2}}})}{m+1}\\ &\qquad {}- \frac{1}{(m+1)\eta (1)} \biggl[{_{ (\frac{1}{{{b_{2}}}} )^{+}}I_{ \Phi }} {\Upsilon \circ \Psi } \biggl( \frac{m{b_{1}}+{{b_{2}}}}{(m+1){b_{1}{b_{2}}}} \biggr) +{_{ ( \frac{1}{{b_{1}}} )^{-}}I_{\Phi }} {\Upsilon \circ \Psi } \biggl( \frac{{b_{1}}+m{{b_{2}}}}{(m+1){b_{1}{b_{2}}}} \biggr) \biggr] \biggr\vert \\ &\quad \leq \frac{{b_{1}{b_{2}}}({{b_{2}}}-{b_{1}})}{\eta (1)}\biggl[ \int _{0}^{1} \frac{\eta ({\tau })}{((m+{\tau }){b_{1}}+(1-{\tau }){{b_{2}}})^{2}} \biggl\vert {\Upsilon }' \biggl( \frac{(m+1){b_{1}{b_{2}}}}{(m+{\tau }){b_{1}}+(1-{\tau }){{b_{2}}}} \biggr) \biggr\vert \,\mathrm{d} {\tau } \\ &\qquad {}+ \int _{0}^{1} \frac{\eta ({\tau })}{((1-{\tau }){b_{1}}+(m+{\tau }){{b_{2}}})^{2}} \biggl\vert {\Upsilon }' \biggl( \frac{(m+1){b_{1}{b_{2}}}}{(1-{\tau }){b_{1}}+(m+{\tau }){{b_{2}}}} \biggr) \biggr\vert \,\mathrm{d} {\tau }\biggr] \\ &\quad \leq \frac{{b_{1}{b_{2}}}({{b_{2}}}-{b_{1}})}{\eta (1)}\biggl[ \biggl( \int _{0}^{1}\eta ({\tau }) \bigl((m+{\tau }){b_{1}}+(1-{\tau }){{b_{2}}}\bigr)^{-2} \,\mathrm{d} {\tau } \biggr)^{1-\frac{1}{q}} \\ &\qquad {}\times \biggl( \int _{0}^{1}{\eta ({\tau })}\bigl((m+{ \tau }){b_{1}}+(1-{\tau }){{b_{2}}}\bigr)^{-2} \biggl\vert {\Upsilon }' \biggl( \frac{(m+1){b_{1}{b_{2}}}}{(m+{\tau }){b_{1}}+(1-{\tau }){{b_{2}}}} \biggr) \biggr\vert ^{q} \,\mathrm{d} { \tau } \biggr)^{\frac{1}{q}} \\ &\qquad {}+ \biggl( \int _{0}^{1}\eta ({\tau }) \bigl((1-{\tau }){b_{1}}+(m+{ \tau }){{b_{2}}}\bigr)^{-2} \,\mathrm{d} {\tau } \biggr)^{1-\frac{1}{q}} \\ &\qquad {} \times \biggl( \int _{0}^{1}{\eta ({\tau })}\bigl((1-{ \tau }){b_{1}}+(m+{\tau }){{b_{2}}}\bigr)^{-2} \biggl\vert {\Upsilon }' \biggl( \frac{(m+1){b_{1}{b_{2}}}}{(1-{\tau }){b_{1}}+(m+{\tau }){{b_{2}}}} \biggr) \biggr\vert ^{q} \,\mathrm{d} {\tau } \biggr)^{\frac{1}{q}}\biggr] \\ &\quad \leq \frac{{b_{1}{b_{2}}}({{b_{2}}}-{b_{1}})}{\eta (1)}\biggl[ \biggl( \int _{0}^{1}\eta ({\tau }) \bigl((m+{\tau }){b_{1}}+(1-{\tau }){{b_{2}}}\bigr)^{-2} \,\mathrm{d} {\tau } \biggr)^{1-\frac{1}{q}} \\ &\qquad {} \times \biggl( \int _{0}^{1}{\eta ({\tau })}\bigl((m+{ \tau }){b_{1}}+(1-{\tau }){{b_{2}}}\bigr)^{-2} \biggl(\frac{1-{\tau }}{m+1} \bigl\vert { \Upsilon }'({b_{1}}) \bigr\vert ^{q}+\frac{m+{\tau }}{m+1} \bigl\vert {\Upsilon }'({{b_{2}}}) \bigr\vert ^{q} \biggr) \,\mathrm{d} {\tau } \biggr)^{\frac{1}{q}} \\ &\qquad {} + \biggl( \int _{0}^{1}\eta ({\tau }) \bigl((1-{\tau }){b_{1}}+(m+{ \tau }){{b_{2}}}\bigr)^{-2} \biggr)^{1-\frac{1}{q}} \\ &\qquad {}\times \biggl( \int _{0}^{1}{\eta ({\tau })}\bigl((1-{ \tau }){b_{1}}+(m+{\tau }){{b_{2}}}\bigr)^{-2}\\ &\qquad {}\times \biggl(\frac{m+{\tau }}{m+1} \bigl\vert { \Upsilon }'({b_{1}}) \bigr\vert ^{q}+\frac{1-{\tau }}{m+1} \bigl\vert {\Upsilon }'({{b_{2}}}) \bigr\vert ^{q} \biggr) \,\mathrm{d} {\tau } \biggr)^{\frac{1}{q}}\biggr]. \end{aligned}$$

After simple calculations, we obtain our required result. □

Corollary 2.16

If we take \(\Phi ({\tau })={\tau }\) in Theorem 2.6, we have

$$\begin{aligned} & \biggl\vert \frac{{\Upsilon }({b_{1}})+{\Upsilon }({{b_{2}}})}{m+1}- \frac{{b_{1}{b_{2}}}}{{{b_{2}}}-{b_{1}}} \int _{\frac{1}{{{b_{2}}}}}^{ \frac{1}{{b_{1}}}}{\Upsilon \circ \Psi }(x) \,\mathrm{d}x \biggr\vert \\ &\quad \leq {{b_{1}{b_{2}}}({{b_{2}}}-{b_{1}})} \bigl[\pi _{3}^{1- \frac{1}{q}} \bigl(\pi _{4} \bigl\vert {\Upsilon }'({b_{1}}) \bigr\vert ^{q}+\pi _{5} \bigl\vert { \Upsilon }'({{b_{2}}}) \bigr\vert ^{q} \bigr)^{\frac{1}{q}}\\ &\qquad{}+\pi _{6}^{1- \frac{1}{q}} \bigl(\pi _{7} \bigl\vert {\Upsilon }'({b_{1}}) \bigr\vert ^{q}+\pi _{8} \bigl\vert { \Upsilon }'({{b_{2}}}) \bigr\vert ^{q} \bigr)^{\frac{1}{q}} \bigr], \end{aligned}$$

where

$$\begin{aligned} &\pi _{3} :=\frac{(m{b_{1}}+{{b_{2}}})^{-2}}{2}{_{2}F_{1} \biggl(2,2,3, \frac{{{b_{2}}}-{b_{1}}}{m{b_{1}}+{{b_{2}}}} \biggr)}, \\ &\pi _{4} :=\frac{(m{b_{1}}+{{b_{2}}})^{-2}}{6(m+1)}{_{2}F_{1} \biggl(2,2,4, \frac{{{b_{2}}}-{b_{1}}}{m{b_{1}}+{{b_{2}}}} \biggr)}, \\ &\pi _{5} :=\frac{m(m{b_{1}}+{{b_{2}}})^{-2}}{2(m+1)}{_{2}F_{1} \biggl(2,2,3, \frac{{{b_{2}}}-{b_{1}}}{m{b_{1}}+{{b_{2}}}} \biggr)}+ \frac{(m{b_{1}}+{{b_{2}}})^{-2}}{6(m+1)}{_{2}F_{1} \biggl(2,3,4, \frac{{{b_{2}}}-{b_{1}}}{m{b_{1}}+{{b_{2}}}} \biggr)}, \\ &\pi _{6} :=\frac{({b_{1}}+m{{b_{2}}})^{-2}}{2}{_{2}F_{1} \biggl(2,2,3, \frac{{b_{1}}-{{b_{2}}}}{{b_{1}}+m{{b_{2}}}} \biggr)}, \\ &\pi _{7} :=\frac{m({b_{1}}+m{{b_{2}}})^{-2}}{2(m+1)}{_{2}F_{1} \biggl(2,2,3, \frac{{b_{1}}-{{b_{2}}}}{{b_{1}}+m{{b_{2}}}} \biggr)}+ \frac{({b_{1}}+m{{b_{2}}})^{-2}}{6(m+1)}{_{2}F_{1} \biggl(2,3,4, \frac{{b_{1}}-{{b_{2}}}}{{b_{1}}+m{{b_{2}}}} \biggr)}, \\ &\pi _{8} :=\frac{({b_{1}}+m{{b_{2}}})^{-2}}{6(m+1)}{_{2}F_{1} \biggl(2,2,4, \frac{{b_{1}}-{{b_{2}}}}{{b_{1}}+m{{b_{2}}}} \biggr)}. \end{aligned}$$

Corollary 2.17

If we choose \(\Phi ({\tau })=\frac{{\tau }^{\alpha }}{\Gamma (\alpha )}\) in Theorem 2.6, we obtain

$$\begin{aligned} &\biggl\vert \frac{{\Upsilon }({b_{1}})+{\Upsilon }({{b_{2}}})}{m+1}- \frac{(m+1)^{\alpha -1}({b_{1}{b_{2}}})^{\alpha }\Gamma (\alpha +1)}{({{b_{2}}}-{b_{1}})^{\alpha }} \biggl[{J_{ (\frac{1}{{{b_{2}}}} )^{+}}^{\alpha }} {\Upsilon \circ \Psi } \biggl( \frac{m{b_{1}}+{{b_{2}}}}{{b_{1}{b_{2}}}(m+1)} \biggr) \\ &\qquad {}+{J_{ (\frac{1}{{b_{1}}} )^{-}}^{\alpha }} { \Upsilon \circ \Psi } \biggl( \frac{{b_{1}}+m{{b_{2}}}}{(m+1){b_{1}{b_{2}}}} \biggr) \biggr]\biggr\vert \\ &\quad \leq {{b_{1}{b_{2}}}({{b_{2}}}-{b_{1}})} \bigl[\pi _{9}^{1- \frac{1}{q}} \bigl(\pi _{10} \bigl\vert {\Upsilon }'({b_{1}}) \bigr\vert ^{q}+\pi _{11} \bigl\vert { \Upsilon }'({{b_{2}}}) \bigr\vert ^{q} \bigr)^{\frac{1}{q}}\\ &\qquad{}+\pi _{12}^{1- \frac{1}{q}} \bigl(\pi _{13} \bigl\vert {\Upsilon }'({b_{1}}) \bigr\vert ^{q}+\pi _{14} \bigl\vert { \Upsilon }'({{b_{2}}}) \bigr\vert ^{q} \bigr)^{\frac{1}{q}} \bigr], \end{aligned}$$

where

$$\begin{aligned} &\pi _{9} :=\frac{(m{b_{1}}+{{b_{2}}})^{-2}}{\alpha +1}{_{2}F_{1} \biggl(2,\alpha +1,\alpha +2, \frac{{{b_{2}}}-{b_{1}}}{m{b_{1}}+{{b_{2}}}} \biggr)}, \\ &\pi _{10} := \frac{(m{b_{1}}+{{b_{2}}})^{-2}}{(\alpha +2)(\alpha +1)(m+1)}{_{2}F_{1} \biggl(2,\alpha +1,\alpha +3, \frac{{{b_{2}}}-{b_{1}}}{m{b_{1}}+{{b_{2}}}} \biggr)}, \\ & \begin{aligned} \pi _{11} :={}&\frac{m(m{b_{1}}+{{b_{2}}})^{-2}}{(\alpha +1)(m+1)}{_{2}F_{1} \biggl(2,\alpha +1,\alpha +2, \frac{{{b_{2}}}-{b_{1}}}{m{b_{1}}+{{b_{2}}}} \biggr)} \\ &{} + \frac{(m{b_{1}}+{{b_{2}}})^{-2}}{(\alpha +2)(\alpha +1)(m+1)}{_{2}F_{1} \biggl(2,\alpha +2, \alpha +3, \frac{{{b_{2}}}-{b_{1}}}{m{b_{1}}+{{b_{2}}}} \biggr)}, \end{aligned} \\ &\pi _{12} :=\frac{({b_{1}}+m{{b_{2}}})^{-2}}{\alpha +1}{_{2}F_{1} \biggl(2,\alpha +1,\alpha +2, \frac{{b_{1}}-{{b_{2}}}}{{b_{1}}+m{{b_{2}}}} \biggr)}, \\ & \begin{aligned} \pi _{13} :={}&\frac{m({b_{1}}+m{{b_{2}}})^{-2}}{(\alpha +1)(m+1)}{_{2}F_{1} \biggl(2,\alpha +1,\alpha +2, \frac{{b_{1}}-{{b_{2}}}}{{b_{1}}+m{{b_{2}}}} \biggr)} \\ & {}+ \frac{({b_{1}}+m{{b_{2}}})^{-2}}{(\alpha +2)(\alpha +1)(m+1)} {_{2}F_{1} \biggl(2,\alpha +2,\alpha +3, \frac{{b_{1}}-{{b_{2}}}}{{b_{1}}+m{{b_{2}}}} \biggr)}, \end{aligned} \\ &\pi _{14} := \frac{({b_{1}}+m{{b_{2}}})^{-2}}{(\alpha +2)(\alpha +1)(m+1)}{_{2}F_{1} \biggl(2,\alpha +1,\alpha +3, \frac{{b_{1}}-{{b_{2}}}}{{b_{1}}+m{{b_{2}}}} \biggr)}. \end{aligned}$$

Corollary 2.18

If we take \(\Phi ({\tau })= \frac{{\tau }^{\frac{\alpha }{k}}}{k\Gamma _{k}(\alpha )}\) in Theorem 2.6, we obtain

$$\begin{aligned} &\biggl\vert \frac{{\Upsilon }({b_{1}})+{\Upsilon }({{b_{2}}})}{m+1}- \frac{(m+1)^{\frac{\alpha }{k}-1}({b_{1}{b_{2}}})^{\frac{\alpha }{k}}\Gamma _{k}(\alpha +k)}{({{b_{2}}}-{b_{1}})^{\frac{\alpha }{k}}} \biggl[{_{k}J_{ (\frac{1}{{{b_{2}}}} )^{+}}^{\alpha }} { \Upsilon \circ \Psi } \biggl( \frac{m{b_{1}}+{{b_{2}}}}{{b_{1}{b_{2}}}(m+1)} \biggr) \\ & \qquad {}+{_{k}J_{ (\frac{1}{{b_{1}}} )^{-}}^{\alpha }} {\Upsilon \circ \Psi } \biggl( \frac{{b_{1}}+m{{b_{2}}}}{(m+1){b_{1}{b_{2}}}} \biggr) \biggr]\biggr\vert \\ &\quad \leq {{b_{1}{b_{2}}}({{b_{2}}}-{b_{1}})} \bigl[\pi _{15}^{1- \frac{1}{q}} \bigl(\pi _{16} \bigl\vert {\Upsilon }'({b_{1}}) \bigr\vert ^{q}+\pi _{17} \bigl\vert { \Upsilon }'({{b_{2}}}) \bigr\vert ^{q} \bigr)^{\frac{1}{q}}\\ &\qquad{}+\pi _{18}^{1- \frac{1}{q}} \bigl(\pi _{19} \bigl\vert {\Upsilon }'({b_{1}}) \bigr\vert ^{q}+\pi _{20} \bigl\vert { \Upsilon }'({{b_{2}}}) \bigr\vert ^{q} \bigr)^{\frac{1}{q}} \bigr], \end{aligned}$$

where

$$\begin{aligned} &\pi _{15} :=\frac{k(m{b_{1}}+{{b_{2}}})^{-2}}{\alpha +k}{_{2}F_{1,k} \biggl(2k,\alpha +k,\alpha +2k, \frac{{{b_{2}}}-{b_{1}}}{m{b_{1}}+{{b_{2}}}} \biggr)}, \\ &\pi _{16} := \frac{k(m{b_{1}}+{{b_{2}}})^{-2}}{(\alpha +2k)(\alpha +k)(m+1)}{_{2}F_{1,k} \biggl(2k,\alpha +k,\alpha +3k, \frac{{{b_{2}}}-{b_{1}}}{m{b_{1}}+{{b_{2}}}} \biggr)}, \\ & \begin{aligned} \pi _{17} :={}&\frac{km(m{b_{1}}+{{b_{2}}})^{-2}}{(\alpha +k)(m+1)}{_{2}F_{1,k} \biggl(2k,\alpha +k,\alpha +2k, \frac{{{b_{2}}}-{b_{1}}}{m{b_{1}}+{{b_{2}}}} \biggr)} \\ &{} + \frac{k(m{b_{1}}+{{b_{2}}})^{-2}}{(\alpha +2k)(\alpha +k)(m+1)}{_{2}F_{1,k} \biggl(2k,\alpha +2k,\alpha +3k, \frac{{{b_{2}}}-{b_{1}}}{m{b_{1}}+{{b_{2}}}} \biggr)}, \end{aligned} \\ &\pi _{18} :=\frac{k({b_{1}}+m{{b_{2}}})^{-2}}{\alpha +k}{_{2}F_{1,k} \biggl(2k,\alpha +k,\alpha +2k, \frac{{b_{1}}-{{b_{2}}}}{{b_{1}}+m{{b_{2}}}} \biggr)}, \\ & \begin{aligned} \pi _{19} :={}&\frac{km({b_{1}}+m{{b_{2}}})^{-2}}{(\alpha +k)(m+1)}{_{2}F_{1,k} \biggl(2k,\alpha +k,\alpha +2k, \frac{{b_{1}}-{{b_{2}}}}{{b_{1}}+m{{b_{2}}}} \biggr)} \\ &{} + \frac{k({b_{1}}+m{{b_{2}}})^{-2}}{(\alpha +2k)(\alpha +k)(m+1)} {_{2}F_{1,k} \biggl(2k,\alpha +2k,\alpha +3k, \frac{{b_{1}}-{{b_{2}}}}{{b_{1}}+m{{b_{2}}}} \biggr)}, \end{aligned} \\ &\pi _{20} := \frac{k({b_{1}}+m{{b_{2}}})^{-2}}{(\alpha +2k)(\alpha +k)(m+1)}{_{2}F_{1,k} \biggl(2k,\alpha +k,\alpha +3k, \frac{{b_{1}}-{{b_{2}}}}{{b_{1}}+m{{b_{2}}}} \biggr)}. \end{aligned}$$

Theorem 2.7

Let \({\Upsilon }:[{b_{1}},{{b_{2}}}]\rightarrow \mathbb{R}\) be a continuous function on \(({b_{1}},{{b_{2}}})\) with \({b_{1}}<{{b_{2}}}\) and \(|{\Upsilon }'|^{q}\) be an harmonic convex function with \(\frac{1}{p}+\frac{1}{q}=1\) and \(\lambda ,\mu \in [0,\infty )\) with \(\lambda +\mu \neq0\), then

$$\begin{aligned} & \biggl\vert \frac{\Omega (\lambda ){\Upsilon }({{b_{2}}})+\Omega (\mu ){\Upsilon }({b_{1}})}{\lambda +\mu }\\ &\qquad{}- \frac{1}{\lambda +\mu } \biggl[{_{ (\frac{1}{{{b_{2}}}} )^{+}}I_{ \Phi }{\Upsilon \circ \Psi } \biggl( \frac{\lambda {{b_{2}}}+\mu {b_{1}}}{{b_{1}{b_{2}}}(\lambda +\mu )} \biggr)} +{_{ (\frac{1}{{b_{1}}} )^{-}}I_{\Phi }{\Upsilon \circ \Psi } \biggl( \frac{\lambda {{b_{2}}}+\mu {b_{1}}}{{b_{1}{b_{2}}}(\lambda +\mu )} \biggr)} \biggr] \biggr\vert \\ &\quad \leq {b_{1}{b_{2}}}({{b_{2}}}-{b_{1}}) \biggl[ \biggl( \int _{0}^{ \lambda }{\Omega ^{p}({\tau })} \,\mathrm{d} {\tau } \biggr)^{\frac{1}{p}} \bigl(\sigma _{1} \bigl\vert {\Upsilon }'({{b_{2}}}) \bigr\vert ^{q}+\sigma _{2} \bigl\vert {\Upsilon }'({b_{1}}) \bigr\vert ^{q} \bigr)^{\frac{1}{q}}\\ &\qquad {}+ \biggl( \int _{0}^{\mu }{\Omega ^{p}({\tau })} \,\mathrm{d} {\tau } \biggr)^{\frac{1}{p}} \bigl(\sigma _{3} \bigl\vert {\Upsilon }'({{b_{2}}}) \bigr\vert ^{q}+\sigma _{4} \bigl\vert { \Upsilon }'({b_{1}}) \bigr\vert ^{q} \bigr)^{\frac{1}{q}} \biggr], \end{aligned}$$

where

$$\begin{aligned}& \begin{aligned} \sigma _{1}& := \int _{0}^{\lambda }\frac{((\lambda -{\tau }){{b_{2}}}+(\mu +{\tau }){b_{1}})^{-2q}(\mu +{\tau })}{\lambda +\mu } \,\mathrm{d} { \tau } \\ &= \frac{\mu (\lambda {{b_{2}}}+\mu {b_{1}})^{1-2q}-(\lambda +\mu )((\lambda +\mu ){b_{1}})^{1-2q}}{(\lambda +\mu )({{b_{2}}}-{b_{1}})(1-2q)}- \frac{((\lambda +\mu ){b_{1}})^{2-2q}-(\lambda {{b_{2}}}+\mu {b_{1}})^{2-2q}}{(\lambda +\mu )({{b_{2}}}-{b_{1}})^{2}(1-2q)(2-2q)}, \end{aligned}\\& \begin{aligned} \sigma _{2}& := \int _{0}^{\lambda }\frac{((\lambda -{\tau }){{b_{2}}}+(\mu +{\tau }){b_{1}})^{-2q}(\lambda -{\tau })}{\lambda +\mu } \,\mathrm{d} { \tau } \\ &= \frac{\lambda (\lambda {{b_{2}}}+\mu {b_{1}})^{1-2q}}{(\lambda +\mu )({{b_{2}}}-{b_{1}})(1-2q)}+ \frac{((\lambda +\mu ){b_{1}})^{2-2q}-(\lambda {{b_{2}}}+\mu {b_{1}})^{2-2q}}{(\lambda +\mu )({{b_{2}}}-{b_{1}})^{2}(1-2q)(2-2q)}, \end{aligned}\\& \begin{aligned} \sigma _{3}& := \int _{0}^{\mu }\frac{((\lambda +{\tau }){{b_{2}}}+(\mu -{\tau }){b_{1}})^{-2q}(\mu -{\tau })}{\lambda +\mu } \,\mathrm{d} { \tau } \\ &= \frac{((\lambda +\mu ){{b_{2}}})^{2-2q}-(\lambda {{b_{2}}}+\mu {b_{1}})^{2-2q}}{(\lambda +\mu )({{b_{2}}}-{b_{1}})^{2}(1-2q)(2-2q)}- \frac{\mu (\lambda {{b_{2}}}+\mu {b_{1}})^{1-2q}}{(\lambda +\mu )({{b_{2}}}-{b_{1}})(1-2q)}, \end{aligned}\\& \begin{aligned} \sigma _{4}& := \int _{0}^{\mu }\frac{((\lambda +{\tau }){{b_{2}}}+(\mu -{\tau }){b_{1}})^{-2q}(\lambda +{\tau })}{\lambda +\mu } \,\mathrm{d} { \tau } \\ &= \frac{(\lambda +\mu )((\lambda +\mu ){{b_{2}}})^{1-2q}-\lambda (\lambda {{b_{2}}}+\mu {b_{1}})^{1-2q}}{(\lambda +\mu )({{b_{2}}}-{b_{1}})(1-2q)}- \frac{((\lambda +\mu ){{b_{2}}})^{2-2q}-(\lambda {{b_{2}}}+\mu {b_{1}})^{2-2q}}{(\lambda +\mu )({{b_{2}}}-{b_{1}})^{2}(1-2q)(2-2q)}. \end{aligned} \end{aligned}$$

Proof

Using Lemma 2.3, the modulus property, Hölder’s inequality and the harmonic convexity of \(|{\Upsilon }'|^{q}\), we have

$$\begin{aligned} & \biggl\vert \frac{\Omega (\lambda ){\Upsilon }({{b_{2}}})+\Omega (\mu ){\Upsilon }({b_{1}})}{\lambda +\mu }\\ &\qquad {}- \frac{1}{\lambda +\mu } \biggl[{_{ (\frac{1}{{{b_{2}}}} )^{+}}I_{ \Phi }{\Upsilon \circ \Psi } \biggl( \frac{\lambda {{b_{2}}}+\mu {b_{1}}}{{b_{1}{b_{2}}}(\lambda +\mu )} \biggr)} +{_{ (\frac{1}{{b_{1}}} )^{-}}I_{\Phi }{\Upsilon \circ \Psi } \biggl( \frac{\lambda {{b_{2}}}+\mu {b_{1}}}{{b_{1}{b_{2}}}(\lambda +\mu )} \biggr)} \biggr] \biggr\vert \\ &\quad \leq {b_{1}{b_{2}}}({{b_{2}}}-{b_{1}}) \biggl[ \biggl( \int _{0}^{ \lambda }{\Omega ^{p}({\tau })} \,\mathrm{d} {\tau } \biggr)^{\frac{1}{p}}\\ &\qquad{}\times \biggl( \int _{0}^{\lambda } \frac{1}{((\lambda -{\tau }){{b_{2}}}+(\mu +{\tau }){b_{1}})^{2p}} \biggl\vert {\Upsilon }' \biggl( \frac{{b_{1}{b_{2}}}(\lambda +\mu )}{(\lambda -{\tau }){{b_{2}}}+(\mu +{\tau }){b_{1}}} \biggr) \biggr\vert ^{q} \,\mathrm{d} {\tau } \biggr)^{\frac{1}{q}} \\ &\qquad {}+ \biggl( \int _{0}^{\mu }{\Omega ^{p}({\tau })} \,\mathrm{d} {\tau } \biggr)^{\frac{1}{p}}\\ &\qquad{}\times \biggl( \int _{0}^{\mu } \frac{1}{((\lambda +{\tau }){{b_{2}}}+(\mu -{\tau }){b_{1}})^{2p}} \biggl\vert {\Upsilon }' \biggl( \frac{{b_{1}{b_{2}}}(\lambda +\mu )}{(\lambda +{\tau }){{b_{2}}}+(\mu -{\tau }){b_{1}}} \biggr) \biggr\vert ^{q} \,\mathrm{d} {\tau } \biggr)^{\frac{1}{q}} \biggr] \\ &\quad \leq {b_{1}{b_{2}}}({{b_{2}}}-{b_{1}}) \biggl[ \biggl( \int _{0}^{ \lambda }{\Omega ^{p}({\tau })} \,\mathrm{d} {\tau } \biggr)^{\frac{1}{p}} \biggl( \int _{0}^{\lambda }{\bigl((\lambda -{\tau }){{b_{2}}}+(\mu +{\tau }){b_{1}}\bigr)^{-2q}}\\ &\qquad{}\times \biggl(\frac{\mu +{\tau }}{\lambda +\mu } \bigl\vert {\Upsilon }'({{b_{2}}}) \bigr\vert ^{q} +\frac{\lambda -{\tau }}{\lambda +\mu } \bigl\vert {\Upsilon }'({b_{1}}) \bigr\vert ^{q} \biggr) \,\mathrm{d} {\tau } \biggr)^{\frac{1}{q}} \\ &\qquad {}+ \biggl( \int _{0}^{\mu }{\Omega ^{p}({\tau })} \,\mathrm{d} {\tau } \biggr)^{\frac{1}{p}} \biggl( \int _{0}^{\mu }{\bigl((\lambda +{\tau }){{b_{2}}}+( \mu -{\tau }){b_{1}}\bigr)^{-2q}}\\ &\qquad{}\times \biggl(\frac{\mu -{\tau }}{\lambda +\mu } \bigl\vert {\Upsilon }'({{b_{2}}}) \bigr\vert ^{q} + \frac{\lambda +{\tau }}{\lambda +\mu } \bigl\vert {\Upsilon }'({b_{1}}) \bigr\vert ^{q} \biggr) \,\mathrm{d} {\tau } \biggr)^{\frac{1}{q}} \biggr]. \end{aligned}$$

After simple calculations, we obtain our required result. □

Corollary 2.19

Choosing \(\Phi ({\tau })={\tau }\) in Theorem 2.7, we have

$$\begin{aligned} & \biggl\vert \frac{\lambda {\Upsilon }({{b_{2}}})+\mu {\Upsilon }({b_{1}})}{\lambda +\mu }- \frac{{b_{1}{b_{2}}}}{{{b_{2}}}-{b_{1}}} \int _{\frac{1}{{{b_{2}}}}}^{ \frac{1}{{b_{1}}}}{\Upsilon \circ \Psi }(x) \,\mathrm{d}x \biggr\vert \\ &\quad \leq {b_{1}{b_{2}}}({{b_{2}}}-{b_{1}}) \biggl[ \biggl( \frac{\lambda ^{p+1}}{p+1} \biggr)^{\frac{1}{p}} \bigl(\sigma _{1} \bigl\vert { \Upsilon }'({{b_{2}}}) \bigr\vert ^{q}+\sigma _{2} \bigl\vert {\Upsilon }'({b_{1}}) \bigr\vert ^{q} \bigr)^{\frac{1}{q}}+ \biggl(\frac{\mu ^{p+1}}{p+1} \biggr)^{ \frac{1}{p}} \\ &\qquad {}\times \bigl(\sigma _{3} \bigl\vert {\Upsilon }'({{b_{2}}}) \bigr\vert ^{q}+\sigma _{4} \bigl\vert { \Upsilon }'({b_{1}}) \bigr\vert ^{q} \bigr)^{\frac{1}{q}} \biggr], \end{aligned}$$

where \(\sigma _{1}\), \(\sigma _{2}\), \(\sigma _{3}\) and \(\sigma _{4}\) are already defined in Theorem 2.7.

Corollary 2.20

Taking \(\Phi ({\tau })=\frac{{\tau }^{\alpha }}{\Gamma (\alpha )}\) in Theorem 2.7, we obtain

$$\begin{aligned} &\biggl\vert \frac{\lambda ^{\alpha }{\Upsilon }({{b_{2}}})+\mu ^{\alpha }{\Upsilon }({b_{1}})}{\lambda +\mu }- \frac{({b_{1}{b_{2}}})^{\alpha }(\lambda +\mu )^{\alpha -1}\Gamma (\alpha +1)}{({{b_{2}}}-{b_{1}})^{\alpha }} \biggl[{J_{ (\frac{1}{{{b_{2}}}} )^{+}}^{\alpha }{\Upsilon \circ \Psi } \biggl( \frac{\lambda {{b_{2}}}+\mu {b_{1}}}{{b_{1}{b_{2}}}(\lambda +\mu )} \biggr)} \\ &\qquad {}+{{J_{ (\frac{1}{{b_{1}}} )^{-}}^{\alpha }} { \Upsilon \circ \Psi } \biggl( \frac{\lambda {{b_{2}}}+\mu {b_{1}}}{{b_{1}{b_{2}}}(\lambda +\mu )} \biggr)} \biggr]\biggr\vert \\ &\quad \leq {b_{1}{b_{2}}}({{b_{2}}}-{b_{1}}) \biggl[ \biggl( \frac{\lambda ^{{\alpha }p+1}}{{\alpha }p+1} \biggr)^{\frac{1}{p}} \bigl(\sigma _{1} \bigl\vert {\Upsilon }'({{b_{2}}}) \bigr\vert ^{q}+\sigma _{2} \bigl\vert {\Upsilon }'({b_{1}}) \bigr\vert ^{q} \bigr)^{\frac{1}{q}}+ \biggl(\frac{\mu ^{{\alpha }p+1}}{{\alpha }p+1} \biggr)^{\frac{1}{p}} \\ &\qquad {}\times \bigl(\sigma _{3} \bigl\vert {\Upsilon }'({{b_{2}}}) \bigr\vert ^{q}+\sigma _{4} \bigl\vert { \Upsilon }'({b_{1}}) \bigr\vert ^{q} \bigr)^{\frac{1}{q}} \biggr], \end{aligned}$$

where \(\sigma _{1}\), \(\sigma _{2}\), \(\sigma _{3}\) and \(\sigma _{4}\) are already defined in Theorem 2.7.

Corollary 2.21

Choosing \(\Phi ({\tau })= \frac{{\tau }^{\frac{\alpha }{k}}}{k\Gamma _{k}(\alpha )}\) in Theorem 2.7, we obtain

$$\begin{aligned} &\biggl\vert \frac{\lambda ^{\frac{\alpha }{k}}{\Upsilon }({{b_{2}}})+\mu ^{\frac{\alpha }{k}}{\Upsilon }({b_{1}})}{\lambda +\mu }- \frac{k({b_{1}{b_{2}}})^{\frac{\alpha }{k}}(\lambda +\mu )^{\frac{\alpha }{k}-1}\Gamma _{k}(\alpha +k)}{({{b_{2}}}-{b_{1}})^{\frac{\alpha }{k}}} \biggl[{_{k}{J_{ ( \frac{1}{{{b_{2}}}} )^{+}}^{\alpha }} {\Upsilon \circ \Psi } \biggl( \frac{\lambda {{b_{2}}}+\mu {b_{1}}}{{b_{1}{b_{2}}}(\lambda +\mu )} \biggr)} \\ &\qquad {}+{{_{k}J_{ (\frac{1}{{b_{1}}} )^{-}}^{ \alpha }} {\Upsilon \circ \Psi } \biggl( \frac{\lambda {{b_{2}}}+\mu {b_{1}}}{{b_{1}{b_{2}}}(\lambda +\mu )} \biggr)} \biggr]\biggr\vert \\ &\quad \leq {b_{1}{b_{2}}}({{b_{2}}}-{b_{1}}) \biggl[ \biggl( \frac{k\lambda ^{\frac{{\alpha }p+k}{k}}}{{\alpha }p+k} \biggr)^{ \frac{1}{p}} \bigl(\sigma _{1} \bigl\vert {\Upsilon }'({{b_{2}}}) \bigr\vert ^{q}+\sigma _{2} \bigl\vert { \Upsilon }'({b_{1}}) \bigr\vert ^{q} \bigr)^{\frac{1}{q}}+ \biggl( \frac{k\mu ^{\frac{{\alpha }p+k}{k}}}{{{\alpha }p}+k} \biggr)^{ \frac{1}{p}} \\ &\qquad {}\times \bigl(\sigma _{3} \bigl\vert {\Upsilon }'({{b_{2}}}) \bigr\vert ^{q}+\sigma _{4} \bigl\vert { \Upsilon }'({b_{1}}) \bigr\vert ^{q} \bigr)^{\frac{1}{q}} \biggr], \end{aligned}$$

where \(\sigma _{1}\), \(\sigma _{2}\), \(\sigma _{3}\) and \(\sigma _{4}\) are already defined in Theorem 2.7.

Theorem 2.8

Let \({\Upsilon }:[{b_{1}},{{b_{2}}}]\rightarrow \mathbb{R}\) be a function on \(({b_{1}},{{b_{2}}})\) with \({b_{1}}<{{b_{2}}}\) and \(|{\Upsilon }'|^{q}\) be an harmonic convex function with \(q\geq 1\) and \(\lambda ,\mu \in [0,\infty )\) with \(\lambda +\mu \neq0\), then

$$\begin{aligned} & \biggl\vert \frac{\Omega (\lambda ){\Upsilon }({{b_{2}}})+\Omega (\mu ){\Upsilon }({b_{1}})}{\lambda +\mu }\\ &\qquad{}- \frac{1}{\lambda +\mu } \biggl[{_{ (\frac{1}{{{b_{2}}}} )^{+}}I_{ \Phi }{\Upsilon \circ \Psi } \biggl( \frac{\lambda {{b_{2}}}+\mu {b_{1}}}{{b_{1}{b_{2}}}(\lambda +\mu )} \biggr)} +{_{ (\frac{1}{{b_{1}}} )^{-}}I_{\Phi }{\Upsilon \circ \Psi } \biggl( \frac{\lambda {{b_{2}}}+\mu {b_{1}}}{{b_{1}{b_{2}}}(\lambda +\mu )} \biggr)} \biggr] \biggr\vert \\ &\quad \leq {b_{1}{b_{2}}}({{b_{2}}}-{b_{1}}) \biggl[ \biggl( \int _{0}^{ \lambda } \frac{\Omega ({\tau })}{((\lambda -{\tau }){{b_{2}}}+(\mu +{\tau }){b_{1}})^{2}} \,\mathrm{d} {\tau } \biggr)^{1-\frac{1}{q}}\\ &\qquad{}\times\biggl( \int _{0}^{\lambda } \frac{\Omega ({\tau })}{((\lambda -{\tau }){{b_{2}}}+(\mu +{\tau }){b_{1}})^{2}} \biggl(\frac{\mu +{\tau }}{\lambda +\mu } \bigl\vert { \Upsilon }'({{b_{2}}}) \bigr\vert ^{q}+ \frac{\lambda -{\tau }}{\lambda +\mu } \bigl\vert { \Upsilon }'({b_{1}}) \bigr\vert ^{q} \biggr) \,\mathrm{d} {\tau }\biggr)^{\frac{1}{q}}\\ &\qquad{}+ \biggl( \int _{0}^{\mu } \frac{\Omega ({\tau })}{((\lambda +{\tau }){{b_{2}}}+(\mu -{\tau }){b_{1}})^{2}} \,\mathrm{d} {\tau } \biggr)^{1-\frac{1}{q}} \\ &\qquad {}\times \biggl( \int _{0}^{\mu } \frac{\Omega ({\tau })}{((\lambda +{\tau }){{b_{2}}}+(\mu -{\tau }){b_{1}})^{2}} \biggl( \frac{\mu -{\tau }}{\lambda +\mu } \bigl\vert {\Upsilon }'({{b_{2}}}) \bigr\vert ^{q}+ \frac{\lambda +{\tau }}{\lambda +\mu } \bigl\vert {\Upsilon }'({b_{1}}) \bigr\vert ^{q} \biggr) \,\mathrm{d} {\tau } \biggr)^{\frac{1}{q}}\biggr]. \end{aligned}$$

Proof

Using Lemma 2.3, the modulus property, the power mean inequality and the harmonic convexity of of \(|{\Upsilon }'|^{q}\), we have

$$\begin{aligned} & \biggl\vert \frac{\Omega (\lambda ){\Upsilon }({{b_{2}}})+\Omega (\mu ){\Upsilon }({b_{1}})}{\lambda +\mu }\\ &\qquad{}- \frac{1}{\lambda +\mu } \biggl[{_{ (\frac{1}{{{b_{2}}}} )^{+}}I_{ \Phi }{\Upsilon \circ \Psi } \biggl( \frac{\lambda {{b_{2}}}+\mu {b_{1}}}{{b_{1}{b_{2}}}(\lambda +\mu )} \biggr)} +{_{ (\frac{1}{{b_{1}}} )^{-}}I_{\Phi }{\Upsilon \circ \Psi } \biggl( \frac{\lambda {{b_{2}}}+\mu {b_{1}}}{{b_{1}{b_{2}}}(\lambda +\mu )} \biggr)} \biggr] \biggr\vert \\ &\quad \leq {b_{1}{b_{2}}}({{b_{2}}}-{b_{1}}) \biggl[ \int _{0}^{\lambda } \frac{\Omega ({\tau })}{((\lambda -{\tau }){{b_{2}}}+(\mu +{\tau }){b_{1}})^{2}} \biggl\vert {\Upsilon }' \biggl( \frac{{b_{1}{b_{2}}}(\lambda +\mu )}{(\lambda -{\tau }){{b_{2}}}+(\mu +{\tau }){b_{1}}} \biggr) \biggr\vert \,\mathrm{d} {\tau } \\ &\qquad {}+ \int _{0}^{\mu } \frac{\Omega ({\tau })}{((\lambda +{\tau }){{b_{2}}}+(\mu -{\tau }){b_{1}})^{2}} \biggl\vert {\Upsilon }' \biggl( \frac{{b_{1}{b_{2}}}(\lambda +\mu )}{(\lambda +{\tau }){{b_{2}}}+(\mu -{\tau }){b_{1}}} \biggr) \biggr\vert \,\mathrm{d} {\tau }\biggr] \\ &\quad \leq {b_{1}{b_{2}}}({{b_{2}}}-{b_{1}}) \biggl[ \biggl( \int _{0}^{ \lambda } \frac{\Omega ({\tau })}{((\lambda -{\tau }){{b_{2}}}+(\mu +{\tau }){b_{1}})^{2}} \,\mathrm{d} {\tau } \biggr)^{1-\frac{1}{q}}\\ &\qquad{}\times\biggl( \int _{0}^{\lambda } \frac{\Omega ({\tau })}{((\lambda -{\tau }){{b_{2}}}+(\mu +{\tau }){b_{1}})^{2}} \\ &\qquad {}\times \biggl\vert {\Upsilon }' \biggl( \frac{{b_{1}{b_{2}}}(\lambda +\mu )}{(\lambda -{\tau }){{b_{2}}}+(\mu +{\tau }){b_{1}}} \biggr) \biggr\vert \,\mathrm{d} {\tau }\biggr)^{\frac{1}{q}}\\ &\qquad{}+ \biggl( \int _{0}^{ \mu } \frac{\Omega ({\tau })}{((\lambda +{\tau }){{b_{2}}}+(\mu -{\tau }){b_{1}})^{2}} \,\mathrm{d} {\tau } \biggr)^{1-\frac{1}{q}} \\ &\qquad {}\times \biggl( \int _{0}^{\mu } \frac{\Omega ({\tau })}{((\lambda +{\tau }){{b_{2}}}+(\mu -{\tau }){b_{1}})^{2}} \biggl\vert {\Upsilon }' \biggl( \frac{{b_{1}{b_{2}}}(\lambda +\mu )}{(\lambda +{\tau }){{b_{2}}}+(\mu -{\tau }){b_{1}}} \biggr) \biggr\vert \,\mathrm{d} {\tau } \biggr)^{\frac{1}{q}}\biggr] \\ &\quad \leq {b_{1}{b_{2}}}({{b_{2}}}-{b_{1}}) \biggl[ \biggl( \int _{0}^{ \lambda } \frac{\Omega ({\tau })}{((\lambda -{\tau }){{b_{2}}}+(\mu +{\tau }){b_{1}})^{2}} \,\mathrm{d} {\tau } \biggr)^{1-\frac{1}{q}}\\ &\qquad{}\times\biggl( \int _{0}^{\lambda } \frac{\Omega ({\tau })}{((\lambda -{\tau }){{b_{2}}}+(\mu +{\tau }){b_{1}})^{2}} \\ &\qquad {} \times \biggl(\frac{\mu +{\tau }}{\lambda +\mu } \bigl\vert { \Upsilon }'({{b_{2}}}) \bigr\vert ^{q}+ \frac{\lambda -{\tau }}{\lambda +\mu } \bigl\vert { \Upsilon }'({b_{1}}) \bigr\vert ^{q} \biggr) \,\mathrm{d} {\tau }\biggr)^{\frac{1}{q}}\\ &\qquad{}+ \biggl( \int _{0}^{\mu } \frac{\Omega ({\tau })}{((\lambda +{\tau }){{b_{2}}}+(\mu -{\tau }){b_{1}})^{2}} \,\mathrm{d} {\tau } \biggr)^{1-\frac{1}{q}} \\ &\qquad {}\times \biggl( \int _{0}^{\mu } \frac{\Omega ({\tau })}{((\lambda +{\tau }){{b_{2}}}+(\mu -{\tau }){b_{1}})^{2}} \biggl( \frac{\mu -{\tau }}{\lambda +\mu } \bigl\vert {\Upsilon }'({{b_{2}}}) \bigr\vert ^{q}+ \frac{\lambda +{\tau }}{\lambda +\mu } \bigl\vert {\Upsilon }'({b_{1}}) \bigr\vert ^{q} \biggr) \,\mathrm{d} {\tau } \biggr)^{\frac{1}{q}}\biggr]. \end{aligned}$$

This completes the proof. □

Corollary 2.22

Choosing \(\Phi ({\tau })={\tau }\) and \(\lambda =\mu =1\) in Theorem 2.8, we have

$$\begin{aligned} & \biggl\vert \frac{ {\Upsilon }({{b_{2}}})+ {\Upsilon }({b_{1}})}{2}- \frac{{b_{1}{b_{2}}}}{{{b_{2}}}-{b_{1}}} \int _{\frac{1}{{{b_{2}}}}}^{ \frac{1}{{b_{1}}}}{\Upsilon \circ \Psi }(x) \,\mathrm{d}x \biggr\vert \\ &\quad \leq {b_{1}{b_{2}}}({{b_{2}}}-{b_{1}}) \bigl[\sigma _{5}^{1- \frac{1}{q}} \bigl(\sigma _{6} \bigl\vert {\Upsilon }'({{b_{2}}}) \bigr\vert ^{q}+\sigma _{7} \bigl\vert { \Upsilon }'({b_{1}}) \bigr\vert ^{q} \bigr)^{\frac{1}{q}}\\ &\qquad{}+ \sigma _{8}^{1- \frac{1}{q}} \bigl(\sigma _{9} \bigl\vert {\Upsilon }'({{b_{2}}}) \bigr\vert ^{q}+\sigma _{10} \bigl\vert { \Upsilon }'({b_{1}}) \bigr\vert ^{q} \bigr)^{\frac{1}{q}} \bigr], \end{aligned}$$

where

$$\begin{aligned} &\sigma _{5} :=\frac{({b_{1}}+{{b_{2}}})^{-2}}{2}{_{2}F_{1} \biggl(2,2,3, \frac{{{b_{2}}}-{b_{1}}}{{b_{1}}+{{b_{2}}}} \biggr)}, \\ &\sigma _{6} :=\frac{({b_{1}}+{{b_{2}}})^{-2}}{4}{_{2}F_{1} \biggl(2,2,3, \frac{{{b_{2}}}-{b_{1}}}{{b_{1}}+{{b_{2}}}} \biggr)}+ \frac{({b_{1}}+{{b_{2}}})^{-2}}{12}{_{2}F_{1} \biggl(2,3,4, \frac{{{b_{2}}}-{b_{1}}}{{b_{1}}+{{b_{2}}}} \biggr)}, \\ &\sigma _{7} :=\frac{({b_{1}}+{{b_{2}}})^{-2}}{12}{_{2}F_{1} \biggl(2,2,4, \frac{{{b_{2}}}-{b_{1}}}{{b_{1}}+{{b_{2}}}} \biggr)}, \\ &\sigma _{8} :=\frac{({b_{1}}+{{b_{2}}})^{-2}}{2}{_{2}F_{1} \biggl(2,2,3, \frac{{b_{1}}-{{b_{2}}}}{{b_{1}}+{{b_{2}}}} \biggr)}, \\ &\sigma _{9} :=\frac{({b_{1}}+{{b_{2}}})^{-2}}{12}{_{2}F_{1} \biggl(2,2,4, \frac{{b_{1}}-{{b_{2}}}}{{b_{1}}+{{b_{2}}}} \biggr)}, \\ &\sigma _{10} :=\frac{({b_{1}}+{{b_{2}}})^{-2}}{4}{_{2}F_{1} \biggl(2,2,3, \frac{{b_{1}}-{{b_{2}}}}{{b_{1}}+{{b_{2}}}} \biggr)}+ \frac{({b_{1}}+{{b_{2}}})^{-2}}{12}{_{2}F_{1} \biggl(2,3,4, \frac{{b_{1}}-{{b_{2}}}}{{b_{1}}+{{b_{2}}}} \biggr)}. \end{aligned}$$

Corollary 2.23

Taking \(\Phi ({\tau })=\frac{{\tau }^{\alpha }}{\Gamma (\alpha )}\) and \(\lambda =\mu =1\) in Theorem 2.8, we obtain

$$\begin{aligned} &\biggl\vert \frac{{\Upsilon }({{b_{2}}})+{\Upsilon }({b_{1}})}{2}- \frac{({b_{1}{b_{2}}})^{\alpha }2^{\alpha -1}\Gamma (\alpha +1)}{({{b_{2}}}-{b_{1}})^{\alpha }} \biggl[{J_{ (\frac{1}{{{b_{2}}}} )^{+}}^{\alpha }{\Upsilon \circ \Psi } \biggl( \frac{{b_{1}}+{{b_{2}}}}{2{b_{1}{b_{2}}}} \biggr)} \\ &\qquad {}+{{J_{ (\frac{1}{{b_{1}}} )^{-}}^{\alpha }} { \Upsilon \circ \Psi } \biggl( \frac{{b_{1}}+{{b_{2}}}}{2{b_{1}{b_{2}}}} \biggr)} \biggr]\biggr\vert \\ &\quad \leq {b_{1}{b_{2}}}({{b_{2}}}-{b_{1}}) \bigl[\sigma _{11}^{1- \frac{1}{q}} \bigl(\sigma _{12} \bigl\vert {\Upsilon }'({{b_{2}}}) \bigr\vert ^{q}+\sigma _{13} \bigl\vert { \Upsilon }'({b_{1}}) \bigr\vert ^{q} \bigr)^{\frac{1}{q}}\\ &\qquad{}+ \sigma _{14}^{1- \frac{1}{q}} \bigl(\sigma _{15} \bigl\vert {\Upsilon }'({{b_{2}}}) \bigr\vert ^{q}+\sigma _{16} \bigl\vert { \Upsilon }'({b_{1}}) \bigr\vert ^{q} \bigr)^{\frac{1}{q}} \bigr], \end{aligned}$$

where

$$\begin{aligned} &\sigma _{11} :=\frac{({b_{1}}+{{b_{2}}})^{-2}}{\alpha +1}{_{2}F_{1} \biggl(2,\alpha +1,\alpha +2, \frac{{{b_{2}}}-{b_{1}}}{{b_{1}}+{{b_{2}}}} \biggr)}, \\ & \begin{aligned} \sigma _{12} :={}&\frac{({b_{1}}+{{b_{2}}})^{-2}}{2(\alpha +1)}{_{2}F_{1} \biggl(2,\alpha +1,\alpha +2, \frac{{{b_{2}}}-{b_{1}}}{{b_{1}}+{{b_{2}}}} \biggr)} \\ &{}+\frac{({b_{1}}+{{b_{2}}})^{-2}}{2(\alpha +1)(\alpha +2)} {_{2}F_{1} \biggl(2,\alpha +2, \alpha +3, \frac{{{b_{2}}}-{b_{1}}}{{b_{1}}+{{b_{2}}}} \biggr)}, \end{aligned} \\ &\sigma _{13} := \frac{({b_{1}}+{{b_{2}}})^{-2}}{2(\alpha +1)(\alpha +2)}{_{2}F_{1} \biggl(2,\alpha +1,\alpha +3, \frac{{{b_{2}}}-{b_{1}}}{{b_{1}}+{{b_{2}}}} \biggr)}, \\ &\sigma _{14} :=\frac{({b_{1}}+{{b_{2}}})^{-2}}{\alpha +1}{_{2}F_{1} \biggl(2,\alpha +1,\alpha +2, \frac{{b_{1}}-{{b_{2}}}}{{b_{1}}+{{b_{2}}}} \biggr)}, \\ &\sigma _{15} := \frac{({b_{1}}+{{b_{2}}})^{-2}}{2(\alpha +1)(\alpha +2)}{_{2}F_{1} \biggl(2,\alpha +1,\alpha +3, \frac{{b_{1}}-{{b_{2}}}}{{b_{1}}+{{b_{2}}}} \biggr)}, \\ &\begin{aligned} \sigma _{16}:={}&\frac{({b_{1}}+{{b_{2}}})^{-2}}{2(\alpha +1)}{_{2}F_{1} \biggl(2,\alpha +1,\alpha +2, \frac{{b_{1}}-{{b_{2}}}}{{b_{1}}+{{b_{2}}}} \biggr)} \\ &{}+\frac{({b_{1}}+{{b_{2}}})^{-2}}{2(\alpha +1)(\alpha +2)}{_{2}F_{1} \biggl(2,\alpha +2, \alpha +3, \frac{{b_{1}}-{{b_{2}}}}{{b_{1}}+{{b_{2}}}} \biggr)}. \end{aligned} \end{aligned}$$

Corollary 2.24

Choosing \(\Phi ({\tau })= \frac{{\tau }^{\frac{\alpha }{k}}}{k\Gamma _{k}(\alpha )}\) and \(\lambda =\mu =1\) in Theorem 2.8, we obtain

$$\begin{aligned} &\biggl\vert \frac{{\Upsilon }({{b_{2}}})+{\Upsilon }({b_{1}})}{2}- \frac{({b_{1}{b_{2}}})^{\frac{\alpha }{k}}2^{\frac{\alpha }{k}-1}\Gamma _{k}(\alpha +k)}{({{b_{2}}}-{b_{1}})^{\frac{\alpha }{k}}} \biggl[{_{k}J_{ (\frac{1}{{{b_{2}}}} )^{+}}^{\alpha }{ \Upsilon \circ \Psi } \biggl(\frac{{b_{1}}+{{b_{2}}}}{2{b_{1}{b_{2}}}} \biggr)} \\ &\qquad {}+{{_{k}J_{ (\frac{1}{{b_{1}}} )^{-}}^{ \alpha }} {\Upsilon \circ \Psi } \biggl( \frac{{b_{1}}+{{b_{2}}}}{2{b_{1}{b_{2}}}} \biggr)} \biggr]\biggr\vert \\ &\quad \leq {b_{1}{b_{2}}}({{b_{2}}}-{b_{1}}) \bigl[\sigma _{16}^{ \frac{1}{p}} \bigl(\sigma _{17} \bigl\vert {\Upsilon }'({{b_{2}}}) \bigr\vert ^{q}+\sigma _{18} \bigl\vert { \Upsilon }'({b_{1}}) \bigr\vert ^{q} \bigr)^{\frac{1}{q}}\\ &\qquad{}+ \sigma _{19}^{ \frac{1}{p}} \bigl(\sigma _{20} \bigl\vert {\Upsilon }'({{b_{2}}}) \bigr\vert ^{q}+\sigma _{21} \bigl\vert { \Upsilon }'({b_{1}}) \bigr\vert ^{q} \bigr)^{\frac{1}{q}} \bigr], \end{aligned}$$

where

$$\begin{aligned} &\sigma _{16} :=\frac{k({b_{1}}+{{b_{2}}})^{-2}}{\alpha +k}{_{2}F_{1} \biggl(2,\alpha +k,\alpha +2k, \frac{{{b_{2}}}-{b_{1}}}{{b_{1}}+{{b_{2}}}} \biggr)}, \\ & \begin{aligned} \sigma _{17} :={}&\frac{k({b_{1}}+{{b_{2}}})^{-2}}{2(\alpha k)}{_{2}F_{1} \biggl(2,\alpha +k,\alpha +2k, \frac{{{b_{2}}}-{b_{1}}}{{b_{1}}+{{b_{2}}}} \biggr)} \\ &{}+\frac{k({b_{1}}+{{b_{2}}})^{-2}}{2(\alpha +k)(\alpha +2k)} {_{2}F_{1} \biggl(2,\alpha +2k, \alpha +3k, \frac{{{b_{2}}}-{b_{1}}}{{b_{1}}+{{b_{2}}}} \biggr)}, \end{aligned} \\ &\sigma _{18} := \frac{k({b_{1}}+{{b_{2}}})^{-2}}{2(\alpha +k)(\alpha +2k)}{_{2}F_{1} \biggl(2,\alpha +k,\alpha +3k, \frac{{{b_{2}}}-{b_{1}}}{{b_{1}}+{{b_{2}}}} \biggr)}, \\ &\sigma _{19} :=\frac{k({b_{1}}+{{b_{2}}})^{-2}}{\alpha +k}{_{2}F_{1} \biggl(2,\alpha +k,\alpha +2k, \frac{{b_{1}}-{{b_{2}}}}{{b_{1}}+{{b_{2}}}} \biggr)}, \\ &\sigma _{20} := \frac{k({b_{1}}+{{b_{2}}})^{-2}}{2(\alpha +k)(\alpha +2k)}{_{2}F_{1} \biggl(2,\alpha +k,\alpha +3k, \frac{{b_{1}}-{{b_{2}}}}{{b_{1}}+{{b_{2}}}} \biggr)}, \\ & \begin{aligned} \sigma _{21}:={}&\frac{k({b_{1}}+{{b_{2}}})^{-2}}{2(\alpha +k)}{_{2}F_{1} \biggl(2,\alpha +k,\alpha +2k, \frac{{b_{1}}-{{b_{2}}}}{{b_{1}}+{{b_{2}}}} \biggr)} \\ &{}+\frac{k({b_{1}}+{{b_{2}}})^{-2}}{2(\alpha +k)(\alpha +2k)}{_{2}F_{1} \biggl(2,\alpha +2k, \alpha +3k, \frac{{b_{1}}-{{b_{2}}}}{{b_{1}}+{{b_{2}}}} \biggr)}. \end{aligned} \end{aligned}$$

Theorem 2.9

Let \({\Upsilon }:[{b_{1}},{{b_{2}}}]\subset (0,+\infty )\rightarrow \mathbb{R}\) be a differentiable function on \(({b_{1}},{{b_{2}}})\) with \({b_{1}}<{{b_{2}}}\). If \(\vert {\Upsilon }' \vert ^{q}\) is an harmonic convex function with \(q>1\) and \(\frac{1}{p}+\frac{1}{q}=1\), then

$$\begin{aligned} & \biggl\vert {\Upsilon } \biggl(\frac{2{b_{1}{b_{2}}}}{{b_{1}}+{{b_{2}}}} \biggr)- \frac{1}{2\Delta (1)} \biggl[_{\frac{1}{{{b_{2}}}}^{+}}I_{ \Phi }{{\Upsilon }\circ {\Psi }} \biggl(\frac{1}{{b_{1}}} \biggr)+ _{ \frac{1}{{b_{1}}}^{-}}I_{\Phi }{{ \Upsilon }\circ {\Psi }} \biggl( \frac{1}{{{b_{2}}}} \biggr) \biggr] \biggr\vert \\ &\quad \leq \frac{{b_{1}{b_{2}}}({{b_{2}}}-{b_{1}})}{2\Delta (1)} \biggl( \biggl( \int _{0}^{\frac{1}{2}} \bigl\vert \Delta ({\tau }) \bigr\vert ^{p} \,\mathrm{d} {\tau } \biggr)^{\frac{1}{p}}+ \biggl( \int _{\frac{1}{2}}^{1} \bigl\vert \delta ({\tau }) \bigr\vert ^{p} \,\mathrm{d} {\tau } \biggr)^{\frac{1}{p}} \biggr) \\ &\qquad {}\times \bigl( \bigl(N_{1} \bigl\vert {\Upsilon }'({b_{1}}) \bigr\vert ^{q}+N_{2} \bigl\vert {\Upsilon }'({{b_{2}}}) \bigr\vert ^{q} \bigr)^{\frac{1}{q}}+ \bigl(N_{3} \bigl\vert {\Upsilon }'({b_{1}}) \bigr\vert ^{q}+N_{4} \bigl\vert {\Upsilon }'({{b_{2}}}) \bigr\vert ^{q} \bigr)^{\frac{1}{q}} \bigr), \end{aligned}$$

where

$$\begin{aligned}& \begin{aligned} N_{1}&:= \int _{0}^{\frac{1}{2}} \frac{(1-{\tau })}{({\tau b_{1}}+(1-{\tau }){{b_{2}}})^{2q}} \,\mathrm{d} { \tau } \\ &= \frac{2^{2-2q}{{b_{2}}}^{2-2q}-({b_{1}}+{{b_{2}}})^{2-2q}}{({{b_{2}}}-{b_{1}})^{2}(2-2q)2^{2-2q}}-{b_{1}} \frac{2^{1-2q}{{b_{2}}}^{1-2q}-({b_{1}}+{{b_{2}}})^{1-2q}}{({{b_{2}}}-{b_{1}})^{2}(1-2q)2^{1-2q}}, \end{aligned}\\& \begin{aligned} N_{2}&:= \int _{0}^{\frac{1}{2}} \frac{{\tau }}{({\tau b_{1}}+(1-{\tau }){{b_{2}}})^{2q}} \,\mathrm{d} { \tau } \\ &={{b_{2}}} \frac{2^{1-2q}{{b_{2}}}^{1-2q}-({b_{1}}+{{b_{2}}})^{1-2q}}{({{b_{2}}}-{b_{1}})^{2}(1-2q)2^{1-2q}}- \frac{2^{2-2q}{{b_{2}}}^{2-2q}-({b_{1}}+{{b_{2}}})^{2-2q}}{({{b_{2}}}-{b_{1}})^{2}(2-2q)2^{2-2q}}, \end{aligned}\\& \begin{aligned} N_{3}&:= \int _{0}^{\frac{1}{2}} \frac{{\tau }}{((1-{\tau }){\tau b_{1}}+{\tau {b_{2}}})^{2q}} \,\mathrm{d} { \tau } \\ &= \frac{({b_{1}}+{{b_{2}}})^{2-2q}-2^{2-2q}{b_{1}}^{2-2q}}{({{b_{2}}}-{b_{1}})^{2}(2-2q)2^{2-2q}}-{b_{1}} \frac{({b_{1}}+{{b_{2}}})^{1-2q}-2^{1-2q}{b_{1}}^{1-2q}}{({{b_{2}}}-{b_{1}})^{2}(1-2q)2^{1-2q}}, \end{aligned}\\& \begin{aligned} N_{4}&:= \int _{0}^{\frac{1}{2}} \frac{1-{\tau }}{((1-{\tau }){\tau b_{1}}+{\tau {b_{2}}})^{2q}} \,\mathrm{d} {\tau } \\ &={{b_{2}}} \frac{({b_{1}}+{{b_{2}}})^{1-2q}-2^{1-2q}{b_{1}}^{1-2q}}{({{b_{2}}}-{b_{1}})^{2}(1-2q)2^{1-2q}}- \frac{({b_{1}}+{{b_{2}}})^{2-2q}-2^{2-2q}{b_{1}}^{2-2q}}{({{b_{2}}}-{b_{1}})^{2}(2-2q)2^{2-2q}}, \end{aligned}\\& \begin{aligned} N_{5}&:= \int _{\frac{1}{2}}^{1} \frac{1-{\tau }}{({\tau b_{1}}+(1-{\tau }){{b_{2}}})^{2q}} \,\mathrm{d} { \tau } \\ &= \frac{({b_{1}}+{{b_{2}}})^{2-2q}-2^{2-2q}{b_{1}}^{2-2q}}{({{b_{2}}}-{b_{1}})^{2}(2-2q)2^{2-2q}}-{b_{1}} \frac{({b_{1}}+{{b_{2}}})^{1-2q}-2^{1-2q}{b_{1}}^{1-2q}}{({{b_{2}}}-{b_{1}})^{2}(1-2q)2^{1-2q}}, \end{aligned}\\& \begin{aligned} N_{6}&:= \int _{\frac{1}{2}}^{1} \frac{{\tau }}{({\tau b_{1}}+(1-{\tau }){{b_{2}}})^{2q}} \,\mathrm{d} { \tau } \\ &={{b_{2}}} \frac{({b_{1}}+{{b_{2}}})^{1-2q}-2^{1-2q}{b_{1}}^{1-2q}}{({{b_{2}}}-{b_{1}})^{2}(1-2q)2^{1-2q}}- \frac{({b_{1}}+{{b_{2}}})^{2-2q}-2^{2-2q}{b_{1}}^{2-2q}}{({{b_{2}}}-{b_{1}})^{2}(2-2q)2^{2-2q}}, \end{aligned}\\& \begin{aligned} N_{7}&:= \int _{\frac{1}{2}}^{1} \frac{{\tau }}{((1-{\tau }){b_{1}}+{\tau {b_{2}}})^{2q}} \,\mathrm{d} { \tau } \\ &= \frac{2^{2-2q}{{b_{2}}}^{2-2q}-({b_{1}}+{{b_{2}}})^{2-2q}}{({{b_{2}}}-{b_{1}})^{2}(2-2q)2^{2-2q}}-{b_{1}} \frac{2^{1-2q}{{b_{2}}}^{1-2q}-({b_{1}}+{{b_{2}}})^{1-2q}}{({{b_{2}}}-{b_{1}})^{2}(1-2q)2^{1-2q}}, \end{aligned}\\& \begin{aligned} N_{8}&:= \int _{\frac{1}{2}}^{1} \frac{1-{\tau }}{((1-{\tau }){b_{1}}+{\tau {b_{2}}})^{2q}} \,\mathrm{d} { \tau } \\ &={{b_{2}}} \frac{2^{1-2q}{{b_{2}}}^{1-2q}-({b_{1}}+{{b_{2}}})^{1-2q}}{({{b_{2}}}-{b_{1}})^{2}(1-2q)2^{1-2q}}- \frac{2^{2-2q}{{b_{2}}}^{2-2q}-({b_{1}}+{{b_{2}}})^{2-2q}}{({{b_{2}}}-{b_{1}})^{2}(2-2q)2^{2-2q}}. \end{aligned} \end{aligned}$$

Also, it is easy to verify that \(N_{1}=N_{7}\), \(N_{2}=N_{8}\), \(N_{3}=N_{5}\) and \(N_{4}=N_{6}\).

Proof

By using Lemma 2.4, the property of modulus, Hölder’s inequality and the harmonic convexity of \(\vert {\Upsilon }' \vert ^{q}\), we obtain the desired result. We omit here the proof. □

Corollary 2.25

Taking \(\Phi ({\tau })={\tau }\) in Theorem 2.9, then

$$\begin{aligned} & \biggl\vert {\Upsilon } \biggl(\frac{2{b_{1}{b_{2}}}}{{b_{1}}+{{b_{2}}}} \biggr)- \frac{{b_{1}{b_{2}}}}{{{b_{2}}}-{b_{1}}} \int _{b_{1}}^{{b_{2}}} \frac{{\Upsilon }(x)}{x^{2}} \,\mathrm{d}x \biggr\vert \\ &\quad \leq \frac{{b_{1}{b_{2}}}({{b_{2}}}-{b_{1}})}{2} \biggl( \frac{1}{2^{p+1}(p+1)} \biggr)^{\frac{1}{p}} \\ &\qquad {}\times \bigl( \bigl(N_{1} \bigl\vert {\Upsilon }'({b_{1}}) \bigr\vert ^{q}+N_{2} \bigl\vert {\Upsilon }'({{b_{2}}}) \bigr\vert ^{q} \bigr)^{\frac{1}{q}}+ \bigl(N_{3} \bigl\vert {\Upsilon }'({b_{1}}) \bigr\vert ^{q}+N_{4} \bigl\vert { \Upsilon }'({{b_{2}}}) \bigr\vert ^{q} \bigr)^{\frac{1}{q}} \bigr). \end{aligned}$$

Corollary 2.26

Choosing \(\Phi ({\tau })=\frac{{\tau }^{\alpha }}{\Gamma (\alpha )}\) in Theorem 2.9, then

$$\begin{aligned} & \biggl\vert {\Upsilon } \biggl(\frac{2{b_{1}{b_{2}}}}{{b_{1}}+{{b_{2}}}} \biggr)- \frac{\Gamma (\alpha +1)}{2} \biggl( \frac{{b_{1}{b_{2}}}}{{{b_{2}}}-{b_{1}}} \biggr)^{\alpha } \biggl[J_{ \frac{1}{{{b_{2}}}}^{+}}^{\alpha }{{\Upsilon }\circ {\Psi }} \biggl( \frac{1}{{b_{1}}} \biggr) +J_{\frac{1}{{b_{1}}}^{-}}^{\alpha }{{ \Upsilon } \circ {\Psi }} \biggl(\frac{1}{{{b_{2}}}} \biggr) \biggr] \biggr\vert \\ &\quad \leq \frac{{b_{1}{b_{2}}}({{b_{2}}}-{b_{1}})}{2} \biggl( \frac{ (1+2^{\alpha p-1}(\alpha p-1) )}{2^{\alpha p}(\alpha p+1)} \biggr)^{\frac{1}{p}} \\ &\qquad {}\times \bigl( \bigl(N_{1} \bigl\vert {\Upsilon }'({b_{1}}) \bigr\vert ^{q}+N_{2} \bigl\vert {\Upsilon }'({{b_{2}}}) \bigr\vert ^{q} \bigr)^{\frac{1}{q}}+ \bigl(N_{3} \bigl\vert {\Upsilon }'({b_{1}}) \bigr\vert ^{q}+N_{4} \bigl\vert { \Upsilon }'({{b_{2}}}) \bigr\vert ^{q} \bigr)^{\frac{1}{q}} \bigr). \end{aligned}$$

Corollary 2.27

Taking \(\Phi ({\tau })= \frac{{\tau }^{\frac{\alpha }{k}}}{k\Gamma _{k}(\alpha )}\) in Theorem 2.9, then

$$\begin{aligned} & \biggl\vert {\Upsilon } \biggl(\frac{2{b_{1}{b_{2}}}}{{b_{1}}+{{b_{2}}}} \biggr)- \frac{\Gamma _{k}(\alpha +k)}{2} \biggl( \frac{{b_{1}{b_{2}}}}{{{b_{2}}}-{b_{1}}} \biggr)^{\frac{\alpha }{k}} \biggl[ _{k}J_{\frac{1}{{{b_{2}}}}^{+}}^{\alpha }{{\Upsilon }\circ { \Psi }} \biggl(\frac{1}{{b_{1}}} \biggr) + _{k}J_{\frac{1}{{b_{1}}}^{-}}^{ \alpha }{{ \Upsilon }\circ {\Psi }} \biggl(\frac{1}{{{b_{2}}}} \biggr) \biggr] \biggr\vert \\ &\quad \leq \frac{{b_{1}{b_{2}}}({{b_{2}}}-{b_{1}})}{2} \biggl( \frac{ (k+2^{\frac{\alpha p}{k}-1}(\alpha p-k) )}{2^{\frac{\alpha p}{k}}(\alpha p+k)} \biggr)^{\frac{1}{p}} \\ &\qquad {}\times \bigl( \bigl(N_{1} \bigl\vert {\Upsilon }'({b_{1}}) \bigr\vert ^{q}+N_{2} \bigl\vert {\Upsilon }'({{b_{2}}}) \bigr\vert ^{q} \bigr)^{\frac{1}{q}}+ \bigl(N_{3} \bigl\vert {\Upsilon }'({b_{1}}) \bigr\vert ^{q}+N_{4} \bigl\vert { \Upsilon }'({{b_{2}}}) \bigr\vert ^{q} \bigr)^{\frac{1}{q}} \bigr). \end{aligned}$$

Theorem 2.10

Let \({\Upsilon }:[{b_{1}},{{b_{2}}}]\subset (0,+\infty )\rightarrow \mathbb{R}\) be a differentiable function on \(({b_{1}},{{b_{2}}})\) with \({b_{1}}<{{b_{2}}}\). If \(\vert {\Upsilon }' \vert ^{q}\) is an harmonic convex function with \(q\geq 1\), then

$$\begin{aligned} & \biggl\vert {\Upsilon } \biggl(\frac{2{b_{1}{b_{2}}}}{{b_{1}}+{{b_{2}}}} \biggr)- \frac{1}{2\Delta (1)} \biggl[_{\frac{1}{{{b_{2}}}}^{+}}I_{ \Phi }{{\Upsilon }\circ {\Psi }} \biggl(\frac{1}{{b_{1}}} \biggr)+ _{ \frac{1}{{b_{1}}}^{-}}I_{\Phi }{{ \Upsilon }\circ {\Psi }} \biggl( \frac{1}{{{b_{2}}}} \biggr) \biggr] \biggr\vert \\ &\quad \leq \frac{{b_{1}{b_{2}}}({{b_{2}}}-{b_{1}})}{2\Delta (1)} \\ &\qquad {}\times \biggl[ \biggl( \int _{0}^{\frac{1}{2}} \bigl\vert \Delta ({\tau }) \bigr\vert \,\mathrm{d} {\tau } \biggr)^{1-\frac{1}{q}} \biggl\{ \biggl( \int _{0}^{ \frac{1}{2}} \frac{(1-{\tau }) \vert \Delta ({\tau }) \vert }{({\tau b_{1}}+(1-{\tau }){{b_{2}}})^{2}} \,\mathrm{d} {\tau } \bigl\vert {\Upsilon }'({b_{1}}) \bigr\vert ^{q}\\ &\qquad{}+ \int _{0}^{ \frac{1}{2}} \frac{{\tau } \vert \Delta ({\tau }) \vert }{({\tau b_{1}}+(1-{\tau }){{b_{2}}})^{2}} \,\mathrm{d} {\tau } \bigl\vert {\Upsilon }'({{b_{2}}}) \bigr\vert ^{q} \biggr)^{ \frac{1}{q}} \\ &\qquad {}+ \biggl( \int _{0}^{\frac{1}{2}} \frac{{\tau } \vert \Delta ({\tau }) \vert }{((1-{\tau }){b_{1}}+{\tau {b_{2}}})^{2}} \,\mathrm{d} {\tau } \bigl\vert {\Upsilon }'({b_{1}}) \bigr\vert ^{q}+ \int _{0}^{ \frac{1}{2}} \frac{(1-{\tau }) \vert \Delta ({\tau }) \vert }{((1-{\tau }){b_{1}}+{\tau {b_{2}}})^{2}} \,\mathrm{d} {\tau } \bigl\vert {\Upsilon }'({{b_{2}}}) \bigr\vert ^{q} \biggr)^{ \frac{1}{q}}\biggr\} \\ &\qquad {}+ \biggl( \int _{\frac{1}{2}}^{1} \bigl\vert \delta ({\tau }) \bigr\vert \,\mathrm{d} {\tau } \biggr)^{1-\frac{1}{q}}\biggl\{ \biggl( \int _{ \frac{1}{2}}^{1} \frac{(1-{\tau }) \vert \delta ({\tau }) \vert }{({\tau b_{1}}+(1-{\tau }){{b_{2}}})^{2}} \,\mathrm{d} {\tau } \bigl\vert {\Upsilon }'({b_{1}}) \bigr\vert ^{q}\\ &\qquad{} + \int _{ \frac{1}{2}}^{1} \frac{{\tau } \vert \delta ({\tau }) \vert }{({\tau b_{1}}+(1-{\tau }){{b_{2}}})^{2}} \,\mathrm{d} {\tau } \bigl\vert {\Upsilon }'({{b_{2}}}) \bigr\vert ^{q} \biggr)^{ \frac{1}{q}} \\ &\qquad {} + \biggl( \int _{\frac{1}{2}}^{1} \frac{{\tau } \vert \delta ({\tau }) \vert }{((1-{\tau }){b_{1}}+{\tau {b_{2}}})^{2}} \,\mathrm{d} {\tau } \bigl\vert {\Upsilon }'({b_{1}}) \bigr\vert ^{q}\\ &\qquad{}+ \int _{ \frac{1}{2}}^{1} \frac{(1-{\tau }) \vert \delta ({\tau }) \vert }{((1-{\tau }){b_{1}}+{\tau {b_{2}}})^{2}} \,\mathrm{d} {\tau } \bigl\vert {\Upsilon }'({{b_{2}}}) \bigr\vert ^{q} \biggr)^{ \frac{1}{q}}\biggr\} \biggr]. \end{aligned}$$

Proof

By using Lemma 2.4, the property of modulus, the power mean inequality and the convexity of \(\vert {\Upsilon }' \vert ^{q}\) we obtain the desired result. We omit here the proof. □

Corollary 2.28

Taking \(\Phi ({\tau })={\tau }\) in Theorem 2.10, then

$$\begin{aligned} & \biggl\vert {\Upsilon } \biggl(\frac{2{b_{1}{b_{2}}}}{{b_{1}}+{{b_{2}}}} \biggr)- \frac{{b_{1}{b_{2}}}}{{{b_{2}}}-{b_{1}}} \int _{b_{1}}^{{b_{2}}} \frac{{\Upsilon }(x)}{x^{2}} \,\mathrm{d}x \biggr\vert \\ &\quad \leq \frac{{b_{1}{b_{2}}}({{b_{2}}}-{b_{1}})}{2} \biggl(\frac{1}{8} \biggr)^{1- \frac{1}{q}} \\ &\qquad {}\times \bigl[ \bigl(M_{1} \bigl\vert {\Upsilon }'({b_{1}}) \bigr\vert ^{q}+M_{2} \bigl\vert {\Upsilon }'({{b_{2}}}) \bigr\vert ^{q} \bigr)^{\frac{1}{q}}+ \bigl(M_{3} \bigl\vert {\Upsilon }'({b_{1}}) \bigr\vert ^{q}+M_{4} \bigl\vert { \Upsilon }'({{b_{2}}}) \bigr\vert ^{q} \bigr)^{\frac{1}{q}} \\ &\qquad {}+ \bigl(M_{2} \bigl\vert {\Upsilon }'({b_{1}}) \bigr\vert ^{q} +M_{1} \bigl\vert {\Upsilon }'({{b_{2}}}) \bigr\vert ^{q} \bigr)^{\frac{1}{q}}+ \bigl(M_{4} \bigl\vert {\Upsilon }'({b_{1}}) \bigr\vert ^{q}+M_{3} \bigl\vert { \Upsilon }'({{b_{2}}}) \bigr\vert ^{q} \bigr)^{\frac{1}{q}} \bigr], \end{aligned}$$

where

$$\begin{aligned}& M_{1}:=\frac{{{b_{2}}}^{-2}}{8}{_{2}F_{1}} \biggl(2,2,3, \frac{{{b_{2}}}-{b_{1}}}{2{{b_{2}}}} \biggr)- \frac{{{b_{2}}}^{-2}}{24}{_{2}F_{1}} \biggl(2,3,4, \frac{{{b_{2}}}-{b_{1}}}{2{{b_{2}}}} \biggr), \\& M_{2}:=\frac{{{b_{2}}}^{-2}}{24}{_{2}F_{1}} \biggl(2,3,4, \frac{{{b_{2}}}-{b_{1}}}{2{{b_{2}}}} \biggr),\qquad M_{3}:= \frac{{b_{1}}^{-2}}{24}{_{2}F_{1}} \biggl(2,3,4, \frac{{b_{1}}-{{b_{2}}}}{2{b_{1}}} \biggr), \\& M_{4}:=\frac{{b_{1}}^{-2}}{8}{_{2}F_{1}} \biggl(2,2,3, \frac{{b_{1}}-{{b_{2}}}}{2{b_{1}}} \biggr)-\frac{{b_{1}}^{-2}}{24}{_{2}F_{1}} \biggl(2,3,4, \frac{{b_{1}}-{{b_{2}}}}{2{b_{1}}} \biggr). \end{aligned}$$

Corollary 2.29

Choosing \(\Phi ({\tau })=\frac{{\tau }^{\alpha }}{\Gamma (\alpha )}\) in Theorem 2.10, then

$$\begin{aligned} & \biggl\vert {\Upsilon } \biggl(\frac{2{b_{1}{b_{2}}}}{{b_{1}}+{{b_{2}}}} \biggr)- \frac{\Gamma (\alpha +1)}{2} \biggl( \frac{{b_{1}{b_{2}}}}{{{b_{2}}}-{b_{1}}} \biggr)^{\alpha } \biggl[J_{ \frac{1}{{{b_{2}}}}^{+}}^{\alpha }{{\Upsilon }\circ {\Psi }} \biggl( \frac{1}{{b_{1}}} \biggr) +J_{\frac{1}{{b_{1}}}^{-}}^{\alpha }{{ \Upsilon } \circ {\Psi }} \biggl(\frac{1}{{{b_{2}}}} \biggr) \biggr] \biggr\vert \\ &\quad \leq \frac{{b_{1}{b_{2}}}({{b_{2}}}-{b_{1}})}{2} \\ &\qquad {}\times \biggl[ \biggl(\frac{1}{2^{\alpha +1}(\alpha +1)} \biggr)^{1- \frac{1}{q}} \bigl\{ \bigl(M_{5} \bigl\vert {\Upsilon }'({b_{1}}) \bigr\vert ^{q}+M_{6} \bigl\vert {\Upsilon }'({{b_{2}}}) \bigr\vert ^{q} \bigr)^{\frac{1}{q}}\\ &\qquad{}+ \bigl(M_{7} \bigl\vert {\Upsilon }'({b_{1}}) \bigr\vert ^{q}+M_{8} \bigl\vert { \Upsilon }'({{b_{2}}}) \bigr\vert ^{q} \bigr)^{\frac{1}{q}} \bigr\} \\ &\qquad {}+ \biggl( \frac{2^{\alpha }(\alpha -1)+1}{2^{\alpha +1}(\alpha +1)} \biggr)^{1- \frac{1}{q}} \bigl\{ \bigl(M_{6} \bigl\vert {\Upsilon }'({b_{1}}) \bigr\vert ^{q} +M_{5} \bigl\vert {\Upsilon }'({{b_{2}}}) \bigr\vert ^{q} \bigr)^{\frac{1}{q}}\\ &\qquad{}+ \bigl(M_{8} \bigl\vert {\Upsilon }'({b_{1}}) \bigr\vert ^{q}+M_{7} \bigl\vert { \Upsilon }'({{b_{2}}}) \bigr\vert ^{q} \bigr)^{\frac{1}{q}} \bigr\} \biggr], \end{aligned}$$

where

$$\begin{aligned}& \begin{aligned} M_{5}:={}&\frac{{{b_{2}}}^{-2}}{2^{\alpha +1}(\alpha +1)}{_{2}F_{1}} \biggl(2,\alpha +1,\alpha +2, \frac{{{b_{2}}}-{b_{1}}}{2{{b_{2}}}} \biggr)\\ &{}-\frac{{{b_{2}}}^{-2}}{2^{\alpha +2}(\alpha +2)}{_{2}F_{1}} \biggl(2,\alpha +2,\alpha +3, \frac{{{b_{2}}}-{b_{1}}}{2{{b_{2}}}} \biggr), \end{aligned} \\& M_{6}:=\frac{{{b_{2}}}^{-2}}{2^{\alpha +2}(\alpha +2)}{_{2}F_{1}} \biggl(2,\alpha +2,\alpha +3, \frac{{{b_{2}}}-{b_{1}}}{2{{b_{2}}}} \biggr), \\& M_{7}:=\frac{{b_{1}}^{-2}}{2^{\alpha +2}(\alpha +2)}{_{2}F_{1}} \biggl(2, \alpha +2,\alpha +3, \frac{{b_{1}}-{{b_{2}}}}{2{b_{1}}} \biggr), \\& \begin{aligned} M_{8}:={}&\frac{{b_{1}}^{-2}}{2^{\alpha +1}(\alpha +1)}{_{2}F_{1}} \biggl(2, \alpha +1,\alpha +2, \frac{{b_{1}}-{{b_{2}}}}{2{b_{1}}} \biggr)\\ &{}- \frac{{b_{1}}^{-2}}{2^{\alpha +2}(\alpha +2)}{_{2}F_{1}} \biggl(2, \alpha +2, \alpha +3, \frac{{b_{1}}-{{b_{2}}}}{2{b_{1}}} \biggr). \end{aligned} \end{aligned}$$

Corollary 2.30

Taking \(\Phi ({\tau })= \frac{{\tau }^{\frac{\alpha }{k}}}{k\Gamma _{k}(\alpha )}\) in Theorem 2.10, then

$$\begin{aligned} & \biggl\vert {\Upsilon } \biggl(\frac{2{b_{1}{b_{2}}}}{{b_{1}}+{{b_{2}}}} \biggr)- \frac{\Gamma _{k}(\alpha +k)}{2} \biggl( \frac{{b_{1}{b_{2}}}}{{{b_{2}}}-{b_{1}}} \biggr)^{\frac{\alpha }{k}} \biggl[_{k}J_{\frac{1}{{{b_{2}}}}^{+}}^{\alpha }{{\Upsilon }\circ {\Psi }} \biggl(\frac{1}{{b_{1}}} \biggr) + _{k}J_{\frac{1}{{b_{1}}}^{-}}^{ \alpha }{{ \Upsilon }\circ {\Psi }} \biggl(\frac{1}{{{b_{2}}}} \biggr) \biggr] \biggr\vert \\ &\qquad {}\times \biggl[ \biggl(\frac{k}{2^{\frac{\alpha }{k}+1}(\alpha +k)} \biggr)^{1-\frac{1}{q}} \bigl\{ \bigl(M_{9} \bigl\vert {\Upsilon }'({b_{1}}) \bigr\vert ^{q}+M_{10} \bigl\vert {\Upsilon }'({{b_{2}}}) \bigr\vert ^{q} \bigr)^{ \frac{1}{q}}\\ &\qquad{}+ \bigl(M_{11} \bigl\vert {\Upsilon }'({b_{1}}) \bigr\vert ^{q}+M_{12} \bigl\vert {\Upsilon }'({{b_{2}}}) \bigr\vert ^{q} \bigr)^{\frac{1}{q}} \bigr\} \\ &\qquad {}+ \biggl( \frac{2^{\frac{\alpha }{k}}(\alpha -k)+k}{2^{\frac{\alpha }{k}+1}(\alpha +k)} \biggr)^{1-\frac{1}{q}} \bigl\{ \bigl(M_{10} \bigl\vert {\Upsilon }'({b_{1}}) \bigr\vert ^{q} +M_{9} \bigl\vert {\Upsilon }'({{b_{2}}}) \bigr\vert ^{q} \bigr)^{ \frac{1}{q}}\\ &\qquad{}+ \bigl(M_{12} \bigl\vert {\Upsilon }'({b_{1}}) \bigr\vert ^{q}+M_{11} \bigl\vert {\Upsilon }'({{b_{2}}}) \bigr\vert ^{q} \bigr)^{\frac{1}{q}} \bigr\} \biggr], \end{aligned}$$

where

$$\begin{aligned}& \begin{aligned} M_{9}:={}&\frac{k{{b_{2}}}^{-2}}{2^{\frac{\alpha }{k}+1}(\alpha +k)}{_{2}F_{1}} \biggl(2,\alpha +k,\alpha +2k, \frac{1}{k} \biggl( \frac{{{b_{2}}}-{b_{1}}}{2{{b_{2}}}} \biggr) \biggr) \\ &{}-\frac{k{{b_{2}}}^{-2}}{2^{\frac{\alpha }{k}+2}(\alpha +2k)}{_{2}F_{1}} \biggl(2,\alpha +2k, \alpha +3k, \frac{1}{k} \biggl( \frac{{{b_{2}}}-{b_{1}}}{2{{b_{2}}}} \biggr) \biggr), \end{aligned}\\& M_{10}:=\frac{k{{b_{2}}}^{-2}}{2^{\frac{\alpha }{k}+2}(\alpha +2k)}{_{2}F_{1}} \biggl(2,\alpha +2k,\alpha +3k, \frac{1}{k} \biggl( \frac{{{b_{2}}}-{b_{1}}}{2{{b_{2}}}} \biggr) \biggr), \\& M_{11}:=\frac{k{b_{1}}^{-2}}{2^{\frac{\alpha }{k}+2}(\alpha +2k)}{_{2}F_{1}} \biggl(2,\alpha +2k,\alpha +3k, \frac{1}{k} \biggl( \frac{{b_{1}}-{{b_{2}}}}{2{b_{1}}} \biggr) \biggr),\\& \begin{aligned} M_{12}:={}&\frac{k{b_{1}}^{-2}}{2^{\frac{\alpha }{k}+1}(\alpha +k)}{_{2}F_{1}} \biggl(2,\alpha +k,\alpha +2k, \frac{1}{k} \biggl( \frac{{b_{1}}-{{b_{2}}}}{2{b_{1}}} \biggr) \biggr) \\ &{}-\frac{k{b_{1}}^{-2}}{2^{\frac{\alpha }{k}+2}(\alpha +2k)}{_{2}F_{1}} \biggl(2,\alpha +2k, \alpha +3k, \frac{1}{k} \biggl( \frac{{b_{1}}-{{b_{2}}}}{2{b_{1}}} \biggr) \biggr). \end{aligned} \end{aligned}$$

Remark

For other suitable choices of function Φ, several new interesting inequalities can be found from our results. We omit here their proofs and the details are left to the interested reader.

Applications

Application to special means

We shall consider the following special means for different positive real numbers \({b_{1}}\) and \({{b_{2}}}\), where \({b_{1}}<{{b_{2}}}\):

  • Arithmetic mean: \(\mathcal{A} ({b_{1}},{{b_{2}}} ) = \frac{{b_{1}}+{{b_{2}}}}{2}\);

  • Harmonic mean: \(\mathcal{H} ({b_{1}},{{b_{2}}} ) = \frac{2{b_{1}}{{b_{2}}}}{{b_{1}}+{{b_{2}}}}\);

  • r–Logarithmic mean: \(\mathcal{L}_{r} ( {b_{1}},{{b_{2}}} ) = ( \frac{{{b_{2}}}^{r+1}-{b_{1}}^{r+1}}{ ( r+1 ) ( {{b_{2}}}-{b_{1}} ) } ) ^{\frac{1}{r}}\), \(r\in \mathbb{R}\backslash \{0, -1\} \).

Using our results, we are in a position to prove the following inequalities regarding the above special means.

Proposition 3.1

Let \(r, {b_{1}}, {{b_{2}}}\in \mathbb{R} \), \(0<{b_{1}}<{{b_{2}}}\) with \(r\geq 1\), \(q>1\) and \(\frac{1}{p}+\frac{1}{q}=1\), then

$$\begin{aligned} &|A \bigl({b_{1}}^{\frac{r}{q}+2},{{b_{2}}}^{\frac{r}{q}+2} \bigr)-\frac{q}{r+q}{b_{1}} {{b_{2}}} \mathcal{L}_{\frac{r}{q}}^{ \frac{r}{q}}({b_{1}},{{b_{2}}})| \\ &\quad \leq {b_{1}{b_{2}}}({{b_{2}}}-{b_{1}}) \biggl(\frac{1}{p+1} \biggr)^{ \frac{1}{p}} \bigl[ \bigl(\sigma _{1}^{\star }{{b_{2}}}^{r+q}+\sigma _{2}^{ \star }{b_{1}}^{r+q} \bigr)^{\frac{1}{q}} + \bigl(\sigma _{3}^{\star }{{b_{2}}}^{r+q}+ \sigma _{4}^{\star }{b_{1}}^{r+q} \bigr)^{\frac{1}{q}} \bigr], \end{aligned}$$

where

$$\begin{aligned}& \begin{aligned} \sigma _{1}^{\star }& := \int _{0}^{1} \frac{((1-{\tau }){{b_{2}}}+(1+{\tau }){b_{1}})^{-2q}(1+{\tau })}{2} \,\mathrm{d} {\tau } \\ &= \frac{({{b_{2}}}+ {b_{1}})^{1-2q}-2(2{b_{1}})^{1-2q}}{2({{b_{2}}}-{b_{1}})(1-2q)}- \frac{(2{b_{1}})^{2-2q}-( {{b_{2}}}+ {b_{1}})^{2-2q}}{2({{b_{2}}}-{b_{1}})^{2}(1-2q)(2-2q)}, \end{aligned}\\& \begin{aligned} \sigma _{2}^{\star } &:= \int _{0}^{1} \frac{((1-{\tau }){{b_{2}}}+(1+{\tau }){b_{1}})^{-2q}(1-{\tau })}{2} \,\mathrm{d} {\tau } \\ &=\frac{( {{b_{2}}}+ {b_{1}})^{1-2q}}{2({{b_{2}}}-{b_{1}})(1-2q)}+ \frac{(2{b_{1}})^{2-2q}-({b_{1}}+{{b_{2}}})^{2-2q}}{2({{b_{2}}}-{b_{1}})^{2}(1-2q)(2-2q)}, \end{aligned}\\& \begin{aligned} \sigma _{3}^{\star }& := \int _{0}^{1} \frac{((1+{\tau }){{b_{2}}}+(1-{\tau }){b_{1}})^{-2q}(1-{\tau })}{2} \,\mathrm{d} {\tau } \\ &= \frac{(2{{b_{2}}})^{2-2q}-( {{b_{2}}}+ {b_{1}})^{2-2q}}{2({{b_{2}}}-{b_{1}})^{2}(1-2q)(2-2q)}- \frac{( {{b_{2}}}+ {b_{1}})^{1-2q}}{2({{b_{2}}}-{b_{1}})(1-2q)}, \end{aligned}\\& \begin{aligned} \sigma _{4}^{\star }& := \int _{0}^{1} \frac{((1+{\tau }){{b_{2}}}+(1-{\tau }){b_{1}})^{-2q}(1+{\tau })}{2} \,\mathrm{d} {\tau } \\ &= \frac{(2{{b_{2}}})^{1-2q}-( {{b_{2}}}+ {b_{1}})^{1-2q}}{2({{b_{2}}}-{b_{1}})(1-2q)}- \frac{(2{{b_{2}}})^{2-2q}-( {{b_{2}}}+ {b_{1}})^{2-2q}}{2({{b_{2}}}-{b_{1}})^{2}(1-2q)(2-2q)}. \end{aligned} \end{aligned}$$

Proof

Taking the harmonic convex function \({\Upsilon }(\tau )=\frac{q}{r+2q}\tau ^{\frac{r}{q}+2}\) for all \(\tau >0\) in Corollary 2.19 and \(\lambda =\mu =1\), we have the desired result. □

Proposition 3.2

Let \(r, {b_{1}}, {{b_{2}}}\in \), \(0<{b_{1}}<{{b_{2}}}\) with \(r\geq 1\), \(q>1\) and \(\frac{1}{p}+\frac{1}{q}=1\), then

$$\begin{aligned} & \biggl\vert \mathcal{H}^{\frac{r}{q}+2} ( {b_{1}}, {{b_{2}}} ) - \frac{q}{r+q}{b_{1}} {{b_{2}}}\mathcal{L}_{\frac{r}{q}}^{ \frac{r}{q}} ( {b_{1}}, {{b_{2}}} ) \biggr\vert \\ &\quad \leq \frac{(r+2q){b_{1}{b_{2}}}({{b_{2}}}-{b_{1}})}{q} \biggl( \frac{1}{2^{p+2}(p+1)} \biggr)^{\frac{1}{p}} \\ &\qquad {}\times \bigl(\mathcal{A}^{\frac{1}{q}} \bigl( N_{1} {b_{1}}^{r+q}, N_{2} {{b_{2}}}^{r+q} \bigr) + \mathcal{A}^{\frac{1}{q}} \bigl( N_{3} {b_{1}}^{r+q}, N_{4} {{b_{2}}}^{r+q} \bigr) \bigr). \end{aligned}$$

Proof

Choosing the harmonic convex function \({\Upsilon }(\tau )=\frac{q}{r+2q}\tau ^{\frac{r}{q}+2}\) for all \(\tau >0\) in Corollary 2.25, we obtain the desired result. □

Proposition 3.3

Let \(r, {b_{1}}, {{b_{2}}}\in \mathbb{R} \), \(0<{b_{1}}<{{b_{2}}}\) with \(r, q\geq 1\), then

$$\begin{aligned} & \biggl\vert \mathcal{H}^{\frac{r}{q}+2} ( {b_{1}}, {{b_{2}}} ) - \frac{q}{r+q}{b_{1}} {{b_{2}}}\mathcal{L}_{\frac{r}{q}}^{ \frac{r}{q}} ( {b_{1}}, {{b_{2}}} ) \biggr\vert \\ &\quad \leq \frac{(r+2q){b_{1}{b_{2}}}({{b_{2}}}-{b_{1}})}{q} \biggl(\frac{1}{16} \biggr)^{1-\frac{1}{q}} \\ &\qquad {}\times \bigl[\mathcal{A}^{\frac{1}{q}} \bigl(M_{1} {b_{1}}^{r+q}, M_{2} {{b_{2}}}^{r+q} \bigr) + \mathcal{A}^{\frac{1}{q}} \bigl(M_{3} {b_{1}}^{r+q}, M_{4} {{b_{2}}}^{r+q} \bigr) \\ &\qquad{}+ \mathcal{A}^{\frac{1}{q}} \bigl(M_{2} {b_{1}}^{r+q}, M_{1} {{b_{2}}}^{r+q} \bigr) + \mathcal{A}^{ \frac{1}{q}} \bigl( M_{4} {b_{1}}^{r+q}, M_{3} {{b_{2}}}^{r+q} \bigr) \bigr]. \end{aligned}$$

Proof

Taking the harmonic convex function \({\Upsilon }(\tau )=\frac{q}{r+2q}\tau ^{\frac{r}{q}+2}\) for all \(\tau >0\) in Corollary 2.28, we obtain the desired result. □

Application to error estimations

Finally, let us consider some applications of the integral inequalities obtained above to find new error bounds for the following quadrature formulas. Let \(\mathcal{P}: b_{1}=x_{0}< x_{1}<\cdots <x_{n-1}<x_{n}={b_{2}}\) be a partition of \([b_{1}, {b_{2}}]\subset (0,+\infty )\). We denote, respectively

$$\begin{aligned} \begin{gathered} \mathcal{T}_{1}(\mathcal{P},{\Upsilon }):=\sum _{i=0}^{n-1} \frac{ ({\Upsilon }(x_{i})+{\Upsilon }(x_{i+1}) )}{(m+1)x_{i}x_{i+1}}h_{i}, \qquad \mathcal{T}_{2}(\mathcal{P},{\Upsilon }):=\sum _{i=0}^{n-1} \frac{{\Upsilon } (\frac{2x_{i}x_{i+1}}{x_{i}+x_{i+1}} )}{x_{i}x_{i+1}}h_{i}, \\ \int _{\frac{1}{{b_{2}}}}^{\frac{1}{b_{1}}}{\Upsilon }\circ \Psi (x)dx := \mathcal{T}_{1}(\mathcal{P},{\Upsilon }) + \mathcal{R}_{1}( \mathcal{P},{\Upsilon }),\\ \int _{b_{1}}^{{b_{2}}} \frac{{\Upsilon }(x)}{x^{2}}dx := \mathcal{T}_{2}(\mathcal{P},{ \Upsilon }) + \mathcal{R}_{2}( \mathcal{P},{\Upsilon }), \end{gathered} \end{aligned}$$
(3.1)

where \(m\in \mathbb{N}\), and \(\mathcal{R}_{1}(\mathcal{P},{\Upsilon })\) and \(\mathcal{R}_{2}(\mathcal{P},{\Upsilon })\) are the remainder terms and \(h_{i}=x_{i+1}-x_{i}\) for \(i=0,1,2,\ldots , n-1\). Using the above notations, we are in a position to prove the following error estimations.

Proposition 3.4

Let \({\Upsilon }:[{b_{1}},{{b_{2}}}]\subset (0,+\infty )\rightarrow \mathbb{R}\) be a differentiable function on \(({b_{1}},{{b_{2}}})\) with \({b_{1}}<{{b_{2}}}\) and \(m\in \mathbb{N}\). If \(\vert {\Upsilon }' \vert ^{q}\) is an harmonic convex function with \(q>1\) and \(\frac{1}{p}+\frac{1}{q}=1\), then

$$\begin{aligned} & \bigl\vert \mathcal{R}_{1}(\mathcal{P},{\Upsilon }) \bigr\vert \\ &\quad \leq\sum_{i=0}^{n-1}h_{i}^{2} \biggl[\pi _{i,1}^{\frac{1}{p}} \biggl(\frac{1}{(m+1)(q+1)(q+2)} \bigl\vert {\Upsilon }'({x_{i}}) \bigr\vert ^{q}+ \frac{m(q+2)+(q+1)}{(m+1)(q+1)(q+2)} \bigl\vert {\Upsilon }'({x_{i+1}}) \bigr\vert ^{q} \biggr)^{\frac{1}{q}} \\ &\qquad {}+ \pi _{i,2}^{\frac{1}{p}} \biggl(\frac{1}{(m+1)(q+1)(q+2)} \bigl\vert { \Upsilon }'({x_{i+1}}) \bigr\vert ^{q}+\frac{m(q+2)+(q+1)}{(m+1)(q+1)(q+2)} \bigl\vert { \Upsilon }'({x_{i}}) \bigr\vert ^{q} \biggr)^{\frac{1}{q}} \biggr], \end{aligned}$$

where

$$\begin{aligned} &\pi _{i,1} :=\frac{(m{x_{i}}+{x_{i+1}})^{1-2p}}{{h_{i}}(1-2p)} \biggl[1- \biggl( \frac{(m+1){x_{i}}}{m{x_{i}}+{x_{i+1}}} \biggr)^{1-2p} \biggr], \\ &\pi _{i,2} :=\frac{({x_{i}}+m{x_{i+1}})^{1-2p}}{{h_{i}}(1-2p)} \biggl[ \biggl( \frac{(m+1){x_{i+1}}}{{x_{i}}+m{x_{i+1}}} \biggr)^{1-2p}-1 \biggr]. \end{aligned}$$

Proof

By using Corollary 2.13 on the subintervals \([x_{i}, x_{i+1}]\;(i=0,1,2,\ldots ,n-1)\) of the partition \(\mathcal{P}\) and summing the obtained inequality over i from 0 to \(n-1\), we have the desired result. □

Proposition 3.5

Let \({\Upsilon }:[{b_{1}},{{b_{2}}}]\subset (0,+\infty )\rightarrow \mathbb{R}\) be a differentiable function on \(({b_{1}},{{b_{2}}})\) with \({b_{1}}<{{b_{2}}}\) and \(m\in \mathbb{N}\). If \(\vert {\Upsilon }' \vert ^{q}\) is an harmonic convex function with \(q\geq 1\), then

$$\begin{aligned} & \bigl\vert \mathcal{R}_{1}(\mathcal{P},{\Upsilon }) \bigr\vert \\ &\quad \leq\sum_{i=0}^{n-1}h_{i}^{2} \bigl[\pi _{i,3}^{1-\frac{1}{q}} \bigl(\pi _{i,4} \bigl\vert {\Upsilon }'({x_{i}}) \bigr\vert ^{q}+\pi _{i,5} \bigl\vert {\Upsilon }'({x_{i+1}}) \bigr\vert ^{q} \bigr)^{\frac{1}{q}}\\ &\qquad{}+\pi _{i,6}^{1-\frac{1}{q}} \bigl(\pi _{i,7} \bigl\vert { \Upsilon }'({x_{i}}) \bigr\vert ^{q}+\pi _{i,8} \bigl\vert {\Upsilon }'({x_{i+1}}) \bigr\vert ^{q} \bigr)^{\frac{1}{q}} \bigr], \end{aligned}$$

where

$$\begin{aligned} &\pi _{i,3} :=\frac{(m{x_{i}}+{x_{i+1}})^{-2}}{2}{_{2}F_{1} \biggl(2,2,3, \frac{h_{i}}{m{x_{i}}+{x_{i+1}}} \biggr)}, \\ &\pi _{i,4} :=\frac{(m{x_{i}}+{x_{i+1}})^{-2}}{6(m+1)}{_{2}F_{1} \biggl(2,2,4,\frac{h_{i}}{m{x_{i}}+{x_{i+1}}} \biggr)}, \\ & \begin{aligned} \pi _{i,5} :={}&\frac{m(m{x_{i}}+{x_{i+1}})^{-2}}{2(m+1)}{_{2}F_{1} \biggl(2,2,3,\frac{h_{i}}{m{x_{i}}+{x_{i+1}}} \biggr)}\\ &{}+ \frac{(m{x_{i}}+{x_{i+1}})^{-2}}{6(m+1)}{_{2}F_{1} \biggl(2,3,4, \frac{h_{i}}{m{x_{i}}+{x_{i+1}}} \biggr)}, \end{aligned} \\ &\pi _{i,6} :=\frac{({x_{i}}+m{x_{i+1}})^{-2}}{2}{_{2}F_{1} \biggl(2,2,3,- \frac{h_{i}}{{x_{i}}+m{x_{i+1}}} \biggr)}, \\ & \begin{aligned} \pi _{i,7} :={}&\frac{m({x_{i}}+m{x_{i+1}})^{-2}}{2(m+1)}{_{2}F_{1} \biggl(2,2,3,-\frac{h_{i}}{{x_{i}}+m{x_{i+1}}} \biggr)}\\ &{}+ \frac{({x_{i}}+m{x_{i+1}})^{-2}}{6(m+1)}{_{2}F_{1} \biggl(2,3,4,- \frac{h_{i}}{{x_{i}}+m{x_{i+1}}} \biggr)}, \end{aligned} \\ &\pi _{i,8} :=\frac{({x_{i}}+m{x_{i+1}})^{-2}}{6(m+1)}{_{2}F_{1} \biggl(2,2,4,-\frac{h_{i}}{{x_{i}}+m{x_{i+1}}} \biggr)}. \end{aligned}$$

Proof

By applying Corollary 2.16 on the subintervals \([x_{i}, x_{i+1}]\;(i=0,1,2,\ldots ,n-1)\) of the partition \(\mathcal{P}\) and summing the obtained inequality over i from 0 to \(n-1\), we obtain the desired result. □

Proposition 3.6

Let \({\Upsilon }:[{b_{1}},{{b_{2}}}]\subset (0,+\infty )\rightarrow \mathbb{R}\) be a differentiable function on \(({b_{1}},{{b_{2}}})\) with \({b_{1}}<{{b_{2}}}\). If \(\vert {\Upsilon }' \vert ^{q}\) is an harmonic convex function with \(q>1\) and \(\frac{1}{p}+\frac{1}{q}=1\), then

$$\begin{aligned} & \bigl\vert \mathcal{R}_{2}(\mathcal{P},{\Upsilon }) \bigr\vert \\ &\quad \leq \frac{1}{2} \biggl(\frac{1}{2^{p+1}(p+1)} \biggr)^{\frac{1}{p}} \\ &\qquad {}\times \sum_{i=0}^{n-1}h_{i}^{2} \bigl( \bigl(N_{i,1} \bigl\vert { \Upsilon }'(x_{i}) \bigr\vert ^{q}+N_{i,2} \bigl\vert {\Upsilon }'(x_{i+1}) \bigr\vert ^{q} \bigr)^{\frac{1}{q}}+ \bigl(N_{i,3} \bigl\vert {\Upsilon }'(x_{i}) \bigr\vert ^{q}+N_{i,4} \bigl\vert {\Upsilon }'(x_{i+1}) \bigr\vert ^{q} \bigr)^{ \frac{1}{q}} \bigr), \end{aligned}$$

where

$$\begin{aligned}& N_{i,1}:= \frac{2^{2-2q}{x_{i+1}}^{2-2q}-({x_{i}}+{x_{i+1}})^{2-2q}}{h_{i}^{2}(2-2q)2^{2-2q}}-{x_{i}} \frac{2^{1-2q}{x_{i+1}}^{1-2q}-({x_{i}}+{x_{i+1}})^{1-2q}}{h_{i}^{2}(1-2q)2^{1-2q}}, \\& N_{i,2}:={x_{i+1}} \frac{2^{1-2q}{x_{i+1}}^{1-2q}-({x_{i}}+{x_{i+1}})^{1-2q}}{h_{i}^{2}(1-2q)2^{1-2q}}- \frac{2^{2-2q}{x_{i+1}}^{2-2q}-({x_{i}}+{x_{i+1}})^{2-2q}}{h_{i}^{2}(2-2q)2^{2-2q}}, \\& N_{i,3}:= \frac{({x_{i}}+{x_{i+1}})^{2-2q}-2^{2-2q}{x_{i}}^{2-2q}}{h_{i}^{2}(2-2q)2^{2-2q}}-{x_{i}} \frac{({x_{i}}+{x_{i+1}})^{1-2q}-2^{1-2q}{x_{i}}^{1-2q}}{h_{i}^{2}(1-2q)2^{1-2q}}, \\& N_{i,4}:={x_{i+1}} \frac{({x_{i}}+{x_{i+1}})^{1-2q}-2^{1-2q}{x_{i}}^{1-2q}}{h_{i}^{2}(1-2q)2^{1-2q}}- \frac{({x_{i}}+{x_{i+1}})^{2-2q}-2^{2-2q}{x_{i}}^{2-2q}}{h_{i}^{2}(2-2q)2^{2-2q}}. \end{aligned}$$

Proof

By using Corollary 2.25 on the subintervals \([x_{i}, x_{i+1}]\;(i=0,1,2,\ldots ,n-1)\) of the partition \(\mathcal{P}\) and summing the obtained inequality over i from 0 to \(n-1\), we have the desired result. □

Proposition 3.7

Let \({\Upsilon }:[{b_{1}},{{b_{2}}}]\subset (0,+\infty )\rightarrow \mathbb{R}\) be a differentiable function on \(({b_{1}},{{b_{2}}})\) with \({b_{1}}<{{b_{2}}}\). If \(\vert {\Upsilon }' \vert ^{q}\) is an harmonic convex function with \(q\geq 1\), then

$$\begin{aligned} & \bigl\vert \mathcal{R}_{2}(\mathcal{P},{\Upsilon }) \bigr\vert \\ &\quad \leq \frac{1}{2} \biggl(\frac{1}{8} \biggr)^{1-\frac{1}{q}} \\ &\qquad {}\times \sum_{i=0}^{n-1}h_{i}^{2} \bigl[ \bigl(M_{i,1} \bigl\vert { \Upsilon }'({x_{i}}) \bigr\vert ^{q}+M_{i,2} \bigl\vert {\Upsilon }'({x_{i+1}}) \bigr\vert ^{q} \bigr)^{\frac{1}{q}}+ \bigl(M_{i,3} \bigl\vert {\Upsilon }'({x_{i}}) \bigr\vert ^{q}+M_{i,4} \bigl\vert {\Upsilon }'({x_{i+1}}) \bigr\vert ^{q} \bigr)^{ \frac{1}{q}} \\ &\qquad {}+ \bigl(M_{i,2} \bigl\vert {\Upsilon }'({x_{i}}) \bigr\vert ^{q} +M_{i,1} \bigl\vert {\Upsilon }'({x_{i+1}}) \bigr\vert ^{q} \bigr)^{\frac{1}{q}}+ \bigl(M_{i,4} \bigl\vert {\Upsilon }'({x_{i}}) \bigr\vert ^{q}+M_{i,3} \bigl\vert { \Upsilon }'({x_{i+1}}) \bigr\vert ^{q} \bigr)^{\frac{1}{q}} \bigr], \end{aligned}$$

where

$$\begin{aligned}& M_{i,1}:=\frac{{x_{i+1}}^{-2}}{8}{_{2}F_{1}} \biggl(2,2,3, \frac{h_{i}}{2{x_{i+1}}} \biggr)-\frac{{x_{i+1}}^{-2}}{24}{_{2}F_{1}} \biggl(2,3,4, \frac{h_{i}}{2{x_{i+1}}} \biggr), \\& M_{i,2}:=\frac{{x_{i+1}}^{-2}}{24}{_{2}F_{1}} \biggl(2,3,4, \frac{h_{i}}{2{x_{i+1}}} \biggr),\qquad M_{i,3}:= \frac{{x_{i}}^{-2}}{24}{_{2}F_{1}} \biggl(2,3,4, - \frac{h_{i}}{2{x_{i}}} \biggr), \\& M_{i,4}:=\frac{{x_{i}}^{-2}}{8}{_{2}F_{1}} \biggl(2,2,3, - \frac{h_{i}}{2{x_{i}}} \biggr)-\frac{{x_{i}}^{-2}}{24}{_{2}F_{1}} \biggl(2,3,4, -\frac{h_{i}}{2{x_{i}}} \biggr). \end{aligned}$$

Proof

By applying Corollary 2.28 on the subintervals \([x_{i}, x_{i+1}]\) (\(i=0,1,2,\ldots ,n-1\)) of the partition \(\mathcal{P}\) and summing the obtained inequality over i from 0 to \(n-1\), we obtain the desired result. □

Remark

For suitable choices of function ϒ, we can obtain new inequalities using special means. Moreover, we can establish new bounds regarding error estimations of the quadrature formulas given above. We omit here their proofs and the details are left to the interested reader.

Conclusion

In this paper, we have derived some generalizations of fractional trapezium-like inequalities using the class of harmonic convex functions. Moreover, three new fractional integral identities are given and on applying them as auxiliary results some interesting integral inequalities are found. Our results unified many known ones, and they related some other unrelated results as well. Finally, some applications to special means for different positive real numbers and error estimations for quadrature formulas are obtained. This shows the efficiency of our results. To the best of our knowledge, these results are new in the literature and we believe that they will have a very deep impact in this field of inequalities, and also in pure and applied sciences.

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Acknowledgements

The authors acknowledge the financial support provided by the Center of Excellence in Theoretical and Computational Science (TaCS-CoE), KMUTT. Also, the third author, Dr. Iftikhar Sabah would like to thank the Postdoctoral Fellowship from King Mongkut’s University of Technology Thonburi (KMUTT), Thailand.

Funding

This research project is supported by the Thailand Science Research and Innovation (TSRI) Basic Research Fund: Fiscal year 2021 under project number 64A306000005.

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Awan, M.U., Kashuri, A., Nisar, K.S. et al. New fractional identities, associated novel fractional inequalities with applications to means and error estimations for quadrature formulas. J Inequal Appl 2022, 3 (2022). https://doi.org/10.1186/s13660-021-02732-6

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MSC

  • 26A51
  • 26A33
  • 26D07
  • 26D10
  • 26D15

Keywords

  • Trapezium inequality
  • Harmonic convex function
  • Riemann–Liouville fractional integrals
  • Hölder inequality
  • Power mean inequality
  • Special means
  • Error estimations