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Hermite–Hadamard type inequalities for F-convex function involving fractional integrals

Journal of Inequalities and Applications20182018:359

https://doi.org/10.1186/s13660-018-1950-1

  • Received: 31 October 2018
  • Accepted: 18 December 2018
  • Published:

Abstract

In this study, the family F and F-convex function are given with its properties. In view of this, we establish some new inequalities of Hermite–Hadamard type for differentiable function. Moreover, we establish some trapezoid type inequalities for functions whose second derivatives in absolute values are F-convex. We also show that through the notion of F-convex we can find some new Hermite–Hadamard type and trapezoid type inequalities for the Riemann–Liouville fractional integrals and classical integrals.

Keywords

  • F-convex
  • \(\lambda _{\varphi }\)-preinvex
  • Integral inequalities

MSC

  • 26A51
  • 26D15
  • 35A23

1 Introduction

A function \(f: I\subseteq \mathbb{R}\to \mathbb{R}\) is said to be convex on the interval I, if for all \(x,y\in I\) and \(t\in (0,1)\) it satisfies the following inequality:
$$\begin{aligned} f\bigl(tx+(1-t)y\bigr)\leq tf(x)+(1-t)f(y). \end{aligned}$$
(1)
Convex functions play an important role in the field of integral inequalities. For convex functions, many equalities and inequalities have been established, but one of the most important ones is the Hermite–Hadamard’ integral inequality, which is defined as follows [1]:
Let \(f: I\subseteq \mathbb{R}\to \mathbb{R}\) be a convex function with \(a< b\) and \(a, b\in I\). Then the Hermite–Hadamard inequality is given by
$$\begin{aligned} f \biggl(\frac{a+b}{2} \biggr) &\leq \frac{1}{b-a} \int _{a}^{b} f(x)\,dx \leq \frac{f(a)+f(b)}{2}. \end{aligned}$$
(2)
In recent years, a number of mathematicians have devoted their efforts to generalizing, refining, counterparting, and extending the Hermite–Hadamard inequality (2) for different classes of convex functions and mappings. The Hermite–Hadamard inequality (2) is established for the classical integral, fractional integrals, conformable fractional integrals and most recently for generalized fractional integrals; see for details and applications [28] and the references therein.

The concepts of classical convex functions have been extended and generalized in several directions, such as quasi-convex [9], pseudo-convex [10], MT-convex [11] strongly convex [12], ϵ-convex [13], s-convex [14], h-convex [15], and \(\lambda _{\varphi }\)-preinvex [16]. Recently, Samet [17] has defined a new concept of convexity that depends on a certain function satisfying some axioms, generalizing different types of convexity, including ϵ-convex functions, α-convex functions, h-convex functions, and so on, as stated in the next section.

2 Review of the family of \(\mathcal{F}\)

We address the family of \(\mathcal{F}\) of mappings \(F:\mathbb{R} \times \mathbb{R}\times \mathbb{R}\times [0,1]\to \mathbb{R}\) satisfying the following axioms:
  1. (A1)
    If \(u_{i}\in L^{1}(0,1)\), \(i=1,2,3\), then, for every \(\lambda \in [0,1]\), we have
    $$\begin{aligned} \int _{0}^{1} F\bigl(u_{1}(t),u_{2}(t),u_{3}(t), \lambda \bigr)\,dt=F \biggl( \int _{0}^{1} u_{1}(t)\,dt, \int _{0}^{1} u_{2}(t)\,dt, \int _{0}^{1} u_{3}(t)\,dt, \lambda \biggr). \end{aligned}$$
     
  2. (A2)
    For every \(u\in L^{1}(0,1)\), \(w\in L^{\infty }(0,1)\) and \((z_{1},z_{2})\in \mathbb{R}^{2}\), we have
    $$\begin{aligned} \int _{0}^{1} F\bigl(w(t) u(t),w(t)z_{1},w(t)z_{2},t \bigr)\,dt=T_{F,w} \biggl( \int _{0}^{1} w(t) u(t)\,dt,z_{1},z_{2} \biggr), \end{aligned}$$
    where \(T_{F,w}:\mathbb{R}\times \mathbb{R}\times \mathbb{R}\to \mathbb{R}\) is a function that depends on \((F,w)\), and it is nondecreasing with respect to the first variable.
     
  3. (A3)
    For any \((w,u_{1},u_{2},u_{3})\in \mathbb{R}^{4}\), \(u_{4} \in [0,1]\), we have
    $$\begin{aligned} wF(u_{1},u_{2},u_{3},u_{4})=F(wu_{1},wu_{2},wu_{3},u_{4})+L_{w}, \end{aligned}$$
    where \(L_{w}\in \mathbb{R}\) is a constant that depends only on w.
     

Definition 2.1

Let \(f: [a,b]\to \mathbb{R}\), \((a,b)\in \mathbb{R}^{2}\), \(a< b\), be a given function. We say that f is a convex function with respect to some \(F\in \mathcal{F}\) (or F-convex function) iff
$$\begin{aligned} F\bigl(f\bigl(tx+(1-t)y\bigr),f(x),f(y),t\bigr)\leq 0, \quad (x,y,t)\in [a,b] \times [a,b] \times [0,1]. \end{aligned}$$

Remark 1

Suppose that \((a,b)\in \mathbb{R}^{2}\) with \(a< b\).
  1. (i)
    Let \(f:[a,b]\to \mathbb{R}\) be an ε-convex function, that is [18],
    $$\begin{aligned} f\bigl(tx+(1-t)y\bigr)\leq tf(x)+(1-t)f(y),\quad (x,y,t)\in [a,b]\times [a,b] \times [0,1]. \end{aligned}$$
    Define the functions \(F:\mathbb{R}\times \mathbb{R}\times \mathbb{R} \times [0,1]\to \mathbb{R}\) by
    $$\begin{aligned} F(u_{1},u_{2},u_{3},u_{4})=u_{1}-u_{4}u_{2}-(1-u_{4})u_{3}- \varepsilon \end{aligned}$$
    (3)
    and \(T_{F,w}:\mathbb{R}\times \mathbb{R}\times \mathbb{R}\times [0,1] \to \mathbb{R}\) by
    $$\begin{aligned} T_{F,w}(u_{1},u_{2},u_{3})=u_{1}- \biggl( \int _{0}^{1} t w(t)\,dt \biggr)u _{2} - \biggl( \int _{0}^{1} (1-t)w(t)\,dt \biggr)u_{3}- \varepsilon . \end{aligned}$$
    (4)
    For
    $$\begin{aligned} L_{w}=(1-w)\varepsilon , \end{aligned}$$
    (5)
    it will be seen that \(F\in \mathcal{F}\) and
    $$\begin{aligned} F\bigl(f\bigl(tx+(1-t)y\bigr),f(x),f(y),t\bigr) &=f\bigl(tx+(1-t)y \bigr)-tf(x)-(1-t)f(y)-\varepsilon \leq 0, \end{aligned}$$
    that is, f is an F-convex function. Particularly, taking \(\varepsilon =0\) we show that if f is a convex function then f is an F-convex function with respect to F defined above.
     
  2. (ii)
    Let \(f:[a,b]\to \mathbb{R}\) be \(\lambda _{\varphi }\)-preinvex function according to φ and bifunction η, \(0\leq \varphi \leq \frac{\pi }{2}\), \(\lambda \in (0,\frac{1}{2} ]\), that is [16],
    $$\begin{aligned} &f \bigl(u+te^{i\varphi }\eta (v,u) \bigr) \\ &\quad \leq \frac{\sqrt{t}}{2 \sqrt{1-t}}f(v) + \frac{(1-\lambda )\sqrt{1-t}}{2\lambda \sqrt{t}}f(u), \quad (u,v,t) \in [a,b]\times [a,b]\times (0,1). \end{aligned}$$
    Define the functions \(F:\mathbb{R}\times \mathbb{R}\times \mathbb{R} \times [0,1]\to \mathbb{R}\) by
    $$\begin{aligned} F(u_{1},u_{2},u_{3},u_{4})=u_{1}- \frac{\sqrt{u_{4}}}{2\sqrt{1-u _{4}}}u_{3} -\frac{(1-\lambda )\sqrt{1-u_{4}}}{2\lambda \sqrt{u _{4}}}u_{2} \end{aligned}$$
    (6)
    and \(T_{F,w}:\mathbb{R}\times \mathbb{R}\times \mathbb{R}\times [0,1] \to \mathbb{R}\) by
    $$\begin{aligned} \begin{aligned}[b] T_{F,w}(u_{1},u_{2},u_{3})&=u_{1}- \biggl( \int _{0}^{1} \frac{\sqrt{t}}{2 \sqrt{1-t}}w(t)\,dt \biggr)u_{3} \\ &\quad {}-\frac{1-\lambda }{\lambda } \biggl( \int _{0}^{1} \frac{\sqrt{1-t}}{2\sqrt{t}}w(t)\,dt \biggr)u_{2}. \end{aligned} \end{aligned}$$
    (7)
    For \(L_{w}=0\), it will be seen that \(F\in \mathcal{F}\) and
    $$\begin{aligned} &F \bigl(f \bigl(u+te^{i\varphi }\eta (v,u) \bigr),f(u),f(v),t \bigr) \\ &\quad =f \bigl(u+te^{i\varphi }\eta (v,u) \bigr)-\frac{\sqrt{t}}{2 \sqrt{1-t}}f(v) - \frac{(1-\lambda )\sqrt{1-t}}{2\lambda \sqrt{t}}f(u)- \varepsilon \leq 0, \end{aligned}$$
    that is f is an F-convex function.
     
  3. (iii)
    Let \(h:I\to \mathbb{R}\) be a given function which is not identical to 0, where I is an interval in \(\mathbb{R}\) such that \((0,1)\subseteq I\). Let \(f:[a,b]\to [0,\infty )\) be an h-convex function, that is,
    $$\begin{aligned} f\bigl(tx+(1-t)y\bigr)\leq h(t)f(x)+\frac{1-\lambda }{\lambda }h(1-t)f(y), \quad (x,y,t) \in [a,b]\times [a,b]\times [0,1]. \end{aligned}$$
    Define the functions \(F:\mathbb{R}\times \mathbb{R}\times \mathbb{R} \times [0,1]\to \mathbb{R}\) by
    $$\begin{aligned} F(u_{1},u_{2},u_{3},u_{4})=u_{1}-h(u_{4})u_{3}- \frac{1-\lambda }{ \lambda }h(1-u_{4})u_{2} \end{aligned}$$
    (8)
    and \(T_{F,w}:\mathbb{R}\times \mathbb{R}\times \mathbb{R}\times [0,1] \to \mathbb{R}\) by
    $$\begin{aligned} T_{F,w}(u_{1},u_{2},u_{3})=u_{1}- \biggl( \int _{0}^{1} h(t)w(t)\,dt \biggr)u _{3} - \frac{1-\lambda }{\lambda } \biggl( \int _{0}^{1} h(1-t)w(t)\,dt \biggr)u _{2}. \end{aligned}$$
    (9)
    For \(L_{w}=(1-w)\varepsilon \), it will be seen that \(F\in \mathcal{F}\) and
    $$\begin{aligned} &F\bigl(f\bigl(tx+(1-t)y\bigr),f(x),f(y),t\bigr)\\ &\quad =f\bigl(tx+(1-t)y \bigr)-h(t)f(x)-\frac{1-\lambda }{ \lambda }h(1-t)f(y) -\varepsilon \leq 0, \end{aligned}$$
    that is f is an F-convex function.
     

Recently Samet [17] established some integral inequalities of Hermite–Hadamard type via F-convex functions.

Theorem 1

([17, Theorem 3.1])

Let \(f: [a,b]\to \mathbb{R}\), \((a,b)\in \mathbb{R}^{2}\), \(a< b\), be an F-convex function, for some \(F\in \mathcal{F}\). Suppose that \(F\in L^{1}(a,b)\). Then
$$\begin{aligned} &F \biggl(f \biggl(\frac{a+b}{2} \biggr),\frac{1}{b-a} \int _{a}^{b} f(x)\,dx, \frac{1}{2} \biggr)\leq 0, \\ &T_{F,1} \biggl(\frac{1}{b-a} \int _{a}^{b} f(x)\,dx,f(a),f(b) \biggr) \leq 0. \end{aligned}$$

Theorem 2

([17, Theorem 3.4])

Let \(f: I^{o}\subseteq \mathbb{R}\to \mathbb{R}\) be a differentiable mapping on \(I^{o}\), \((a,b)\in I^{o}\times I^{o}\), \(a< b\). Suppose that
  1. (i)

    \(|f'|\) is F-convex on \([a,b]\), for some \(F\in \mathcal{F;}\)

     
  2. (ii)

    the function \(t\in (0,1)\to L_{w(t)}\) belongs to \(L^{1}(a,b)\), where \(w(t)=|1-2t|\).

     
Then
$$\begin{aligned} &T_{F,w} \biggl(\frac{2}{b-a} \biggl\vert \frac{f(a)+f(b)}{2}- \frac{1}{b-a} \int _{a}^{b} f(x)\,dx \biggr\vert , \bigl\vert f'(a) \bigr\vert , \bigl\vert f'(b) \bigr\vert \biggr)+ \int _{0}^{1} L _{w(t)}\,dt\leq 0. \end{aligned}$$

Theorem 3

([17, Theorem 3.5])

Let \(f: I^{o}\subseteq \mathbb{R}\to \mathbb{R}\) be a differentiable mapping on \(I^{o}\), \((a,b)\in I^{o}\times I^{o}\), \(a< b\) and let \(p>1\). Suppose that \(|f'|^{p/(p-1)}\) is F-convex on \([a,b]\), for some \(F\in \mathcal{F}\) and \(F\in L^{p/(p-1)}(a,b)\). Then
$$\begin{aligned} &T_{F,1} \bigl(A(p,f), \bigl\vert f'(a) \bigr\vert ^{\frac{p}{p-1}}, \bigl\vert f'(b) \bigr\vert ^{\frac{p}{p-1}} \bigr) \leq 0, \end{aligned}$$
where
$$\begin{aligned} A(p,f) &= \biggl(\frac{2}{b-a} \biggr)^{\frac{p}{p-1}}(p+1)^{ \frac{1}{p-1}} \biggl\vert \frac{f(a)+f(b)}{2}-\frac{1}{b-a} \int _{a}^{b} f(x)\,dx \biggr\vert ^{\frac{p}{p-1}}. \end{aligned}$$

As consequences of the above theorems, the author obtained some integral inequalities for ε-convexity, α-convexity, and h-convexity.

Theorem 4

([17, Corollary 4.3])

Let \(f: I^{o}\subseteq \mathbb{R}\to \mathbb{R}\) be a differentiable mapping on \(I^{o}\), \((a,b)\in I^{o}\times I^{o}\), \(a< b\). Suppose that the function \(|f'|\) is ε-convex on \([a,b]\), \(\varepsilon \geq 0\). Then
$$\begin{aligned} & \biggl\vert \frac{f(a)+f(b)}{2}-\frac{1}{b-a} \int _{a}^{b} f(x)\,dx \biggr\vert \leq (b-a) \biggl[\frac{ \vert f'(a) \vert + \vert f'(b) \vert }{8}+\frac{\varepsilon }{4} \biggr]. \end{aligned}$$

Theorem 5

([17, Corollary 4.9])

Let \(f: I^{o}\subseteq \mathbb{R}\to \mathbb{R}\) be a differentiable mapping on \(I^{o}\), \((a,b)\in I^{o}\times I^{o}\), \(a< b\). Suppose that the function \(|f'|\) is α-convex on \([a,b]\), \(\alpha \in (0,1]\). Then
$$\begin{aligned} & \biggl\vert \frac{f(a)+f(b)}{2}-\frac{1}{b-a} \int _{a}^{b} f(x)\,dx \biggr\vert \\ &\quad \leq \frac{(b-a)}{2(\alpha +1)(\alpha +2)} \biggl[ \bigl(2^{-\alpha }+ \alpha \bigr) \bigl\vert f'(a) \bigr\vert \frac{\alpha (\alpha +1)+2 (1-2^{-\alpha } )}{2} \bigl\vert f'(b) \bigr\vert \biggr]. \end{aligned}$$

Theorem 6

([17, Corollary 4.14])

Let \(f: I^{o}\subseteq \mathbb{R}\to \mathbb{R}\) be a differentiable mapping on \(I^{o}\), \((a,b)\in I^{o}\times I^{o}\), \(a< b\). Suppose that the function \(|f'|\) is h-convex on \([a,b]\). Then
$$\begin{aligned} & \biggl\vert \frac{f(a)+f(b)}{2}-\frac{1}{b-a} \int _{a}^{b} f(x)\,dx \biggr\vert \leq (b-a) \biggl( \int _{0}^{1} h(t)\,dt \biggr) \biggl( \frac{ \vert f'(a) \vert + \vert f'(b) \vert }{2} \biggr). \end{aligned}$$

For more recent results on integral inequalities of Hermite–Hadamard type concerning the F-convex functions, we refer the interested reader to [19] and the references therein.

In the sequel, we recall the concepts of the left-sided and right-sided Riemann- Liouville fractional integrals of the order \(\alpha >0\).

Definition 2.2

([20])

Suppose that \(f\in L([a,b])\). The left and right Riemann–Liouville fractional integrals denoted by \(J_{a^{+}}^{\alpha }f\) and \(J_{b^{-}}^{\alpha }f\) of order \(\alpha >0\) are defined by
$$\begin{aligned} J_{a^{+}}^{\alpha }f(x) &=\frac{1}{\varGamma (\alpha )} \int _{a}^{x} (x-t)^{ \alpha -1}f(t)\,dt,\quad x>a, \end{aligned}$$
and
$$\begin{aligned} J_{b^{-}}^{\alpha }f(x) &=\frac{1}{\varGamma (\alpha )} \int _{x}^{b} (t-x)^{ \alpha -1}f(t)\,dt,\quad x< b, \end{aligned}$$
respectively, where \(\varGamma (\alpha )\) is the gamma function defined by \(\varGamma (\alpha )=\int _{0}^{\infty }e^{-t}t^{\alpha -1}\,dt\) and \(J_{b^{-}}^{0}f(x)=J_{b^{-}}^{0}f(x)=f(x)\).

In [21], authors established the following Hermite–Hadamard type inequalities for F-convex functions involving a Riemann–Liouville fractional:

Theorem 7

Let \(I\subseteq \mathbb{R}\) be an interval, \(f: I^{o}\subseteq \mathbb{R}\to \mathbb{R}\) be a differentiable mapping on \(I^{o}\), \(a,b \in I^{o}\), \(a< b\). If f is F-convex on \([a,b]\), for some \(F\in \mathcal{F}\), then we have
$$\begin{aligned} &F \biggl(f \biggl(\frac{a+b}{2} \biggr),\frac{\varGamma (\alpha +1)}{(b-a)^{ \alpha }}J_{a^{+}}^{\alpha }f(b), \frac{\varGamma (\alpha +1)}{(b-a)^{ \alpha }}J_{b^{-}}^{\alpha }f(a),\frac{1}{2} \biggr)+ \int _{0}^{1} L _{w(t)}\,dt\leq 0, \\ &T_{F,w} \biggl(\frac{\varGamma (\alpha +1)}{(b-a)^{\alpha }} \bigl[J _{a^{+}}^{\alpha }f(b)+J_{b^{-}}^{\alpha }f(a) \bigr],f(a)+f(b),f(a)+f(b) \biggr)+ \int _{0}^{1} L_{w(t)}\,dt\leq 0, \end{aligned}$$
where \(w(t)=\alpha t^{\alpha -1}\).

Theorem 8

Let \(I\subseteq \mathbb{R}\) be an interval, \(f: I^{o}\subseteq \mathbb{R}\to \mathbb{R}\) be a differentiable mapping on \(I^{o}\), \(a,b \in I^{o}\), \(a< b\). If f is F-convex on \([a,b]\), for some \(F\in \mathcal{F}\) and the function \(t\in (0,1)\to L_{w(t)}\) belongs to \(L_{1}(a,b)\), where \(w(t)= \vert (1-t)^{\alpha }-t^{\alpha } \vert \). Then we have the inequality
$$\begin{aligned} &T_{F,w} \biggl(\frac{2}{b-a} \biggl\vert \frac{f(a)+f(b)}{2}- \frac{\varGamma ( \alpha +1)}{2(b-a)^{\alpha }} \bigl[J_{a^{+}}^{\alpha }f(b)+J_{b^{-}} ^{\alpha }f(a) \bigr] \biggr\vert , \bigl\vert f'(a) \bigr\vert , \bigl\vert f'(b) \bigr\vert \biggr)\\ &\quad {}+ \int _{0}^{1} L _{w(t)}\,dt\leq 0. \end{aligned}$$

The following definitions will be useful for this study [20].

Definition 2.3

The Euler beta function is defined as follows:
$$\begin{aligned} B(a,b)= \int _{0}^{1} t^{a-1}(1-t)^{b-1}\,dt, \quad a,b>0. \end{aligned}$$
The incomplete beta function is defined by
$$\begin{aligned} B_{x}(a,b)= \int _{0}^{x} t^{a-1}(1-t)^{b-1}\,dt, \quad a,b>0. \end{aligned}$$

Note that, for \(x=1\), the incomplete beta function reduces to the Euler beta function.

Also, the following three lemmas are important to obtain our main results.

Lemma 1

([22, Lemma 4])

Let \(f: [a,b]\to \mathbb{R}\) be a once differentiable mappings on \((a,b)\) with \(a< b\), \(\eta (b,a)>0\). If \(f'\in L [a,a+e^{i \varphi }\eta (b,a) ]\), then the following equality for the fractional integral holds:
$$\begin{aligned} &\frac{f(a)+f (a+e^{i \varphi }\eta (b,a) )}{2} -\frac{ \varGamma (\alpha +1))}{2 (e^{i \varphi }\eta (b,a) )^{\alpha }} \bigl[J_{a^{+}}^{\alpha }f \bigl(a+e^{i \varphi }\eta (b,a) \bigr) +J_{ (a+e^{i \varphi }\eta (b,a) )^{-}}^{\alpha }f(a) \bigr] \\ &\quad =\frac{e^{i \varphi }\eta (b,a)}{2} \int _{0}^{1} \bigl[(1-t)^{ \alpha }-t^{\alpha } \bigr] f' \bigl(a+(1-t)e^{i \varphi }\eta (b,a) \bigr)\,dt. \end{aligned}$$

Lemma 2

([16, Lemma 5])

Let \(f: [a,b]\to \mathbb{R}\) be a once differentiable mappings on \((a,b)\) with \(a< b\), \(\eta (b,a)>0\). If \(f''\in L [a,a+e^{i \varphi }\eta (b,a) ]\), then the following equality for the fractional integral holds:
$$\begin{aligned} &\frac{f(a)+f (a+e^{i \varphi }\eta (b,a) )}{2} -\frac{ \varGamma (\alpha +1))}{2 (e^{i \varphi }\eta (b,a) )^{\alpha }} \bigl[J_{a^{+}}^{\alpha }f \bigl(a+e^{i \varphi }\eta (b,a) \bigr) +J_{ (a+e^{i \varphi }\eta (b,a) )^{-}}^{\alpha }f(a) \bigr] \\ &\quad =\frac{ (e^{i \varphi }\eta (b,a) )^{2}}{2(\alpha +1)} \int _{0}^{1} \bigl[1-(1-t)^{\alpha +1}-t^{\alpha +1} \bigr]f'' \bigl(a+(1-t)e ^{i \varphi }\eta (b,a) \bigr)\,dt. \end{aligned}$$

Lemma 3

([22])

For \(t\in [0,1]\), we have
$$\begin{aligned} (1-t)^{m} &\leq 2^{1-m}-t^{m}, \quad \textit{for }m \in [0,1], \\ (1-t)^{m} &\geq 2^{1-m}-t^{m}, \quad \textit{for }m \in [1,\infty ). \end{aligned}$$

In this study, using the \(\lambda _{\varphi }\)-preinvexity of the function, we establish new inequalities of Hermite–Hadamard type for differentiable function and some trapezoid type inequalities for function whose second derivatives absolutely values are F-convex.

3 Hermite–Hadamard type inequalities for differentiable functions

In this section, we establish some inequalities of Hermite–Hadamard type for F-convex functions in fractional integral forms.

Theorem 9

Let \(I\subseteq \mathbb{R}\) be an open invex set with respect to bifunction \(\eta : I\times I\to \mathbb{R}\), where \(\eta (b,a)>0\). Let \(f: [0,b]\to \mathbb{R}\) be a differentiable mapping. Suppose that \(|f'|\) is measurable, decreasing, \(\lambda _{\varphi }\)-preinvex function on I, and F-convex on \([a,b]\), for some \(F\in \mathcal{F}\) and the function \(t\in (0,1)\to L_{w(t)}\) belongs to \(L^{1}(0,1)\), where \(w(t)= \vert (1-t)^{\alpha }-t^{\alpha } \vert \). Then
$$\begin{aligned} \begin{aligned}[b] &T_{F,w} \biggl(\frac{2}{e^{i \varphi }\eta (b,a)} \biggl\vert \frac{f(a)+f (a+e^{i \varphi } \eta (b,a) )}{2} \\ &\quad {}-\frac{\varGamma (\alpha +1)}{2 (e^{i \varphi }\eta (b,a) )^{\alpha }} \bigl[J_{a^{+}} ^{\alpha }f \bigl(a+e^{i \varphi }\eta (b,a) \bigr) +J_{ (a+e ^{i \varphi }\eta (b,a) )^{-}}^{\alpha }f(a) \bigr] \biggr\vert , \bigl\vert f'(a) \bigr\vert , \bigl\vert f'(b) \bigr\vert \biggr)\\ &\quad {} + \int _{0}^{1} L _{w(t)}\,dt\leq 0. \end{aligned} \end{aligned}$$
(10)

Proof

Since \(|f'|\) is F-convex, we have
$$\begin{aligned} F \bigl( \bigl\vert f' \bigl(a+(1-t)e^{i \varphi }\eta (b,a) \bigr) \bigr\vert , \bigl\vert f'(a) \bigr\vert , \bigl\vert f'(b) \bigr\vert ,t \bigr)\leq 0,\quad t \in [0,1]. \end{aligned}$$
Multiplying this inequality by \(w(t)= \vert (1-t)^{\alpha }-t^{\alpha } \vert \) and using axiom (A3), we have
$$\begin{aligned} F \bigl(w(t) \bigl\vert f' \bigl(a+(1-t)e^{i \varphi }\eta (b,a) \bigr) \bigr\vert ,w(t) \bigl\vert f'(a) \bigr\vert , w(t) \bigl\vert f'(b) \bigr\vert ,t \bigr)+L_{w(t)} \leq 0,\quad t\in [0,1]. \end{aligned}$$
Integrating over \([0,1]\) and using axiom (A2), we get
$$\begin{aligned} &T_{F,w} \biggl( \int _{0}^{1} \bigl\vert (1-t)^{\alpha }-t^{\alpha } \bigr\vert \bigl\vert f' \bigl(a+(1-t)e^{i \varphi } \eta (b,a) \bigr) \bigr\vert \,dt, \bigl\vert f'(a) \bigr\vert , \bigl\vert f'(b) \bigr\vert ,t \biggr)\\ &\quad {}+ \int _{0}^{1} L _{w(t)}\,dt \leq 0,\quad t\in [0,1]. \end{aligned}$$
But from Lemma 1 we have
$$\begin{aligned} &\frac{2}{e^{i \varphi }\eta (b,a)} \biggl\vert \frac{f(a)+f (a+e^{i \varphi }\eta (b,a) )}{2} \\ &\qquad {}-\frac{\varGamma (\alpha +1))}{2 (e ^{i \varphi }\eta (b,a) )^{\alpha }} \bigl[J_{a^{+}}^{\alpha }f \bigl(a+e^{i \varphi }\eta (b,a) \bigr) +J_{ (a+e^{i \varphi } \eta (b,a) )^{-}}^{\alpha }f(a) \bigr] \biggr\vert \\ &\quad \leq \int _{0}^{1} \bigl\vert (1-t)^{\alpha }-t^{\alpha } \bigr\vert \bigl\vert f' \bigl(a+(1-t)e^{i \varphi } \eta (b,a) \bigr) \bigr\vert \,dt. \end{aligned}$$
Because \(T_{F,w}\) is nondecreasing with respect to the first variable so that
$$\begin{aligned} &T_{F,w} \biggl(\frac{2}{e^{i \varphi }\eta (b,a)} \biggl\vert \frac{f(a)+f (a+e^{i \varphi } \eta (b,a) )}{2} \\ &\quad {}- \frac{\varGamma (\alpha +1)}{2 (e^{i \varphi }\eta (b,a) )^{\alpha }} \bigl[J_{a^{+}} ^{\alpha }f \bigl(a+e^{i \varphi } \eta (b,a) \bigr) +J_{ (a+e ^{i \varphi }\eta (b,a) )^{-}}^{\alpha }f(a) \bigr] \biggr\vert , \bigl\vert f'(a) \bigr\vert , \bigl\vert f'(b) \bigr\vert \biggr) \\ &\quad {}+ \int _{0}^{1} L _{w(t)}\,dt\leq 0,\quad t\in [0,1]. \end{aligned}$$
This proves (10). □

Remark 2

If we choose \(\eta (b,a)=b-a\) and \(\varphi =0\) in Theorem 9, we get
$$\begin{aligned} &T_{F,w} \biggl(\frac{2}{b-a} \biggl\vert \frac{f(a)+f(b)}{2}- \frac{\varGamma ( \alpha +1))}{2(b-a)^{\alpha }} \bigl[J_{a^{+}}^{\alpha }f(b)+J_{b ^{-}}^{\alpha }f(a) \bigr] \biggr\vert , \bigl\vert f'(a) \bigr\vert , \bigl\vert f'(b) \bigr\vert \biggr)\\ &\quad {} + \int _{0}^{1} L_{w(t)}\,dt\leq 0. \end{aligned}$$

Corollary 1

Under the assumptions of Theorem 9, if \(|f'|\) is ε-convex, then we have
$$\begin{aligned} & \biggl\vert \frac{f(a)+f (a+e^{i \varphi } \eta (b,a) )}{2}-\frac{ \varGamma (\alpha +1))}{2 (e^{i \varphi }\eta (b,a) )^{\alpha }} \bigl[J_{a^{+}}^{\alpha }f \bigl(a+e^{i \varphi }\eta (b,a) \bigr) +J_{ (a+e^{i \varphi }\eta (b,a) )^{-}}^{\alpha }f(a) \bigr] \biggr\vert \\ &\quad \leq \frac{e^{i \varphi }\eta (b,a)}{2(\alpha +1)} \biggl(1-\frac{1}{2^{ \alpha }} \biggr) \bigl( \bigl\vert f'(a) \bigr\vert + \bigl\vert f'(b) \bigr\vert +2 \varepsilon \bigr). \end{aligned}$$

Proof

Using (5) with \(w(t)= \vert (1-t)^{\alpha }-t^{\alpha } \vert \), we find
$$\begin{aligned} \int _{0}^{1} L_{w(t)}\,dt &=\varepsilon \int _{0}^{1} \bigl(1- \bigl\vert (1-t)^{ \alpha }-t^{\alpha } \bigr\vert \bigr)\,dt \\ &=\varepsilon \biggl[ \int _{0}^{\frac{1}{2}} \bigl(1-(1-t)^{\alpha }+t ^{\alpha } \bigr)\,dt + \int _{1}^{\frac{1}{2}} \bigl(1-(1-t)^{\alpha }+t ^{\alpha } \bigr)\,dt \biggr] \\ &=\varepsilon \biggl[1-\frac{2}{\alpha +1} \biggl(1-\frac{1}{\alpha } \biggr) \biggr]. \end{aligned}$$
From (4) with \(w(t)= \vert (1-t)^{\alpha }-t^{\alpha } \vert \), we have
$$\begin{aligned} &T_{F,w}(u_{1},u_{2},u_{3}) \\ &\quad =u_{1}- \biggl( \int _{0}^{1} t \bigl\vert (1-t)^{\alpha }-t^{\alpha } \bigr\vert \,dt \biggr)u _{2} - \biggl( \int _{0}^{1} (1-t) \bigl\vert (1-t)^{\alpha }-t^{\alpha } \bigr\vert \,dt \biggr)u _{3}- \varepsilon \\ &\quad =u_{1}-\frac{1}{\alpha +1} \biggl(1-\frac{1}{\alpha } \biggr) (u_{2}+u _{3})-\varepsilon , \end{aligned}$$
for \(u_{1}, u_{2}, u_{3}\in \mathbb{R}\). Hence, by Theorem 9, we have
$$\begin{aligned} 0 &\geq T_{F,w} \biggl(\frac{2}{e^{i \varphi }\eta (b,a)} \biggl\vert \frac{f(a)+f (a+e^{i \varphi } \eta (b,a) )}{2} \\ &\quad {}-\frac{\varGamma (\alpha +1)}{2 (e^{i \varphi }\eta (b,a) )^{\alpha }} \bigl[J_{a^{+}} ^{\alpha }f \bigl(a+e^{i \varphi }\eta (b,a) \bigr) +J_{ (a+e ^{i \varphi }\eta (b,a) )^{-}}^{\alpha }f(a) \bigr] \biggr\vert , \bigl\vert f'(a) \bigr\vert , \bigl\vert f'(b) \bigr\vert \biggr)\\ &\quad {}+ \int _{0}^{1} L_{w(t)}\,dt \\ &=\frac{2}{e^{i \varphi }\eta (b,a)} \biggl\vert \frac{f(a)+f (a+e ^{i \varphi } \eta (b,a) )}{2}\\ &\quad {}-\frac{\varGamma (\alpha +1)}{2 (e ^{i \varphi }\eta (b,a) )^{\alpha }} \bigl[J_{a^{+}}^{\alpha }f \bigl(a+e^{i \varphi }\eta (b,a) \bigr) +J_{ (a+e^{i \varphi } \eta (b,a) )^{-}}^{\alpha }f(a) \bigr] \biggr\vert \\ &\quad{}-\frac{1}{\alpha +1} \biggl(1-\frac{1}{\alpha } \biggr) \bigl( \bigl\vert f'(a) \bigr\vert + \bigl\vert f'(b) \bigr\vert +2\varepsilon \bigr). \end{aligned}$$
This completes the proof. □

Remark 3

In Corollary 1, if we choose
  1. (a)
    \(\eta (b,a)=b-a\) and \(\varphi =0\), we get
    $$\begin{aligned} & \biggl\vert \frac{f(a)+f(b)}{2}-\frac{\varGamma (\alpha +1))}{2(b-a)^{\alpha }} \bigl[J_{a^{+}}^{\alpha }f(b)+J_{b^{-}}^{\alpha }f(a) \bigr] \biggr\vert \\ &\quad \leq \frac{b-a}{2(\alpha +1)} \biggl(1-\frac{1}{2^{\alpha }} \biggr) \bigl( \bigl\vert f'(a) \bigr\vert + \bigl\vert f'(b) \bigr\vert +2\varepsilon \bigr). \end{aligned}$$
     
  2. (b)
    \(\eta (b,a)=b-a\), \(\varphi =0\), and \(\varepsilon =0\), we get
    $$\begin{aligned} & \biggl\vert \frac{f(a)+f(b)}{2}-\frac{\varGamma (\alpha +1))}{2(b-a)^{\alpha }} \bigl[J_{a^{+}}^{\alpha }f(b)+J_{b^{-}}^{\alpha }f(a) \bigr] \biggr\vert \\ &\quad \leq \frac{b-a}{2(\alpha +1)} \biggl(1-\frac{1}{2^{\alpha }} \biggr) \bigl( \bigl\vert f'(a) \bigr\vert + \bigl\vert f'(b) \bigr\vert \bigr) \end{aligned}$$
    which is given by [18].
     

Corollary 2

Under the assumptions of Theorem 9, if \(|f'|\) is \(\lambda _{\varphi }\)-preinvex, then we have
$$\begin{aligned} & \biggl\vert \frac{f(a)+f (a+e^{i \varphi } \eta (b,a) )}{2}-\frac{ \varGamma (\alpha +1))}{2 (e^{i \varphi }\eta (b,a) )^{\alpha }} \bigl[J_{a^{+}}^{\alpha }f \bigl(a+e^{i \varphi }\eta (b,a) \bigr) +J_{ (a+e^{i \varphi }\eta (b,a) )^{-}}^{\alpha }f(a) \bigr] \biggr\vert \\ &\quad \leq \frac{e^{i \varphi }\eta (b,a)}{4} \biggl[B_{\frac{1}{2}} \biggl(\frac{1}{2}, \alpha +\frac{1}{2} \biggr) -B_{\frac{1}{2}} \biggl(\alpha + \frac{1}{2},\frac{1}{2} \biggr) \biggr] \biggl( \bigl\vert f'(a) \bigr\vert +\frac{1- \lambda }{\lambda } \bigl\vert f'(b) \bigr\vert \biggr). \end{aligned}$$

Proof

Using (7) with \(w(t)= \vert (1-t)^{\alpha }-t^{\alpha } \vert \), we have
$$\begin{aligned} &T_{F,w}(u_{1},u_{2},u_{3}) \\ &\quad =u_{1}- \biggl( \int _{0}^{1} \frac{\sqrt{t}}{2\sqrt{1-t}} \bigl\vert (1-t)^{ \alpha }-t^{\alpha } \bigr\vert \,dt \biggr)u_{3}- \frac{1-\lambda }{\lambda } \biggl( \int _{0}^{1} \frac{\sqrt{1-t}}{2\sqrt{t}} \bigl\vert (1-t)^{ \alpha }-t^{\alpha } \bigr\vert \,dt \biggr)u_{2} \\ &\quad =u_{1}-\frac{1}{2} \biggl[B_{\frac{1}{2}} \biggl( \frac{1}{2},\alpha + \frac{1}{2} \biggr) -B_{\frac{1}{2}} \biggl( \alpha +\frac{1}{2}, \frac{1}{2} \biggr) \biggr] \biggl(u_{2}+ \frac{1-\lambda }{\lambda }u _{3} \biggr) \end{aligned}$$
for \(u_{1}, u_{2}, u_{3}\in \mathbb{R}\). Hence, by Theorem 9, we have
$$\begin{aligned} 0 &\geq T_{F,w} \biggl(\frac{2}{e^{i \varphi }\eta (b,a)} \biggl\vert \frac{f(a)+f (a+e^{i \varphi } \eta (b,a) )}{2} \\ &\quad {}-\frac{\varGamma (\alpha +1)}{2 (e^{i \varphi }\eta (b,a) )^{\alpha }} \bigl[J_{a^{+}} ^{\alpha }f \bigl(a+e^{i \varphi }\eta (b,a) \bigr) +J_{ (a+e ^{i \varphi }\eta (b,a) )^{-}}^{\alpha }f(a) \bigr] \biggr\vert , \bigl\vert f'(a) \bigr\vert , \bigl\vert f'(b) \bigr\vert \biggr)\\ &\quad {}+ \int _{0}^{1} L_{w(t)}\,dt \\ &=\frac{2}{e^{i \varphi }\eta (b,a)} \biggl\vert \frac{f(a)+f (a+e ^{i \varphi } \eta (b,a) )}{2}\\ &\quad {}-\frac{\varGamma (\alpha +1)}{2 (e ^{i \varphi }\eta (b,a) )^{\alpha }} \bigl[J_{a^{+}}^{\alpha }f \bigl(a+e^{i \varphi }\eta (b,a) \bigr) +J_{ (a+e^{i \varphi } \eta (b,a) )^{-}}^{\alpha }f(a) \bigr] \biggr\vert \\ &\quad{}-\frac{1}{2} \biggl[B_{\frac{1}{2}} \biggl(\frac{1}{2}, \alpha + \frac{1}{2} \biggr) -B_{\frac{1}{2}} \biggl(\alpha + \frac{1}{2}, \frac{1}{2} \biggr) \biggr] \biggl( \bigl\vert f'(a) \bigr\vert +\frac{1- \lambda }{\lambda } \bigl\vert f'(b) \bigr\vert \biggr). \end{aligned}$$
This leads to
$$\begin{aligned} & \biggl\vert \frac{f(a)+f (a+e^{i \varphi } \eta (b,a) )}{2}-\frac{ \varGamma (\alpha +1)}{2 (e^{i \varphi }\eta (b,a) )^{\alpha }} \bigl[J_{a^{+}}^{\alpha }f \bigl(a+e^{i \varphi }\eta (b,a) \bigr) +J_{ (a+e^{i \varphi }\eta (b,a) )^{-}}^{\alpha }f(a) \bigr] \biggr\vert \\ &\quad \leq \frac{e^{i \varphi }\eta (b,a)}{4} \biggl[B_{\frac{1}{2}} \biggl(\frac{1}{2}, \alpha +\frac{1}{2} \biggr) -B_{\frac{1}{2}} \biggl(\alpha + \frac{1}{2},\frac{1}{2} \biggr) \biggr] \biggl( \bigl\vert f'(a) \bigr\vert +\frac{1- \lambda }{\lambda } \bigl\vert f'(b) \bigr\vert \biggr). \end{aligned}$$
Thus, the proof is done. □

Remark 4

In Corollary 2, if we choose
  1. (a)
    \(\eta (b,a)=b-a\) and \(\varphi =0\), we get
    $$\begin{aligned} & \biggl\vert \frac{f(a)+f(b)}{2}-\frac{\varGamma (\alpha +1))}{2(b-a)^{\alpha }} \bigl[J_{a^{+}}^{\alpha }f(b)+J_{b^{-}}^{\alpha }f(a) \bigr] \biggr\vert \\ &\quad \leq \frac{b-a}{4} \biggl[B_{\frac{1}{2}} \biggl(\frac{1}{2}, \alpha + \frac{1}{2} \biggr) -B_{\frac{1}{2}} \biggl(\alpha + \frac{1}{2}, \frac{1}{2} \biggr) \biggr] \biggl( \bigl\vert f'(a) \bigr\vert +\frac{1- \lambda }{\lambda } \bigl\vert f'(b) \bigr\vert \biggr). \end{aligned}$$
     
  2. (b)
    \(\eta (b,a)=b-a\), \(\varphi =0\), and \(\lambda =\frac{1}{2}\), we get
    $$\begin{aligned} & \biggl\vert \frac{f(a)+f(b)}{2}-\frac{\varGamma (\alpha +1))}{2(b-a)^{\alpha }} \bigl[J_{a^{+}}^{\alpha }f(b)+J_{b^{-}}^{\alpha }f(a) \bigr] \biggr\vert \\ &\quad \leq \frac{b-a}{4} \biggl[B_{\frac{1}{2}} \biggl( \frac{1}{2},\alpha + \frac{1}{2} \biggr) -B_{\frac{1}{2}} \biggl( \alpha +\frac{1}{2}, \frac{1}{2} \biggr) \biggr] \bigl( \bigl\vert f'(a) \bigr\vert + \bigl\vert f'(b) \bigr\vert \bigr). \end{aligned}$$
     

Corollary 3

Under the assumptions of Theorem 9, if \(|f'|\) is h-convex, then we have
$$\begin{aligned} & \biggl\vert \frac{f(a)+f (a+e^{i \varphi } \eta (b,a) )}{2}-\frac{ \varGamma (\alpha +1))}{2 (e^{i \varphi }\eta (b,a) )^{\alpha }} \bigl[J_{a^{+}}^{\alpha }f \bigl(a+e^{i \varphi }\eta (b,a) \bigr) +J_{ (a+e^{i \varphi }\eta (b,a) )^{-}}^{\alpha }f(a) \bigr] \biggr\vert \\ &\quad \leq \frac{e^{i \varphi }\eta (b,a)}{2} \biggl( \int _{0}^{1} h(t) \bigl\vert (1-t)^{\alpha }-t^{\alpha } \bigr\vert \,dt \biggr) \biggl( \bigl\vert f'(a) \bigr\vert + \frac{1- \lambda }{\lambda } \bigl\vert f'(b) \bigr\vert \biggr). \end{aligned}$$

Proof

Using (9) with \(w(t)= \vert (1-t)^{\alpha }-t^{\alpha } \vert \), we have
$$\begin{aligned} &T_{F,w}(u_{1},u_{2},u_{3}) \\ &\quad =u_{1}- \biggl( \int _{0}^{1} h(t) \bigl\vert (1-t)^{\alpha }-t^{\alpha } \bigr\vert \,dt \biggr)u _{3} -\frac{1-\lambda }{\lambda } \biggl( \int _{0}^{1} h(1-t) \bigl\vert (1-t)^{ \alpha }-t^{\alpha } \bigr\vert \,dt \biggr)u_{2} \\ &\quad =u_{1}- \biggl( \int _{0}^{1} h(t) \bigl\vert (1-t)^{\alpha }-t^{\alpha } \bigr\vert \,dt \biggr)u _{3} -\frac{1-\lambda }{\lambda } \biggl( \int _{0}^{1} h(t) \bigl\vert (1-t)^{ \alpha }-t^{\alpha } \bigr\vert \,dt \biggr)u_{2} \\ &\quad =u_{1}- \biggl( \int _{0}^{1} h(t) \bigl\vert (1-t)^{\alpha }-t^{\alpha } \bigr\vert \,dt \biggr) \biggl(u_{2}+\frac{1- \lambda }{\lambda }u_{3} \biggr) \end{aligned}$$
for \(u_{1}, u_{2}, u_{3}\in \mathbb{R}\). So, by Theorem 9, we have
$$\begin{aligned} 0 &\geq T_{F,w} \biggl(\frac{2}{e^{i \varphi }\eta (b,a)} \biggl\vert \frac{f(a)+f (a+e^{i \varphi } \eta (b,a) )}{2}\\ &\quad {}-\frac{\varGamma (\alpha +1)}{2 (e^{i \varphi }\eta (b,a) )^{\alpha }} \bigl[J_{a^{+}} ^{\alpha }f \bigl(a+e^{i \varphi }\eta (b,a) \bigr) +J_{ (a+e ^{i \varphi }\eta (b,a) )^{-}}^{\alpha }f(a) \bigr] \biggr\vert ,\bigl\vert f'(a) \bigr\vert , \bigl\vert f'(b) \bigr\vert \biggr) \\ & \quad {}+ \int _{0}^{1} L_{w(t)}\,dt \\ &=\frac{2}{e^{i \varphi }\eta (b,a)} \biggl\vert \frac{f(a)+f (a+e ^{i \varphi } \eta (b,a) )}{2}\\ &\quad {}-\frac{\varGamma (\alpha +1)}{2 (e ^{i \varphi }\eta (b,a) )^{\alpha }} \bigl[J_{a^{+}}^{\alpha }f \bigl(a+e^{i \varphi }\eta (b,a) \bigr) +J_{ (a+e^{i \varphi } \eta (b,a) )^{-}}^{\alpha }f(a) \bigr] \biggr\vert \\ &\quad {}- \biggl( \int _{0}^{1} h(t) \bigl\vert (1-t)^{\alpha }-t^{\alpha } \bigr\vert \,dt \biggr) \biggl( \bigl\vert f'(a) \bigr\vert + \frac{1- \lambda }{\lambda } \bigl\vert f'(b) \bigr\vert \biggr). \end{aligned}$$
This leads to
$$\begin{aligned} & \biggl\vert \frac{f(a)+f (a+e^{i \varphi } \eta (b,a) )}{2}-\frac{ \varGamma (\alpha +1)}{2 (e^{i \varphi }\eta (b,a) )^{\alpha }} \bigl[J_{a^{+}}^{\alpha }f \bigl(a+e^{i \varphi }\eta (b,a) \bigr) +J_{ (a+e^{i \varphi }\eta (b,a) )^{-}}^{\alpha }f(a) \bigr] \biggr\vert \\ &\quad \leq \frac{e^{i \varphi }\eta (b,a)}{2} \biggl( \int _{0}^{1} h(t) \bigl\vert (1-t)^{\alpha }-t^{\alpha } \bigr\vert \,dt \biggr) \biggl( \bigl\vert f'(a) \bigr\vert + \frac{1- \lambda }{\lambda } \bigl\vert f'(b) \bigr\vert \biggr). \end{aligned}$$
Thus, the proof is done. □

Theorem 10

Let \(I\subseteq \mathbb{R}\) be an open invex set with respect to bifunction \(\eta : I\times I\to \mathbb{R}\), where \(\eta (b,a)>0\). Let \(f: [0,b]\to \mathbb{R}\) be a differentiable mapping. Suppose that \(|f'|^{{\frac{p}{p-1}}}\) is measurable, decreasing, \(\lambda _{\varphi }\)-preinvex function on I, and F-convex on \([a,b]\), for some \(F\in \mathcal{F}\) and \(|f'|\in L^{{\frac{p}{p-1}}}(a,b)\). Then
$$\begin{aligned} &T_{F,1} \bigl(G_{1}(f,p), \bigl\vert f'(a) \bigr\vert ^{\frac{p}{p-1}}, \bigl\vert f'(b) \bigr\vert ^{\frac{p}{p-1}} \bigr)\leq 0, \end{aligned}$$
(11)
where
$$\begin{aligned} G_{1}(f,p) &= \biggl(\frac{2}{e^{i \varphi }\eta (b,a)} \biggr)^{ \frac{p}{p-1}} \biggl( \frac{\alpha p+1}{2-2^{1-\alpha p}} \biggr)^{ \frac{1}{p-1}} \biggl\vert \frac{f(a)+f (a+e^{i \varphi } \eta (b,a) )}{2} \\ &\quad {}-\frac{ \varGamma (\alpha +1)}{2 (e^{i \varphi }\eta (b,a) )^{\alpha }} \bigl[J_{a^{+}}^{\alpha }f \bigl(a+e^{i \varphi }\eta (b,a) \bigr) +J_{ (a+e^{i \varphi }\eta (b,a) )^{-}}^{\alpha }f(a) \bigr] \biggr\vert ^{\frac{p}{p-1}}. \end{aligned}$$

Proof

Since \(|f'|^{\frac{p}{p-1}}\) is F-convex, we have
$$\begin{aligned} F \bigl( \bigl\vert f' \bigl(a+(1-t)e^{i \varphi }\eta (b,a) \bigr) \bigr\vert ^{\frac{p}{p-1}}, \bigl\vert f'(a) \bigr\vert ^{\frac{p}{p-1}}, \bigl\vert f'(b) \bigr\vert ^{\frac{p}{p-1}},t \bigr) \leq 0,\quad t\in [0,1]. \end{aligned}$$
With \(w(t)=1\) in (A2), we have
$$\begin{aligned} T_{F,1} \biggl( \int _{0}^{1} \bigl\vert f' \bigl(a+(1-t)e^{i \varphi }\eta (b,a) \bigr) \bigr\vert ^{\frac{p}{p-1}}\,dt, \bigl\vert f'(a) \bigr\vert ^{\frac{p}{p-1}}, \bigl\vert f'(b) \bigr\vert ^{\frac{p}{p-1}} \biggr)\leq 0, \quad t\in [0,1]. \end{aligned}$$
Using Lemma 1 and the Hölder inequality, we get
$$\begin{aligned} & \biggl\vert \frac{f(a)+f (a+e^{i \varphi } \eta (b,a) )}{2}-\frac{ \varGamma (\alpha +1)}{2 (e^{i \varphi }\eta (b,a) )^{\alpha }} \bigl[J_{a^{+}}^{\alpha }f \bigl(a+e^{i \varphi }\eta (b,a) \bigr) +J_{ (a+e^{i \varphi }\eta (b,a) )^{-}}^{\alpha }f(a) \bigr] \biggr\vert \\ &\quad =\frac{e^{i \varphi }\eta (b,a)}{2} \int _{0}^{1} \bigl\vert (1-t)^{ \alpha }-t^{\alpha } \bigr\vert \bigl\vert f' \bigl(a+(1-t)e^{i \varphi } \eta (b,a) \bigr) \bigr\vert \,dt \\ &\quad \leq \frac{e^{i \varphi }\eta (b,a)}{2} \biggl( \int _{0}^{1} \bigl\vert (1-t)^{ \alpha }-t^{\alpha } \bigr\vert \,dt \biggr)^{\frac{1}{p}} \biggl( \int _{0} ^{1} \bigl\vert f' \bigl(a+(1-t)e^{i \varphi }\eta (b,a) \bigr) \bigr\vert ^{\frac{p}{p-1}}\,dt \biggr)^{\frac{p-1}{p}} \\ &\quad = \frac{e^{i \varphi }\eta (b,a)}{2} \biggl(\frac{2-2^{1-\alpha p}}{ \alpha p+1} \biggr)^{\frac{1}{p}} \biggl( \int _{0}^{1} \bigl\vert f' \bigl(a+(1-t)e ^{i \varphi }\eta (b,a) \bigr) \bigr\vert ^{\frac{p}{p-1}}\,dt \biggr) ^{\frac{p-1}{p}} \end{aligned}$$
or, equivalently,
$$\begin{aligned} & \biggl(\frac{2}{e^{i \varphi }\eta (b,a)} \biggr)^{\frac{p}{p-1}} \biggl(\frac{\alpha p+1}{2-2^{1-\alpha p}} \biggr)^{\frac{1}{p-1}} \biggl\vert \frac{f(a)+f (a+e^{i \varphi } \eta (b,a) )}{2} \\ &\qquad {}-\frac{ \varGamma (\alpha +1)}{2 (e^{i \varphi }\eta (b,a) )^{\alpha }} \bigl[J_{a^{+}}^{\alpha }f \bigl(a+e^{i \varphi }\eta (b,a) \bigr) +J_{ (a+e^{i \varphi }\eta (b,a) )^{-}}^{\alpha }f(a) \bigr] \biggr\vert ^{\frac{p}{p-1}} \\ &\quad \leq \int _{0}^{1} \bigl\vert f' \bigl(a+(1-t)e^{i \varphi }\eta (b,a) \bigr) \bigr\vert ^{\frac{p}{p-1}}\,dt. \end{aligned}$$
Because \(T_{F,1}\) is nondecreasing with respect to the first variable, we get
$$\begin{aligned} &T_{F,1} \bigl(G_{1}(f,p), \bigl\vert f'(a) \bigr\vert ^{{\frac{p}{p-1}}}, \bigl\vert f'(b) \bigr\vert ^{{\frac{p}{p-1}}} \bigr)\leq 0. \end{aligned}$$
Thus, the proof is completed. □

Remark 5

If we choose \(\eta (b,a)=b-a\) and \(\varphi =0\) in Theorem 10, we get
$$\begin{aligned} &T_{F,1} \biggl( \biggl(\frac{2}{b-a} \biggr)^{\frac{p}{p-1}} \biggl(\frac{ \alpha p+1}{2-2^{1-\alpha p}} \biggr)^{{\frac{1}{p-1}}} \biggl\vert \frac{f(a)+f(b)}{2} - \frac{\varGamma (\alpha +1))}{2(b-a)^{\alpha }} \bigl[J_{a^{+}}^{\alpha }f(b)+J_{b^{-}}^{\alpha }f(a) \bigr] \biggr\vert ,\\ &\quad \bigl\vert f'(a) \bigr\vert , \bigl\vert f'(b) \bigr\vert \biggr)\leq 0. \end{aligned}$$

Corollary 4

Under the assumptions of Theorem 10, if \(|f'|^{\frac{p}{p-1}}\) is ε-convex, we have
$$\begin{aligned} & \biggl\vert \frac{f(a)+f (a+e^{i \varphi } \eta (b,a) )}{2}-\frac{ \varGamma (\alpha +1))}{2 (e^{i \varphi }\eta (b,a) )^{\alpha }} \bigl[J_{a^{+}}^{\alpha }f \bigl(a+e^{i \varphi }\eta (b,a) \bigr) +J_{ (a+e^{i \varphi }\eta (b,a) )^{-}}^{\alpha }f(a) \bigr] \biggr\vert \\ &\quad \leq \frac{e^{i \varphi }\eta (b,a)}{2} \biggl(\frac{2-2^{1-\alpha p}}{ \alpha p+1} \biggr)^{\frac{1}{p}} \biggl(\frac{ \vert f'(a) \vert ^{\frac{p}{p-1}}+ \vert f'(b) \vert ^{\frac{p}{p-1}}}{2} +\varepsilon \biggr)^{\frac{p-1}{p}}. \end{aligned}$$

Proof

Using (5) with \(w(t)=1\), we have
$$\begin{aligned} \int _{0}^{1} L_{w(t)}\,dt &=\varepsilon \int _{0}^{1} \bigl(1-w(t)\bigr)\,dt=0. \end{aligned}$$
(12)
From (4) with \(w(t)=1\), we have
$$\begin{aligned} &T_{F,1}(u_{1},u_{2},u_{3}) \\ &\quad =u_{1}- \biggl( \int _{0}^{1} t\,dt \biggr)u_{2}- \biggl( \int _{0}^{1} (1-t)\,dt \biggr)u _{3}- \varepsilon \\ &\quad =u_{1}-\frac{u_{2}+u_{3}}{2}-\varepsilon , \end{aligned}$$
(13)
for \(u_{1}, u_{2}, u_{3}\in \mathbb{R}\). Hence, by Theorem 10, we have
$$\begin{aligned} 0 &\geq T_{F,1} \bigl(G_{1}(f,p), \bigl\vert f'(a) \bigr\vert ^{\frac{p}{p-1}}, \bigl\vert f'(b) \bigr\vert ^{\frac{p}{p-1}} \bigr) \\ &=G_{1}(f,p)-\frac{ \vert f'(a) \vert ^{\frac{p}{p-1}}+ \vert f'(b) \vert ^{\frac{p}{p-1}}}{2} -\varepsilon . \end{aligned}$$
This leads to
$$\begin{aligned} & \biggl(\frac{2}{e^{i \varphi }\eta (b,a)} \biggr)^{\frac{p}{p-1}} \biggl(\frac{\alpha p+1}{2-2^{1-\alpha p}} \biggr)^{\frac{1}{p-1}} \biggl\vert \frac{f(a)+f (a+e^{i \varphi } \eta (b,a) )}{2} \\ &\quad {}-\frac{ \varGamma (\alpha +1)}{2 (e^{i \varphi }\eta (b,a) )^{\alpha }} \bigl[J_{a^{+}}^{\alpha }f \bigl(a+e^{i \varphi }\eta (b,a) \bigr) +J_{ (a+e^{i \varphi }\eta (b,a) )^{-}}^{\alpha }f(a) \bigr] \biggr\vert ^{\frac{p}{p-1}} \\ &\quad {}-\frac{ \vert f'(a) \vert ^{\frac{p}{p-1}}+ \vert f'(b) \vert ^{\frac{p}{p-1}}}{2} \leq 0 \end{aligned}$$
or, equivalently,
$$\begin{aligned} & \biggl\vert \frac{f(a)+f (a+e^{i \varphi } \eta (b,a) )}{2}-\frac{ \varGamma (\alpha +1))}{2 (e^{i \varphi }\eta (b,a) )^{\alpha }} \bigl[J_{a^{+}}^{\alpha }f \bigl(a+e^{i \varphi }\eta (b,a) \bigr) +J_{ (a+e^{i \varphi }\eta (b,a) )^{-}}^{\alpha }f(a) \bigr] \biggr\vert \\ &\quad \leq \frac{e^{i \varphi }\eta (b,a)}{2} \biggl(\frac{2-2^{1-\alpha p}}{ \alpha p+1} \biggr)^{\frac{1}{p}} \biggl(\frac{ \vert f'(a) \vert ^{\frac{p}{p-1}}+ \vert f'(b) \vert ^{\frac{p}{p-1}}}{2} +\varepsilon \biggr)^{\frac{p-1}{p}}. \end{aligned}$$
This completes the proof. □

Remark 6

In Corollary 4, if we choose
  1. (a)
    \(\eta (b,a)=b-a\) and \(\varphi =0\), we get
    $$\begin{aligned} & \biggl\vert \frac{f(a)+f(b)}{2}-\frac{\varGamma (\alpha +1))}{2(b-a)^{\alpha }} \bigl[J_{a^{+}}^{\alpha }f(b)+J_{b^{-}}^{\alpha }f(a) \bigr] \biggr\vert \\ &\quad \leq \frac{b-a}{2} \biggl(\frac{2-2^{1-\alpha p}}{\alpha p+1} \biggr) ^{\frac{1}{p}} \biggl(\frac{ \vert f'(a) \vert ^{\frac{p}{p-1}}+ \vert f'(b) \vert ^{\frac{p}{p-1}}}{2}+\varepsilon \biggr)^{\frac{p-1}{p}}. \end{aligned}$$
     
  2. (b)
    \(\eta (b,a)=b-a\), \(\varphi =0\), and \(\varepsilon =0\), we get
    $$\begin{aligned} & \biggl\vert \frac{f(a)+f(b)}{2}-\frac{\varGamma (\alpha +1))}{2(b-a)^{\alpha }} \bigl[J_{a^{+}}^{\alpha }f(b)+J_{b^{-}}^{\alpha }f(a) \bigr] \biggr\vert \\ &\quad \leq \frac{b-a}{2} \biggl(\frac{2-2^{1-\alpha p}}{\alpha p+1} \biggr) ^{\frac{1}{p}} \biggl(\frac{ \vert f'(a) \vert ^{\frac{p}{p-1}}+ \vert f'(b) \vert ^{\frac{p}{p-1}}}{2} \biggr)^{\frac{p-1}{p}}. \end{aligned}$$
     

Corollary 5

Under the assumptions of Theorem 10. If \(|f'|^{\frac{p-1}{p}}\) is \(\lambda _{\varphi }\)-preinvex, we have
$$\begin{aligned} & \biggl\vert \frac{f(a)+f (a+e^{i \varphi } \eta (b,a) )}{2}-\frac{ \varGamma (\alpha +1))}{2 (e^{i \varphi }\eta (b,a) )^{\alpha }} \bigl[J_{a^{+}}^{\alpha }f \bigl(a+e^{i \varphi }\eta (b,a) \bigr) +J_{ (a+e^{i \varphi }\eta (b,a) )^{-}}^{\alpha }f(a) \bigr] \biggr\vert \\ &\quad \leq \frac{e^{i \varphi }\eta (b,a)}{2} \biggl(\frac{2-2^{1-\alpha p}}{ \alpha p+1} \biggr)^{\frac{1}{p}} \biggl[\frac{\pi }{4} \biggl( \bigl\vert f'(a) \bigr\vert ^{\frac{p-1}{p}} +\frac{1-\lambda }{\lambda } \bigl\vert f'(b) \bigr\vert ^{ \frac{p-1}{p}} \biggr) \biggr]^{\frac{p-1}{p}}. \end{aligned}$$

Proof

Using (7) with \(w(t)=1\), we have
$$\begin{aligned} &T_{F,1}(u_{1},u_{2},u_{3}) \\ &\quad =u_{1}- \biggl( \int _{0}^{1} \frac{\sqrt{t}}{2\sqrt{1-t}}\,dt \biggr)u _{3}-\frac{1-\lambda }{\lambda } \biggl( \int _{0}^{1} \frac{ \sqrt{1-t}}{2\sqrt{t}}\,dt \biggr)u_{2} \\ &\quad =u_{1}-\frac{1}{2}\beta \biggl(\frac{1}{2}, \frac{3}{2} \biggr) \biggl(u_{2}+\frac{1-\lambda }{\lambda }u_{3} \biggr) =u_{1}-\frac{ \pi }{4} \biggl(u_{2}+ \frac{1-\lambda }{\lambda }u_{3} \biggr) \end{aligned}$$
(14)
for \(u_{1}, u_{2}, u_{3}\in \mathbb{R}\). So, by Theorem 10, we have
$$\begin{aligned} 0 &\geq T_{F,1} \bigl(G_{1}(f,p), \bigl\vert f'(a) \bigr\vert ^{\frac{p-1}{p}}, \bigl\vert f'(b) \bigr\vert ^{\frac{p-1}{p}} \bigr) \\ &= \biggl(\frac{2}{e^{i \varphi }\eta (b,a)} \biggr)^{\frac{p}{p-1}} \biggl(\frac{\alpha p+1}{2-2^{1-\alpha p}} \biggr)^{\frac{1}{p-1}} \biggl\vert \frac{f(a)+f (a+e^{i \varphi } \eta (b,a) )}{2} \\ &\quad {}-\frac{ \varGamma (\alpha +1)}{2 (e^{i \varphi }\eta (b,a) )^{\alpha }} \bigl[J_{a^{+}}^{\alpha }f \bigl(a+e^{i \varphi }\eta (b,a) \bigr) +J_{ (a+e^{i \varphi }\eta (b,a) )^{-}}^{\alpha }f(a) \bigr] \biggr\vert ^{\frac{p}{p-1}} \\ &\quad {}-\frac{\pi }{4} \biggl( \bigl\vert f'(a) \bigr\vert ^{\frac{p-1}{p}} +\frac{1- \lambda }{\lambda } \bigl\vert f'(b) \bigr\vert ^{\frac{p-1}{p}} \biggr). \end{aligned}$$
This leads to
$$\begin{aligned} & \biggl\vert \frac{f(a)+f (a+e^{i \varphi } \eta (b,a) )}{2}-\frac{ \varGamma (\alpha +1)}{2 (e^{i \varphi }\eta (b,a) )^{\alpha }} \bigl[J_{a^{+}}^{\alpha }f \bigl(a+e^{i \varphi }\eta (b,a) \bigr) +J_{ (a+e^{i \varphi }\eta (b,a) )^{-}}^{\alpha }f(a) \bigr] \biggr\vert \\ &\quad \leq \frac{e^{i \varphi }\eta (b,a)}{2} \biggl(\frac{2-2^{1-\alpha p}}{\alpha p+1} \biggr)^{\frac{1}{p}} \biggl[\frac{\pi }{4} \biggl( \bigl\vert f'(a) \bigr\vert ^{\frac{p-1}{p}} +\frac{1-\lambda }{\lambda } \bigl\vert f'(b) \bigr\vert ^{ \frac{p-1}{p}} \biggr) \biggr]^{\frac{p-1}{p}}. \end{aligned}$$
Thus, the proof is done. □

Remark 7

In Corollary 5, if we choose
  1. (a)
    \(\eta (b,a)=b-a\) and \(\varphi =0\), we get
    $$\begin{aligned} & \biggl\vert \frac{f(a)+f(b)}{2}-\frac{\varGamma (\alpha +1))}{2(b-a)^{\alpha }} \bigl[J_{a^{+}}^{\alpha }f(b)+J_{b^{-}}^{\alpha }f(a) \bigr] \biggr\vert \\&\quad\leq \frac{b-a}{2} \biggl(\frac{2-2^{1-\alpha p}}{\alpha p+1} \biggr) ^{\frac{1}{p}} \biggl[\frac{\pi }{4} \biggl( \bigl\vert f'(a) \bigr\vert ^{ \frac{p-1}{p}} +\frac{1-\lambda }{\lambda } \bigl\vert f'(b) \bigr\vert ^{ \frac{p-1}{p}} \biggr) \biggr]^{\frac{p-1}{p}}. \end{aligned}$$
     
  2. (b)
    \(\eta (b,a)=b-a\), \(\varphi =0\), and \(\lambda =\frac{1}{2}\), we get
    $$\begin{aligned} & \biggl\vert \frac{f(a)+f(b)}{2}-\frac{\varGamma (\alpha +1))}{2(b-a)^{\alpha }} \bigl[J_{a^{+}}^{\alpha }f(b)+J_{b^{-}}^{\alpha }f(a) \bigr] \biggr\vert \\&\quad\leq \frac{b-a}{2} \biggl(\frac{2-2^{1-\alpha p}}{\alpha p+1} \biggr) ^{\frac{1}{p}} \biggl[\frac{\pi }{4} \bigl( \bigl\vert f'(a) \bigr\vert ^{ \frac{p-1}{p}} + \bigl\vert f'(b) \bigr\vert ^{\frac{p-1}{p}} \bigr) \biggr] ^{\frac{p-1}{p}}. \end{aligned}$$
     

Corollary 6

Under the assumptions of Theorem 10. If \(|f'|^{\frac{p}{p-1}}\) is h-convex, we have
$$\begin{aligned} & \biggl\vert \frac{f(a)+f (a+e^{i \varphi } \eta (b,a) )}{2}-\frac{ \varGamma (\alpha +1))}{2 (e^{i \varphi }\eta (b,a) )^{\alpha }} \bigl[J_{a^{+}}^{\alpha }f \bigl(a+e^{i \varphi }\eta (b,a) \bigr) +J_{ (a+e^{i \varphi }\eta (b,a) )^{-}}^{\alpha }f(a) \bigr] \biggr\vert \\ &\quad \leq \frac{e^{i \varphi }\eta (b,a)}{2} \biggl(\frac{2-2^{1-\alpha p}}{\alpha p+1} \biggr)^{\frac{1}{p}} \biggl( \int _{0}^{1} h(t) \biggr) ^{\frac{p-1}{p}} \biggl( \bigl\vert f'(a) \bigr\vert ^{\frac{p-1}{p}} +\frac{1- \lambda }{\lambda } \bigl\vert f'(b) \bigr\vert ^{\frac{p-1}{p}} \biggr). \end{aligned}$$

Proof

From (9) with \(w(t)=1\), we have
$$\begin{aligned} &T_{F,1}(u_{1},u_{2},u_{3}) \\ &\quad =u_{1}- \biggl( \int _{0}^{1} h(t)\,dt \biggr)u_{3} - \frac{1-\lambda }{ \lambda } \biggl( \int _{0}^{1} h(1-t)\,dt \biggr)u_{2} \\ &\quad =u_{1}- \biggl( \int _{0}^{1} h(t)\,dt \biggr)u_{3} - \frac{1-\lambda }{ \lambda } \biggl( \int _{0}^{1} h(t)\,dt \biggr)u_{2} \\ &\quad =u_{1}- \biggl( \int _{0}^{1} h(t)\,dt \biggr) \biggl(u_{2}+ \frac{1- \lambda }{\lambda }u_{3} \biggr) \end{aligned}$$
for \(u_{1}, u_{2}, u_{3}\in \mathbb{R}\). So, by Theorem 9, we have
$$\begin{aligned} 0 &\geq T_{F,1} \bigl(G_{1}(f,p), \bigl\vert f'(a) \bigr\vert ^{\frac{p-1}{p}}, \bigl\vert f'(b) \bigr\vert ^{\frac{p-1}{p}} \bigr) \\ &= \biggl(\frac{2}{e^{i \varphi }\eta (b,a)} \biggr)^{\frac{p}{p-1}} \biggl(\frac{\alpha p+1}{2-2^{1-\alpha p}} \biggr)^{\frac{1}{p-1}} \\ &\quad {}\times \biggl\vert \frac{f(a)+f (a+e^{i \varphi } \eta (b,a) )}{2}-\frac{ \varGamma (\alpha +1)}{2 (e^{i \varphi }\eta (b,a) )^{\alpha }} \bigl[J_{a^{+}}^{\alpha }f \bigl(a+e^{i \varphi }\eta (b,a) \bigr) +J_{ (a+e^{i \varphi }\eta (b,a) )^{-}}^{\alpha }f(a) \bigr] \biggr\vert \\ &\quad {}- \biggl( \int _{0}^{1} h(t)\,dt \biggr) \biggl( \bigl\vert f'(a) \bigr\vert ^{ \frac{p-1}{p}} +\frac{1-\lambda }{\lambda } \bigl\vert f'(b) \bigr\vert ^{ \frac{p-1}{p}} \biggr), \end{aligned}$$
that is,
$$\begin{aligned} & \biggl\vert \frac{f(a)+f (a+e^{i \varphi } \eta (b,a) )}{2}-\frac{ \varGamma (\alpha +1)}{2 (e^{i \varphi }\eta (b,a) )^{\alpha }} \bigl[J_{a^{+}}^{\alpha }f \bigl(a+e^{i \varphi }\eta (b,a) \bigr) +J_{ (a+e^{i \varphi }\eta (b,a) )^{-}}^{\alpha }f(a) \bigr] \biggr\vert \\ &\quad \leq \frac{e^{i \varphi }\eta (b,a)}{2} \biggl(\frac{2-2^{1-\alpha p}}{ \alpha p+1} \biggr)^{\frac{1}{p}}\\&\qquad{}\times \biggl( \int _{0}^{1} h(t) \bigl\vert (1-t)^{ \alpha }-t^{\alpha } \bigr\vert \,dt \biggr) \biggl( \bigl\vert f'(a) \bigr\vert ^{\frac{p-1}{p}} +\frac{1-\lambda }{\lambda } \bigl\vert f'(b) \bigr\vert ^{ \frac{p-1}{p}} \biggr). \end{aligned}$$
This completes the proof. □

Theorem 11

Let \(I\subseteq \mathbb{R}\) be an open invex set with respect to bifunction \(\eta : I\times I\to \mathbb{R}\), where \(\eta (b,a)>0\). Let \(f: [0,b]\to \mathbb{R}\) be a differentiable mapping. Suppose that \(|f'|^{{\frac{p}{p-1}}}\) is measurable, decreasing, \(\lambda _{\varphi }\)-preinvex function on I, and F-convex on \([a,b]\), for some \(F\in \mathcal{F}\) and \(|f'|\in L^{{\frac{p}{p-1}}}(a,b)\). Then
$$\begin{aligned} &T_{F,w} \bigl(G_{2}(f,p), \bigl\vert f'(a) \bigr\vert ^{\frac{p}{p-1}}, \bigl\vert f'(b) \bigr\vert ^{\frac{p}{p-1}} \bigr)+ \int _{0}^{1} L_{w(t)}\,dt\leq 0, \end{aligned}$$
(15)
where
$$\begin{aligned} G_{2}(f,p) &= \biggl(\frac{2}{e^{i \varphi }\eta (b,a)} \biggr)^{ \frac{p}{p-1}} \biggl( \frac{\alpha +1}{2-2^{1-\alpha }} \biggr)^{ \frac{1}{p-1}} \biggl\vert \frac{f(a)+f (a+e^{i \varphi } \eta (b,a) )}{2} \\ &\quad {}-\frac{ \varGamma (\alpha +1)}{2 (e^{i \varphi }\eta (b,a) )^{\alpha }} \bigl[J_{a^{+}}^{\alpha }f \bigl(a+e^{i \varphi }\eta (b,a) \bigr) +J_{ (a+e^{i \varphi }\eta (b,a) )^{-}}^{\alpha }f(a) \bigr] \biggr\vert ^{\frac{p}{p-1}} \end{aligned}$$
for \(w(t)= \vert (1-t)^{\alpha }-t^{\alpha } \vert \).

Proof

Since \(|f'|^{\frac{p}{p-1}}\) is F-convex, we have
$$\begin{aligned} F \bigl( \bigl\vert f' \bigl(a+(1-t)e^{i \varphi }\eta (b,a) \bigr) \bigr\vert ^{\frac{p}{p-1}}, \bigl\vert f'(a) \bigr\vert ^{\frac{p}{p-1}}, \bigl\vert f'(b) \bigr\vert ^{\frac{p}{p-1}},t \bigr) \leq 0,\quad t\in [0,1]. \end{aligned}$$
Using (A3) with \(w(t)= \vert (1-t)^{\alpha }-t^{\alpha } \vert \), we obtain
$$\begin{aligned} &F \bigl(w(t) \bigl\vert f' \bigl(a+(1-t)e^{i \varphi }\eta (b,a) \bigr) \bigr\vert ^{\frac{p}{p-1}}, w(t) \bigl\vert f'(a) \bigr\vert ^{\frac{p}{p-1}},w(t) \bigl\vert f'(b) \bigr\vert ^{\frac{p}{p-1}},t \bigr) +L_{w(t)}\leq 0,\\ &\quad t\in [0,1]. \end{aligned}$$
Integrating over \([0,1]\) and using axiom (A2), we obtain
$$\begin{aligned} &T_{F,w} \biggl( \int _{0}^{1} w(t) \bigl\vert f' \bigl(a+(1-t)e^{i \varphi } \eta (b,a) \bigr) \bigr\vert ^{\frac{p}{p-1}}\,dt, \bigl\vert f'(a) \bigr\vert ^{\frac{p}{p-1}}, \bigl\vert f'(b) \bigr\vert ^{\frac{p}{p-1}} \biggr)+ \int _{0}^{1} L_{w(t)}\,dt\leq 0,\\ &\quad t\in [0,1]. \end{aligned}$$
Using Lemma 1 and the power mean inequality, we get
$$\begin{aligned} & \biggl\vert \frac{f(a)+f (a+e^{i \varphi } \eta (b,a) )}{2}-\frac{ \varGamma (\alpha +1)}{2 (e^{i \varphi }\eta (b,a) )^{\alpha }} \bigl[J_{a^{+}}^{\alpha }f \bigl(a+e^{i \varphi }\eta (b,a) \bigr) +J_{ (a+e^{i \varphi }\eta (b,a) )^{-}}^{\alpha }f(a) \bigr] \biggr\vert \\ &\quad =\frac{e^{i \varphi }\eta (b,a)}{2} \int _{0}^{1} \bigl\vert (1-t)^{ \alpha }-t^{\alpha } \bigr\vert \bigl\vert f' \bigl(a+(1-t)e^{i \varphi } \eta (b,a) \bigr) \bigr\vert \,dt \\ &\quad \leq \frac{e^{i \varphi }\eta (b,a)}{2} \biggl( \int _{0}^{1} \bigl\vert (1-t)^{ \alpha }-t^{\alpha } \bigr\vert \,dt \biggr)^{\frac{1}{p}} \biggl( \int _{0} ^{1} w(t) \bigl\vert f' \bigl(a+(1-t)e^{i \varphi }\eta (b,a) \bigr) \bigr\vert ^{\frac{p}{p-1}}\,dt \biggr)^{\frac{p-1}{p}} \\ &\quad = \frac{e^{i \varphi }\eta (b,a)}{2} \biggl(\frac{2-2^{1-\alpha }}{ \alpha +1} \biggr)^{\frac{1}{p}} \biggl( \int _{0}^{1} w(t) \bigl\vert f' \bigl(a+(1-t)e^{i \varphi }\eta (b,a) \bigr) \bigr\vert ^{\frac{p}{p-1}}\,dt \biggr) ^{ \frac{p-1}{p}} \end{aligned}$$
or, equivalently,
$$\begin{aligned} & \biggl(\frac{2}{e^{i \varphi }\eta (b,a)} \biggr)^{\frac{p}{p-1}} \biggl(\frac{\alpha +1}{2-2^{1-\alpha }} \biggr)^{\frac{1}{p-1}} \biggl\vert \frac{f(a)+f (a+e^{i \varphi } \eta (b,a) )}{2} \\ &\qquad {}-\frac{ \varGamma (\alpha +1)}{2 (e^{i \varphi }\eta (b,a) )^{\alpha }} \bigl[J_{a^{+}}^{\alpha }f \bigl(a+e^{i \varphi }\eta (b,a) \bigr) +J_{ (a+e^{i \varphi }\eta (b,a) )^{-}}^{\alpha }f(a) \bigr] \biggr\vert ^{\frac{p}{p-1}} \\ &\quad \leq \int _{0}^{1} w(t) \bigl\vert f' \bigl(a+(1-t)e^{i \varphi }\eta (b,a) \bigr) \bigr\vert ^{\frac{p}{p-1}}\,dt. \end{aligned}$$
Because \(T_{F,w}\) is nondecreasing with respect to the first variable, we find
$$\begin{aligned} &T_{F,w} \bigl(G_{2}(f,p), \bigl\vert f'(a) \bigr\vert ^{{\frac{p}{p-1}}}, \bigl\vert f'(b) \bigr\vert ^{{\frac{p}{p-1}}} \bigr)+ \int _{0}^{1} L_{w(t)}\,dt \leq 0. \end{aligned}$$
This completes the proof. □

Remark 8

If we choose \(\eta (b,a)=b-a\) and \(\varphi =0\) in Theorem 11, we get
$$\begin{aligned} &T_{F,w} \biggl( \biggl(\frac{2}{b-a} \biggr)^{\frac{p}{p-1}} \biggl(\frac{ \alpha +1}{2-2^{1-\alpha }} \biggr)^{{\frac{1}{p-1}}} \biggl\vert \frac{f(a)+f(b)}{2} - \frac{\varGamma (\alpha +1))}{2(b-a)^{\alpha }} \bigl[J_{a^{+}}^{\alpha }f(b)+J_{b^{-}}^{\alpha }f(a) \bigr] \biggr\vert , \\ &\quad \bigl\vert f'(a) \bigr\vert ,\bigl\vert f'(b) \bigr\vert \biggr)+ \int _{0}^{1} L_{w(t)}\,dt\leq 0. \end{aligned}$$

Corollary 7

Under the assumptions of Theorem 11, if \(|f'|^{\frac{p}{p-1}}\) is ε-convex, we have
$$\begin{aligned} & \biggl\vert \frac{f(a)+f (a+e^{i \varphi } \eta (b,a) )}{2}-\frac{ \varGamma (\alpha +1))}{2 (e^{i \varphi }\eta (b,a) )^{\alpha }} \bigl[J_{a^{+}}^{\alpha }f \bigl(a+e^{i \varphi }\eta (b,a) \bigr) +J_{ (a+e^{i \varphi }\eta (b,a) )^{-}}^{\alpha }f(a) \bigr] \biggr\vert \\ &\quad \leq \frac{e^{i \varphi }\eta (b,a)}{2} \biggl(\frac{2-2^{1-\alpha }}{ \alpha +1} \biggr)^{\frac{1}{p}} \biggl[\frac{2^{\alpha }-1}{2^{ \alpha }(\alpha +1)} \bigl( \bigl\vert f'(a) \bigr\vert ^{\frac{p}{p-1}} + \bigl\vert f'(b) \bigr\vert ^{\frac{p}{p-1}} \bigr) +2\varepsilon \biggr] ^{\frac{p-1}{p}}. \end{aligned}$$

Proof

Using (5) with \(w(t)= \vert (1-t)^{\alpha }-t^{\alpha } \vert \), we get
$$\begin{aligned} \int _{0}^{1} L_{w(t)}\,dt &=\varepsilon \biggl(1-2\frac{2^{\alpha }-1}{2^{ \alpha }(\alpha +1)} \biggr). \end{aligned}$$
From (4) with \(w(t)= \vert (1-t)^{\alpha }-t^{\alpha } \vert \), we get
$$\begin{aligned} &T_{F,w}(u_{1},u_{2},u_{3}) \\ &\quad =u_{1}- \biggl( \int _{0}^{1} \bigl\vert (1-t)^{\alpha }-t^{\alpha } \bigr\vert t\,dt \biggr)u _{2} - \biggl( \int _{0}^{1} \bigl\vert (1-t)^{\alpha }-t^{\alpha } \bigr\vert (1-t)\,dt \biggr)u _{3}-\varepsilon \\ &\quad =u_{1}-\frac{2^{\alpha }-1}{2^{\alpha }(\alpha +1)}(u_{2}+u_{3})- \varepsilon , \end{aligned}$$
for \(u_{1}, u_{2}, u_{3}\in \mathbb{R}\). Hence, by Theorem 10, we have
$$\begin{aligned} 0 &\geq T_{F,w} \bigl(G_{2}(f,p), \bigl\vert f'(a) \bigr\vert ^{\frac{p}{p-1}}, \bigl\vert f'(b) \bigr\vert ^{\frac{p}{p-1}} \bigr)+ \int _{0}^{1} L_{w(t)}\,dt \\ &=G_{2}(f,p)-\frac{2^{\alpha }-1}{2^{\alpha }(\alpha +1)} \bigl( \bigl\vert f'(a) \bigr\vert ^{\frac{p}{p-1}}+ \bigl\vert f'(b) \bigr\vert ^{\frac{p}{p-1}} \bigr) -\varepsilon +\varepsilon \biggl(1-2\frac{2^{\alpha }-1}{2^{\alpha }(\alpha +1)} \biggr). \end{aligned}$$
This implies that
$$\begin{aligned} & \biggl\vert \frac{f(a)+f (a+e^{i \varphi } \eta (b,a) )}{2}-\frac{ \varGamma (\alpha +1))}{2 (e^{i \varphi }\eta (b,a) )^{\alpha }} \bigl[J_{a^{+}}^{\alpha }f \bigl(a+e^{i \varphi }\eta (b,a) \bigr) +J_{ (a+e^{i \varphi }\eta (b,a) )^{-}}^{\alpha }f(a) \bigr] \biggr\vert \\ &\quad \leq \frac{e^{i \varphi }\eta (b,a)}{2} \biggl(\frac{2-2^{1-\alpha }}{\alpha +1} \biggr)^{\frac{1}{p}} \biggl[\frac{2^{\alpha }-1}{2^{ \alpha }(\alpha +1)} \bigl( \bigl\vert f'(a) \bigr\vert ^{\frac{p}{p-1}} + \bigl\vert f'(b) \bigr\vert ^{\frac{p}{p-1}} \bigr)+2\varepsilon \biggr] ^{\frac{p-1}{p}}. \end{aligned}$$
This completes the proof. □

Remark 9

In Corollary 7, if we choose
  1. (a)
    \(\eta (b,a)=b-a\) and \(\varphi =0\), we get
    $$\begin{aligned} & \biggl\vert \frac{f(a)+f(b)}{2}-\frac{\varGamma (\alpha +1))}{2(b-a)^{\alpha }} \bigl[J_{a^{+}}^{\alpha }f(b)+J_{b^{-}}^{\alpha }f(a) \bigr] \biggr\vert \\ &\quad \leq \frac{b-a}{2} \biggl(\frac{2-2^{1-\alpha }}{\alpha +1} \biggr) ^{\frac{1}{p}} \biggl[\frac{2^{\alpha }-1}{2^{\alpha }(\alpha +1)} \bigl( \bigl\vert f'(a) \bigr\vert ^{\frac{p}{p-1}} + \bigl\vert f'(b) \bigr\vert ^{ \frac{p}{p-1}} \bigr)+2\varepsilon \biggr]^{\frac{p-1}{p}}. \end{aligned}$$
     
  2. (b)
    \(\eta (b,a)=b-a\), \(\varphi =0\), and \(\varepsilon =0\), we get
    $$\begin{aligned} & \biggl\vert \frac{f(a)+f(b)}{2}-\frac{\varGamma (\alpha +1))}{2(b-a)^{\alpha }} \bigl[J_{a^{+}}^{\alpha }f(b)+J_{b^{-}}^{\alpha }f(a) \bigr] \biggr\vert \\ &\quad \leq \frac{b-a}{2} \biggl(\frac{2-2^{1-\alpha }}{\alpha +1} \biggr) ^{\frac{1}{p}} \biggl[\frac{2^{\alpha }-1}{2^{\alpha }(\alpha +1)} \bigl( \bigl\vert f'(a) \bigr\vert ^{\frac{p}{p-1}} + \bigl\vert f'(b) \bigr\vert ^{ \frac{p}{p-1}} \bigr) \biggr]^{\frac{p-1}{p}}. \end{aligned}$$
     

Corollary 8

Under the assumptions of Theorem 11. If \(|f'|^{\frac{p-1}{p}}\) is \(\lambda _{\varphi }\)-preinvex, we have
$$\begin{aligned} & \biggl\vert \frac{f(a)+f (a+e^{i \varphi } \eta (b,a) )}{2}-\frac{ \varGamma (\alpha +1))}{2 (e^{i \varphi }\eta (b,a) )^{\alpha }} \bigl[J_{a^{+}}^{\alpha }f \bigl(a+e^{i \varphi }\eta (b,a) \bigr) +J_{ (a+e^{i \varphi }\eta (b,a) )^{-}}^{\alpha }f(a) \bigr] \biggr\vert \\ &\quad \leq \frac{e^{i \varphi }\eta (b,a)}{2} \biggl(\frac{2-2^{1-\alpha }}{ \alpha +1} \biggr)^{\frac{1}{p}}\\&\qquad{}\times \biggl[\frac{B_{\frac{1}{2}} (\frac{1}{2}, \alpha +\frac{1}{2} ) -B_{\frac{1}{2}} (\alpha + \frac{1}{2},\frac{1}{2} )}{2} \biggl( \bigl\vert f'(a) \bigr\vert ^{ \frac{p}{p-1}}+\frac{1-\lambda }{\lambda } \bigl\vert f'(b) \bigr\vert ^{ \frac{p}{p-1}} \biggr) \biggr]^{\frac{p-1}{p}}. \end{aligned}$$

Proof

Using (7) with \(w(t)= \vert (1-t)^{\alpha }-t^{\alpha } \vert \), we have
$$\begin{aligned} &T_{F,w}(u_{1},u_{2},u_{3}) \\ &\quad =u_{1}- \biggl( \int _{0}^{1} \frac{\sqrt{t}}{2\sqrt{1-t}} \bigl\vert (1-t)^{ \alpha }-t^{\alpha } \bigr\vert \,dt \biggr)u_{3}- \frac{1-\lambda }{\lambda } \biggl( \int _{0}^{1} \frac{\sqrt{1-t}}{2\sqrt{t}} \bigl\vert (1-t)^{ \alpha }-t^{\alpha } \bigr\vert \,dt \biggr)u_{2} \\ &\quad =u_{1}-\frac{1}{2} \biggl(B_{\frac{1}{2}} \biggl( \frac{1}{2},\alpha + \frac{1}{2} \biggr) -B_{\frac{1}{2}} \biggl( \alpha +\frac{1}{2}, \frac{1}{2} \biggr) \biggr) \biggl(u_{2}+ \frac{1-\lambda }{\lambda }u _{3} \biggr) \end{aligned}$$
for \(u_{1}, u_{2}, u_{3}\in \mathbb{R}\). Now, by Theorem 11, we have
$$\begin{aligned} 0 &\geq T_{F,w} \bigl(G_{2}(f,p), \bigl\vert f'(a) \bigr\vert ^{\frac{p-1}{p}}, \bigl\vert f'(b) \bigr\vert ^{\frac{p-1}{p}} \bigr) \\ &= \biggl(\frac{2}{e^{i \varphi }\eta (b,a)} \biggr)^{\frac{p}{p-1}} \biggl(\frac{\alpha +1}{2-2^{1-\alpha }} \biggr)^{\frac{1}{p-1}} \\ &\quad {}\times \biggl\vert \frac{f(a)+f (a+e^{i \varphi } \eta (b,a) )}{2}-\frac{ \varGamma (\alpha +1)}{2 (e^{i \varphi }\eta (b,a) )^{\alpha }} \bigl[J_{a^{+}}^{\alpha }f \bigl(a+e^{i \varphi }\eta (b,a) \bigr) +J_{ (a+e^{i \varphi }\eta (b,a) )^{-}}^{\alpha }f(a) \bigr] \biggr\vert \\ &\quad {}-\frac{1}{2} \biggl(B_{\frac{1}{2}} \biggl(\frac{1}{2}, \alpha + \frac{1}{2} \biggr) -B_{\frac{1}{2}} \biggl(\alpha + \frac{1}{2}, \frac{1}{2} \biggr) \biggr) \biggl( \bigl\vert f'(a) \bigr\vert ^{ \frac{p-1}{p}} +\frac{1-\lambda }{\lambda } \bigl\vert f'(b) \bigr\vert ^{ \frac{p-1}{p}} \biggr). \end{aligned}$$
This leads to
$$\begin{aligned} & \biggl\vert \frac{f(a)+f (a+e^{i \varphi } \eta (b,a) )}{2}-\frac{ \varGamma (\alpha +1)}{2 (e^{i \varphi }\eta (b,a) )^{\alpha }} \bigl[J_{a^{+}}^{\alpha }f \bigl(a+e^{i \varphi }\eta (b,a) \bigr) +J_{ (a+e^{i \varphi }\eta (b,a) )^{-}}^{\alpha }f(a) \bigr] \biggr\vert \\ &\quad \leq \frac{e^{i \varphi }\eta (b,a)}{2} \biggl(\frac{2-2^{1-\alpha }}{ \alpha +1} \biggr)^{\frac{1}{p}}\\&\qquad{}\times \biggl[\frac{B_{\frac{1}{2}} (\frac{1}{2}, \alpha +\frac{1}{2} ) -B_{\frac{1}{2}} (\alpha + \frac{1}{2},\frac{1}{2} )}{2} \biggl( \bigl\vert f'(a) \bigr\vert ^{ \frac{p}{p-1}}+\frac{1-\lambda }{\lambda } \bigl\vert f'(b) \bigr\vert ^{ \frac{p}{p-1}} \biggr) \biggr]^{\frac{p-1}{p}}. \end{aligned}$$
Thus, the proof is done. □

Remark 10

In Corollary 8, if we choose
  1. (a)
    \(\eta (b,a)=b-a\) and \(\varphi =0\), we get
    $$\begin{aligned} & \biggl\vert \frac{f(a)+f(b)}{2}-\frac{\varGamma (\alpha +1))}{2(b-a)^{\alpha }} \bigl[J_{a^{+}}^{\alpha }f(b)+J_{b^{-}}^{\alpha }f(a) \bigr] \biggr\vert \\ &\quad \leq \frac{b-a}{2} \biggl(\frac{2-2^{1-\alpha }}{\alpha +1} \biggr) ^{\frac{1}{p}}\\&\qquad{}\times \biggl[\frac{B_{\frac{1}{2}} (\frac{1}{2},\alpha +\frac{1}{2} ) -B_{\frac{1}{2}} (\alpha +\frac{1}{2}, \frac{1}{2} )}{2} \biggl( \bigl\vert f'(a) \bigr\vert ^{\frac{p}{p-1}}+\frac{1- \lambda }{\lambda } \bigl\vert f'(b) \bigr\vert ^{\frac{p}{p-1}} \biggr) \biggr] ^{\frac{p-1}{p}}. \end{aligned}$$
     
  2. (b)
    \(\eta (b,a)=b-a\), \(\varphi =0\), and \(\lambda =\frac{1}{2}\), we get
    $$\begin{aligned} & \biggl\vert \frac{f(a)+f(b)}{2}-\frac{\varGamma (\alpha +1))}{2(b-a)^{\alpha }} \bigl[J_{a^{+}}^{\alpha }f(b)+J_{b^{-}}^{\alpha }f(a) \bigr] \biggr\vert \\ &\quad \leq \frac{b-a}{2} \biggl(\frac{2-2^{1-\alpha }}{\alpha +1} \biggr) ^{\frac{1}{p}}\\&\qquad{}\times \biggl[\frac{B_{\frac{1}{2}} (\frac{1}{2},\alpha +\frac{1}{2} ) -B_{\frac{1}{2}} (\alpha +\frac{1}{2}, \frac{1}{2} )}{2} \bigl( \bigl\vert f'(a) \bigr\vert ^{\frac{p}{p-1}}+ \bigl\vert f'(b) \bigr\vert ^{\frac{p}{p-1}} \bigr) \biggr]^{\frac{p-1}{p}}. \end{aligned}$$
     

Corollary 9

Under the assumptions of Theorem 11. If \(|f'|^{\frac{p}{p-1}}\) is h-convex, we have
$$\begin{aligned} & \biggl\vert \frac{f(a)+f (a+e^{i \varphi } \eta (b,a) )}{2}-\frac{ \varGamma (\alpha +1))}{2 (e^{i \varphi }\eta (b,a) )^{\alpha }} \bigl[J_{a^{+}}^{\alpha }f \bigl(a+e^{i \varphi }\eta (b,a) \bigr) +J_{ (a+e^{i \varphi }\eta (b,a) )^{-}}^{\alpha }f(a) \bigr] \biggr\vert \\ &\quad \leq \frac{e^{i \varphi }\eta (b,a)}{2} \biggl(\frac{2-2^{1-\alpha }}{\alpha +1} \biggr)^{\frac{1}{p}}\\&\qquad{}\times \biggl( \int _{0}^{1} h(t) \bigl\vert (1-t)^{ \alpha }-t^{\alpha } \bigr\vert \,dt \biggr)^{\frac{p-1}{p}} \biggl( \bigl\vert f'(a) \bigr\vert ^{\frac{p}{p-1}}+\frac{1-\lambda }{\lambda } \bigl\vert f'(b) \bigr\vert ^{ \frac{p}{p-1}} \biggr)^{\frac{p-1}{p}}. \end{aligned}$$

Proof

From (9) with \(w(t)= \vert (1-t)^{\alpha }-t^{\alpha } \vert \), we have
$$\begin{aligned} &T_{F,w}(u_{1},u_{2},u_{3}) \\ &\quad =u_{1}- \biggl( \int _{0}^{1} h(t) \bigl\vert (1-t)^{\alpha }-t^{\alpha } \bigr\vert \,dt \biggr)u _{3} -\frac{1-\lambda }{\lambda } \biggl( \int _{0}^{1} h(1-t) \bigl\vert (1-t)^{ \alpha }-t^{\alpha } \bigr\vert \,dt \biggr)u_{2} \\ &\quad =u_{1}- \biggl( \int _{0}^{1} h(t) \bigl\vert (1-t)^{\alpha }-t^{\alpha } \bigr\vert \,dt \biggr)u _{3} -\frac{1-\lambda }{\lambda } \biggl( \int _{0}^{1} h(t) \bigl\vert (1-t)^{ \alpha }-t^{\alpha } \bigr\vert \,dt \biggr)u_{2} \\ &\quad =u_{1}- \biggl( \int _{0}^{1} h(t) \bigl\vert (1-t)^{\alpha }-t^{\alpha } \bigr\vert \,dt \biggr) \biggl(u_{2}+\frac{1- \lambda }{\lambda }u_{3} \biggr) \end{aligned}$$
for \(u_{1}, u_{2}, u_{3}\in \mathbb{R}\). So, by Theorem 11, we have
$$\begin{aligned} 0 &\geq T_{F,w} \bigl(G_{2}(f,p), \bigl\vert f'(a) \bigr\vert ^{\frac{p-1}{p}}, \bigl\vert f'(b) \bigr\vert ^{\frac{p-1}{p}} \bigr) \\ &= \biggl(\frac{2}{e^{i \varphi }\eta (b,a)} \biggr)^{\frac{p}{p-1}} \biggl(\frac{\alpha +1}{2-2^{1-\alpha }} \biggr)^{\frac{1}{p-1}} \\ &\quad {}\times \biggl\vert \frac{f(a)+f (a+e^{i \varphi } \eta (b,a) )}{2}-\frac{ \varGamma (\alpha +1)}{2 (e^{i \varphi }\eta (b,a) )^{\alpha }} \bigl[J_{a^{+}}^{\alpha }f \bigl(a+e^{i \varphi }\eta (b,a) \bigr) +J_{ (a+e^{i \varphi }\eta (b,a) )^{-}}^{\alpha }f(a) \bigr] \biggr\vert \\ &\quad {}- \biggl( \int _{0}^{1} h(t)\,dt \biggr) \biggl( \bigl\vert f'(a) \bigr\vert ^{ \frac{p-1}{p}} +\frac{1-\lambda }{\lambda } \bigl\vert f'(b) \bigr\vert ^{ \frac{p-1}{p}} \biggr), \end{aligned}$$
that is,
$$\begin{aligned} & \biggl\vert \frac{f(a)+f (a+e^{i \varphi } \eta (b,a) )}{2}-\frac{ \varGamma (\alpha +1)}{2 (e^{i \varphi }\eta (b,a) )^{\alpha }} \bigl[J_{a^{+}}^{\alpha }f \bigl(a+e^{i \varphi }\eta (b,a) \bigr) +J_{ (a+e^{i \varphi }\eta (b,a) )^{-}}^{\alpha }f(a) \bigr] \biggr\vert \\ &\quad \leq \frac{e^{i \varphi }\eta (b,a)}{2} \biggl(\frac{2-2^{1-\alpha }}{\alpha +1} \biggr)^{\frac{1}{p}}\\&\qquad{}\times \biggl( \int _{0}^{1} h(t) \bigl\vert (1-t)^{ \alpha }-t^{\alpha } \bigr\vert \,dt \biggr)^{\frac{p-1}{p}} \biggl( \bigl\vert f'(a) \bigr\vert ^{\frac{p}{p-1}}+\frac{1-\lambda }{\lambda } \bigl\vert f'(b) \bigr\vert ^{ \frac{p}{p-1}} \biggr)^{\frac{p-1}{p}}. \end{aligned}$$
This completes the proof. □

4 Trapezoid type inequalities for twice differentiable functions

In this section, we establish some trapezoid type inequalities for functions whose second derivatives absolutely values are

Theorem 12

Let \(f: [0,b]\to \mathbb{R}\) be a differentiable mapping and \(|f''|\) is measurable, decreasing, \(\lambda _{\varphi }\)-preinvex function on \([0,b]\) for \(0\leq a< b\), \(\eta (b,a)>0\) and \(\alpha >0\). Suppose that F-convex on \([0,b]\), for some \(F\in \mathcal{F}\) and the function \(t\in (0,1)\to L_{w(t)}\) belongs to \(L^{1}(0,1)\), where \(w(t)=1-(1-t)^{\alpha +1}-t^{\alpha +1}\). Then
$$\begin{aligned} \begin{aligned}[b] &T_{F,w} \biggl(\frac{2(\alpha +1)}{ (e^{i \varphi }\eta (b,a) ) ^{2}} \biggl\vert \frac{f(a)+f (a+e^{i \varphi }\eta (b,a) )}{2} \\&\quad{}-\frac{\varGamma (\alpha +1)}{2 (e^{i \varphi }\eta (b,a) ) ^{\alpha }} \bigl[J_{a^{+}}^{\alpha }f \bigl(a+e^{i \varphi }\eta (b,a) \bigr) +J_{ (a+e^{i \varphi }\eta (b,a) ) ^{-}}^{\alpha }f(a) \bigr] \biggr\vert ,\bigl\vert f''(a) \bigr\vert , \bigl\vert f''(b) \bigr\vert \biggr) \\ &\quad {} + \int _{0}^{1} L _{w(t)}\,dt\leq 0. \end{aligned} \end{aligned}$$
(16)

Proof

Since \(|f''|\) is F-convex, we can see that
$$\begin{aligned} F \bigl( \bigl\vert f'' \bigl(a+(1-t)e^{i \varphi } \eta (b,a) \bigr) \bigr\vert , \bigl\vert f''(a) \bigr\vert , \bigl\vert f''(b) \bigr\vert ,t \bigr)\leq 0,\quad t \in [0,1]. \end{aligned}$$
Multiplying this inequality by \(w(t)=1-(1-t)^{\alpha +1}-t^{\alpha +1}\) and using axiom (A3), we have
$$\begin{aligned} F \bigl(w(t) \bigl\vert f'' \bigl(a+(1-t)e^{i \varphi } \eta (b,a) \bigr) \bigr\vert ,w(t) \bigl\vert f''(a) \bigr\vert ,w(t) \bigl\vert f''(b) \bigr\vert ,t \bigr)+L_{w(t)}\leq 0,\quad t\in [0,1]. \end{aligned}$$
Integrating over \([0,1]\) and using axiom (A2), we get
$$\begin{aligned} &T_{F,w} \biggl( \int _{0}^{1} w(t) \bigl\vert f'' \bigl(a+(1-t)e^{i \varphi } \eta (b,a) \bigr) \bigr\vert \,dt, \bigl\vert f''(a) \bigr\vert , \bigl\vert f''(b) \bigr\vert ,t \biggr) + \int _{0}^{1} L_{w(t)}\,dt\leq 0,\\ &\quad t\in [0,1]. \end{aligned}$$
Using Lemma 2, we have
$$\begin{aligned} &\frac{2(\alpha +1)}{ (e^{i \varphi }\eta (b,a) )^{2}} \biggl\vert \frac{f(a)+f (a+e^{i \varphi }\eta (b,a) )}{2} \\ &\qquad {}-\frac{ \varGamma (\alpha +1))}{2 (e^{i \varphi }\eta (b,a) )^{\alpha }} \bigl[J_{a^{+}}^{\alpha }f \bigl(a+e^{i \varphi }\eta (b,a) \bigr) +J_{ (a+e^{i \varphi }\eta (b,a) )^{-}}^{\alpha }f(a) \bigr] \biggr\vert \\ &\quad \leq \int _{0}^{1} \bigl[1-(1-t)^{\alpha +1}-t^{\alpha +1} \bigr] \bigl\vert f'' \bigl(a+(1-t)e^{i \varphi } \eta (b,a) \bigr) \bigr\vert \,dt. \end{aligned}$$
Because \(T_{F,w}\) is nondecreasing with respect to the first variable so that
$$\begin{aligned} &T_{F,w} \biggl(\frac{2(\alpha +1)}{ (e^{i \varphi }\eta (b,a) ) ^{2}} \biggl\vert \frac{f(a)+ f (a+e^{i \varphi }\eta (b,a) )}{2}\\ &\quad {}- \frac{ \varGamma (\alpha +1)}{2 (e^{i \varphi } \eta (b,a) )^{\alpha }} \bigl[J_{a^{+}}^{\alpha }f \bigl(a+e^{i \varphi } \eta (b,a) \bigr) +J_{ (a+e^{i \varphi }\eta (b,a) )^{-}}^{\alpha }f(a) \bigr] \biggr\vert ,\bigl\vert f''(a) \bigr\vert , \bigl\vert f''(b) \bigr\vert \biggr) \\ &\quad {} + \int _{0}^{1} L _{w(t)}\,dt\leq 0,\quad t\in [0,1]. \end{aligned}$$
This completes the proof. □

Remark 11

By taking \(\eta (b,a)=b-a\) and \(\varphi =0\) in Theorem 12, we obtain
$$\begin{aligned} &T_{F,w} \biggl(\frac{2(\alpha +1)}{(b-a)^{2}} \biggl\vert \frac{f(a)+f(b)}{2} - \frac{\varGamma (\alpha +1))}{2(b-a)^{\alpha }} \bigl[J_{a^{+}}^{\alpha }f(b)+J_{b^{-}}^{\alpha }f(a) \bigr] \biggr\vert , \bigl\vert f''(a) \bigr\vert , \bigl\vert f''(b) \bigr\vert \biggr)\\ &\quad {}+ \int _{0}^{1} L_{w(t)}\,dt\leq 0. \end{aligned}$$

Corollary 10

Under the assumptions of Theorem 12, if \(|f''|\) is ε-convex, then
$$\begin{aligned} & \biggl\vert \frac{f(a)+f (a+e^{i \varphi } \eta (b,a) )}{2}-\frac{ \varGamma (\alpha +1))}{2 (e^{i \varphi }\eta (b,a) )^{\alpha }} \bigl[J_{a^{+}}^{\alpha }f \bigl(a+e^{i \varphi }\eta (b,a) \bigr) +J_{ (a+e^{i \varphi }\eta (b,a) )^{-}}^{\alpha }f(a) \bigr] \biggr\vert \\ &\quad \leq \frac{\alpha (e^{i \varphi }\eta (b,a) )^{2}}{4( \alpha +1)(\alpha +2)} \bigl( \bigl\vert f''(a) \bigr\vert + \bigl\vert f''(b) \bigr\vert +2 \varepsilon \bigr). \end{aligned}$$

Proof

Using (5) with \(w(t)=1-(1-t)^{\alpha +1}-t^{\alpha +1}\), we find
$$\begin{aligned} \int _{0}^{1} L_{w(t)}\,dt &=\varepsilon \int _{0}^{1} \bigl((1-t)^{ \alpha +1}+t^{\alpha +1} \bigr)\,dt =\frac{2\varepsilon }{\alpha +2}. \end{aligned}$$
(17)
With \(w(t)=1-(1-t)^{\alpha +1}-t^{\alpha +1}\), Eq. (4) gives
$$\begin{aligned} &T_{F,w}(u_{1},u_{2},u_{3}) \\ &\quad =u_{1}- \biggl( \int _{0}^{1} t \bigl[1-(1-t)^{\alpha +1}-t^{\alpha +1} \bigr]\,dt \biggr)u_{2} \\ &\qquad{}- \biggl( \int _{0}^{1} (1-t) \bigl[1-(1-t)^{ \alpha +1}-t^{\alpha +1} \bigr]\,dt \biggr)u_{3}-\varepsilon \\ &\quad =u_{1}-\frac{\alpha }{2(\alpha +2)}(u_{2}+u_{3})- \varepsilon , \end{aligned}$$
(18)
for \(u_{1}, u_{2}, u_{3}\in \mathbb{R}\). Hence, by Theorem 12, we have
$$\begin{aligned} 0 &\geq T_{F,w} \biggl(\frac{2(\alpha +1)}{ (e^{i \varphi }\eta (b,a) ) ^{2}} \biggl\vert \frac{f(a)+f (a+e^{i \varphi }\eta (b,a) )}{2} \\ &\quad {}-\frac{\varGamma (\alpha +1)}{2 (e^{i \varphi }\eta (b,a) ) ^{\alpha }} \bigl[J_{a^{+}}^{\alpha }f \bigl(a+e^{i \varphi }\eta (b,a) \bigr) +J_{ (a+e^{i \varphi }\eta (b,a) ) ^{-}}^{\alpha }f(a) \bigr] \biggr\vert ,\bigl\vert f''(a) \bigr\vert , \bigl\vert f''(b) \bigr\vert \biggr) \\ &\quad {}+ \int _{0}^{1} L _{w(t)}\,dt \\ &=\frac{2(\alpha +1)}{ (e^{i \varphi }\eta (b,a) )^{2}} \biggl\vert \frac{f(a)+f (a+e^{i \varphi }\eta (b,a) )}{2}\\ &\quad {} -\frac{ \varGamma (\alpha +1)}{2 (e^{i \varphi }\eta (b,a) )^{\alpha }} \bigl[J_{a^{+}}^{\alpha }f \bigl(a+e^{i \varphi }\eta (b,a) \bigr) +J_{ (a+e^{i \varphi }\eta (b,a) )^{-}}^{\alpha }f(a) \bigr] \biggr\vert \\ &\quad{}-\frac{\alpha }{2(\alpha +2)} \bigl( \bigl\vert f''(a) \bigr\vert + \bigl\vert f''(b) \bigr\vert \bigr) - \varepsilon +\frac{2\varepsilon }{\alpha +2} \\ &=\frac{2(\alpha +1)}{ (e^{i \varphi }\eta (b,a) )^{2}} \biggl\vert \frac{f(a)+f (a+e^{i \varphi }\eta (b,a) )}{2}\\ &\quad {} -\frac{ \varGamma (\alpha +1)}{2 (e^{i \varphi }\eta (b,a) )^{\alpha }} \bigl[J_{a^{+}}^{\alpha }f \bigl(a+e^{i \varphi }\eta (b,a) \bigr) +J_{ (a+e^{i \varphi }\eta (b,a) )^{-}}^{\alpha }f(a) \bigr] \biggr\vert \\ &\quad{}-\frac{\alpha }{2(\alpha +2)} \bigl( \bigl\vert f''(a) \bigr\vert + \bigl\vert f''(b) \bigr\vert \bigr) -\frac{ \alpha }{\alpha +2}\varepsilon . \end{aligned}$$
This completes the proof. □

Remark 12

In Corollary 10, if we take
  1. (a)
    \(\eta (b,a)=b-a\) and \(\varphi =0\), we get
    $$\begin{aligned} & \biggl\vert \frac{f(a)+f(b)}{2}-\frac{\varGamma (\alpha +1))}{2(b-a)^{\alpha }} \bigl[J_{a^{+}}^{\alpha }f(b)+J_{b^{-}}^{\alpha }f(a) \bigr] \biggr\vert \\ &\quad \leq \frac{\alpha (b-a)^{2}}{4(\alpha +1)(\alpha +2)} \bigl( \bigl\vert f''(a) \bigr\vert + \bigl\vert f''(b) \bigr\vert +2\varepsilon \bigr). \end{aligned}$$
     
  2. (b)
    \(\eta (b,a)=b-a\), \(\varphi =0\), and \(\varepsilon =0\), we get
    $$\begin{aligned} & \biggl\vert \frac{f(a)+f(b)}{2}-\frac{\varGamma (\alpha +1))}{2(b-a)^{\alpha }} \bigl[J_{a^{+}}^{\alpha }f(b)+J_{b^{-}}^{\alpha }f(a) \bigr] \biggr\vert \leq \frac{\alpha (b-a)^{2}}{4(\alpha +1)(\alpha +2)} \bigl( \bigl\vert f''(a) \bigr\vert + \bigl\vert f''(b) \bigr\vert \bigr). \end{aligned}$$
     

Corollary 11

Under the assumptions of Theorem 12, if \(|f''|\) is \(\lambda _{\varphi }\)-preinvex, then
$$\begin{aligned} & \biggl\vert \frac{f(a)+f (a+e^{i \varphi } \eta (b,a) )}{2}-\frac{ \varGamma (\alpha +1))}{2 (e^{i \varphi }\eta (b,a) )^{\alpha }} \bigl[J_{a^{+}}^{\alpha }f \bigl(a+e^{i \varphi }\eta (b,a) \bigr) +J_{ (a+e^{i \varphi }\eta (b,a) )^{-}}^{\alpha }f(a) \bigr] \biggr\vert \\ &\quad \leq \frac{ (e^{i \varphi }\eta (b,a) )^{2}}{2(\alpha +1)} \biggl[\frac{\pi }{2}-\beta \biggl( \frac{1}{2},\alpha +\frac{5}{2} \biggr) -\beta \biggl( \frac{3}{2},\alpha +\frac{3}{2} \biggr) \biggr] \biggl( \bigl\vert f''(a) \bigr\vert +\frac{1- \lambda }{\lambda } \bigl\vert f''(b) \bigr\vert \biggr). \end{aligned}$$

Proof

Using (7) with \(w(t)=1-(1-t)^{\alpha +1}-t^{\alpha +1}\), we have
$$\begin{aligned} \begin{aligned}[b] &T_{F,w}(u_{1},u_{2},u_{3}) \\ &\quad =u_{1}- \biggl( \int _{0}^{1} \frac{\sqrt{t}}{2\sqrt{1-t}} \bigl[1-(1-t)^{ \alpha +1}-t^{\alpha +1} \bigr]\,dt \biggr)u_{3}\\ &\qquad{}- \frac{1-\lambda }{ \lambda } \biggl( \int _{0}^{1} \frac{\sqrt{1-t}}{2\sqrt{t}} \bigl[1-(1-t)^{ \alpha +1}-t^{\alpha +1} \bigr]\,dt \biggr)u_{2} \\ &\quad =u_{1}- \biggl[\frac{\pi }{2}-\beta \biggl(\frac{1}{2}, \alpha + \frac{5}{2} \biggr) -\beta \biggl(\frac{3}{2},\alpha + \frac{3}{2} \biggr) \biggr] \biggl(u _{2}+\frac{1-\lambda }{\lambda }u_{3} \biggr) \end{aligned} \end{aligned}$$
(19)
for \(u_{1}, u_{2}, u_{3}\in \mathbb{R}\). Hence, by Theorem 12, we get
$$\begin{aligned} 0 &\geq T_{F,w} \biggl(\frac{2(\alpha +1)}{ (e^{i \varphi }\eta (b,a) ) ^{2}} \biggl\vert \frac{f(a)+f (a+e^{i \varphi }\eta (b,a) )}{2} \\ &\quad {}-\frac{\varGamma (\alpha +1)}{2 (e^{i \varphi }\eta (b,a) ) ^{\alpha }} \bigl[J_{a^{+}}^{\alpha }f \bigl(a+e^{i \varphi }\eta (b,a) \bigr) +J_{ (a+e^{i \varphi }\eta (b,a) ) ^{-}}^{\alpha }f(a) \bigr] \biggr\vert ,\bigl\vert f''(a) \bigr\vert , \bigl\vert f''(b) \bigr\vert \biggr) \\ & \quad {}+ \int _{0}^{1} L _{w(t)}\,dt \\ &=\frac{2(\alpha +1)}{ (e^{i \varphi }\eta (b,a) )^{2}} \biggl\vert \frac{f(a)+f (a+e^{i \varphi }\eta (b,a) )}{2} \\ &\quad {}-\frac{ \varGamma (\alpha +1)}{2 (e^{i \varphi }\eta (b,a) )^{\alpha }} \bigl[J_{a^{+}}^{\alpha }f \bigl(a+e^{i \varphi }\eta (b,a) \bigr) +J_{ (a+e^{i \varphi }\eta (b,a) )^{-}}^{\alpha }f(a) \bigr] \biggr\vert \\ &\quad{}- \biggl[\frac{\pi }{2}-\beta \biggl(\frac{1}{2},\alpha + \frac{5}{2} \biggr) - \beta \biggl(\frac{3}{2},\alpha + \frac{3}{2} \biggr) \biggr] \biggl( \bigl\vert f''(a ) \bigr\vert +\frac{1- \lambda }{\lambda } \bigl\vert f''(b) \bigr\vert \biggr). \end{aligned}$$
This leads to
$$\begin{aligned} & \biggl\vert \frac{f(a)+f (a+e^{i \varphi } \eta (b,a) )}{2}-\frac{ \varGamma (\alpha +1)}{2 (e^{i \varphi }\eta (b,a) )^{\alpha }} \bigl[J_{a^{+}}^{\alpha }f \bigl(a+e^{i \varphi }\eta (b,a) \bigr) +J_{ (a+e^{i \varphi }\eta (b,a) )^{-}}^{\alpha }f(a) \bigr] \biggr\vert \\ &\leq \frac{ (e^{i \varphi }\eta (b,a) )^{2}}{2(\alpha +1)} \biggl[\frac{\pi }{2}-\beta \biggl( \frac{1}{2},\alpha +\frac{5}{2} \biggr) -\beta \biggl( \frac{3}{2},\alpha +\frac{3}{2} \biggr) \biggr] \biggl( \bigl\vert f''(a) \bigr\vert +\frac{1- \lambda }{\lambda } \bigl\vert f''(b) \bigr\vert \biggr). \end{aligned}$$
Thus, the proof is completed. □

Remark 13

In Corollary 11, if we choose
  1. (a)
    \(\eta (b,a)=b-a\) and \(\varphi =0\), we get
    $$\begin{aligned} & \biggl\vert \frac{f(a)+f(b)}{2}-\frac{\varGamma (\alpha +1))}{2(b-a)^{\alpha }} \bigl[J_{a^{+}}^{\alpha }f(b)+J_{b^{-}}^{\alpha }f(a) \bigr] \biggr\vert \\ &\leq \frac{(b-a)^{2}}{2(\alpha +1)} \biggl[\frac{\pi }{2}-\beta \biggl( \frac{1}{2},\alpha +\frac{5}{2} \biggr) -\beta \biggl( \frac{3}{2}, \alpha +\frac{3}{2} \biggr) \biggr] \biggl( \bigl\vert f''(a) \bigr\vert +\frac{1- \lambda }{\lambda } \bigl\vert f''(b) \bigr\vert \biggr). \end{aligned}$$
     
  2. (b)
    \(\eta (b,a)=b-a\), \(\varphi =0\), and \(\lambda =\frac{1}{2}\), we get
    $$\begin{aligned} & \biggl\vert \frac{f(a)+f(b)}{2}-\frac{\varGamma (\alpha +1))}{2(b-a)^{\alpha }} \bigl[J_{a^{+}}^{\alpha }f(b)+J_{b^{-}}^{\alpha }f(a) \bigr] \biggr\vert \\ &\quad \leq \frac{(b-a)^{2}}{2(\alpha +1)} \biggl[\frac{\pi }{2}-\beta \biggl(\frac{1}{2}, \alpha +\frac{5}{2} \biggr) -\beta \biggl( \frac{3}{2},\alpha + \frac{3}{2} \biggr) \biggr] \bigl( \bigl\vert f''(a) \bigr\vert + \bigl\vert f''(b) \bigr\vert \bigr). \end{aligned}$$
     

Corollary 12

Under the assumptions of Theorem 12, if \(|f''|\) is h-convex, then we have
$$\begin{aligned} & \biggl\vert \frac{f(a)+f (a+e^{i \varphi } \eta (b,a) )}{2}-\frac{ \varGamma (\alpha +1))}{2 (e^{i \varphi }\eta (b,a) )^{\alpha }} \bigl[J_{a^{+}}^{\alpha }f \bigl(a+e^{i \varphi }\eta (b,a) \bigr) +J_{ (a+e^{i \varphi }\eta (b,a) )^{-}}^{\alpha }f(a) \bigr] \biggr\vert \\ &\quad \leq \frac{ (e^{i \varphi }\eta (b,a) )^{2}}{2(\alpha +1)} \biggl( \int _{0}^{1} h(t) \bigl[1-(1-t)^{\alpha +1}-t^{\alpha +1} \bigr]\,dt \biggr) \biggl( \bigl\vert f''(a) \bigr\vert +\frac{1- \lambda }{\lambda } \bigl\vert f''(b) \bigr\vert \biggr). \end{aligned}$$

Proof

Using (9) with \(w(t)=1-(1-t)^{\alpha +1}-t^{\alpha +1}\), we obtain
$$\begin{aligned} &T_{F,w}(u_{1},u_{2},u_{3}) \\ &\quad =u_{1}- \biggl( \int _{0}^{1} h(t) \bigl[1-(1-t)^{\alpha +1}-t^{\alpha +1} \bigr]\,dt \biggr)u_{3} \\ &\qquad {}-\frac{1-\lambda }{\lambda } \biggl( \int _{0}^{1} h(1-t) \bigl[1-(1-t)^{\alpha +1}-t^{\alpha +1} \bigr]\,dt \biggr)u _{2} \\ &\quad =u_{1}- \biggl( \int _{0}^{1} h(t) \bigl[1-(1-t)^{\alpha +1}-t^{\alpha +1} \bigr]\,dt \biggr)u_{3} \\ &\qquad {}-\frac{1-\lambda }{\lambda } \biggl( \int _{0}^{1} h(t) \bigl[1-(1-t)^{\alpha +1}-t^{\alpha +1} \bigr]\,dt \biggr)u _{2} \\ &\quad =u_{1}- \biggl( \int _{0}^{1} h(t) \bigl[1-(1-t)^{\alpha +1}-t^{\alpha +1} \bigr]\,dt \biggr) \biggl(u_{2}+\frac{1-\lambda }{\lambda }u_{3} \biggr) \end{aligned}$$
for \(u_{1}, u_{2}, u_{3}\in \mathbb{R}\), so Theorem 12 implies that
$$\begin{aligned} 0 &\geq T_{F,w} \biggl(\frac{2(\alpha +1)}{ (e^{i \varphi }\eta (b,a) ) ^{2}} \biggl\vert \frac{f(a)+f (a+e^{i \varphi } \eta (b,a) )}{2}\\ &\quad {}-\frac{ \varGamma (\alpha +1)}{2 (e^{i \varphi }\eta (b,a) )^{\alpha }} \bigl[J_{a^{+}}^{\alpha }f \bigl(a+e^{i \varphi }\eta (b,a) \bigr) +J_{ (a+e^{i \varphi }\eta (b,a) )^{-}}^{\alpha }f(a) \bigr] \biggr\vert ,\bigl\vert f''(a) \bigr\vert , \bigl\vert f''(b) \bigr\vert \biggr) \\ & \quad {}+ \int _{0}^{1} L _{w(t)}\,dt \\ &=\frac{2(\alpha +1)}{ (e^{i \varphi }\eta (b,a) )^{2}} \biggl\vert \frac{f(a)+f (a+e^{i \varphi } \eta (b,a) )}{2}\\ &\quad {}-\frac{ \varGamma (\alpha +1)}{2 (e^{i \varphi }\eta (b,a) )^{\alpha }} \bigl[J_{a^{+}}^{\alpha }f \bigl(a+e^{i \varphi }\eta (b,a) \bigr) +J_{ (a+e^{i \varphi }\eta (b,a) )^{-}}^{\alpha }f(a) \bigr] \biggr\vert \\ &\quad{}- \biggl( \int _{0}^{1} h(t) \bigl[1-(1-t)^{\alpha +1}-t^{\alpha +1} \bigr]\,dt \biggr) \biggl( \bigl\vert f''(a) \bigr\vert +\frac{1- \lambda }{\lambda } \bigl\vert f''(b) \bigr\vert \biggr), \end{aligned}$$
which can be written as
$$\begin{aligned} & \biggl\vert \frac{f(a)+f (a+e^{i \varphi } \eta (b,a) )}{2}-\frac{ \varGamma (\alpha +1)}{2 (e^{i \varphi }\eta (b,a) )^{\alpha }} \bigl[J_{a^{+}}^{\alpha }f \bigl(a+e^{i \varphi }\eta (b,a) \bigr) +J_{ (a+e^{i \varphi }\eta (b,a) )^{-}}^{\alpha }f(a) \bigr] \biggr\vert \\ &\quad \leq \frac{ (e^{i \varphi }\eta (b,a) )^{2}}{2(\alpha +1)} \biggl( \int _{0}^{1} h(t) \bigl[1-(1-t)^{\alpha +1}-t^{\alpha +1} \bigr]\,dt \biggr) \biggl( \bigl\vert f''(a) \bigr\vert +\frac{1- \lambda }{\lambda } \bigl\vert f''(b) \bigr\vert \biggr). \end{aligned}$$
Thus, the proof is done. □

Theorem 13

Let \(f: [0,b]\to \mathbb{R}\) be a differentiable mapping and \(|f''|^{{\frac{p}{p-1}}}\) is measurable, decreasing, \(\lambda _{\varphi }\)-preinvex function on \([0,b]\) for \(\eta (b,a)>0\) and \(0\leq a< b\). Suppose that \(|f''|^{{\frac{p}{p-1}}}\) is F-convex on \([a,b]\), for some \(F\in \mathcal{F}\) and \(|f''|\in L^{{\frac{p}{p-1}}}(a,b)\), \(p>1\). Then we have
$$\begin{aligned} &T_{F,1} \bigl(H_{1}(f,p), \bigl\vert f''(a) \bigr\vert ^{\frac{p}{p-1}}, \bigl\vert f''(b) \bigr\vert ^{\frac{p}{p-1}} \bigr)\leq 0, \end{aligned}$$
(20)
where
$$\begin{aligned} H_{1}(f,p) &= \biggl(\frac{2^{\alpha +1}(\alpha +1)}{ (2^{\alpha }-1 ) (e^{i \varphi }\eta (b,a) )^{2}} \biggr)^{\frac{p}{p-1}} \biggl\vert \frac{f(a)+f (a+e^{i \varphi } \eta (b,a) )}{2} \\ &\quad{}-\frac{ \varGamma (\alpha +1)}{2 (e^{i \varphi }\eta (b,a) )^{\alpha }} \bigl[J_{a^{+}}^{\alpha }f \bigl(a+e^{i \varphi }\eta (b,a) \bigr) +J_{ (a+e^{i \varphi }\eta (b,a) )^{-}}^{\alpha }f(a) \bigr] \biggr\vert ^{\frac{p}{p-1}}. \end{aligned}$$

Proof

Since \(|f''|^{\frac{p}{p-1}}\) is F-convex, we have
$$\begin{aligned} F \bigl( \bigl\vert f'' \bigl(a+(1-t)e^{i \varphi } \eta (b,a) \bigr) \bigr\vert ^{\frac{p}{p-1}}, \bigl\vert f''(a) \bigr\vert ^{\frac{p}{p-1}}, \bigl\vert f''(b) \bigr\vert ^{\frac{p}{p-1}},t \bigr) \leq 0,\quad t\in [0,1]. \end{aligned}$$
Using (A2) with \(w(t)=1\), we have
$$\begin{aligned} T_{F,1} \biggl( \int _{0}^{1} \bigl\vert f \bigl(a+(1-t)e^{i \varphi } \eta (b,a) \bigr) \bigr\vert ^{\frac{p}{p-1}} \,dt, \bigl\vert f''(a) \bigr\vert ^{\frac{p}{p-1}}, \bigl\vert f''(b) \bigr\vert ^{\frac{p}{p-1}} \biggr) \leq 0, \quad t\in [0,1]. \end{aligned}$$
Using Lemma 2, Lemma 3 and the Hölder inequality, we get
$$\begin{aligned} & \biggl\vert \frac{f(a)+f (a+e^{i \varphi } \eta (b,a) )}{2}-\frac{ \varGamma (\alpha +1)}{2 (e^{i \varphi }\eta (b,a) )^{\alpha }} \bigl[J_{a^{+}}^{\alpha }f \bigl(a+e^{i \varphi }\eta (b,a) \bigr) +J_{ (a+e^{i \varphi }\eta (b,a) )^{-}}^{\alpha }f(a) \bigr] \biggr\vert \\ &\quad =\frac{ (e^{i \varphi }\eta (b,a) )^{2}}{2(\alpha +1)} \int _{0}^{1} \bigl[1-(1-t)^{\alpha +1}-t^{\alpha +1} \bigr] \bigl\vert f'' \bigl(a+e^{i \varphi }\eta (b,a) \bigr) \bigr\vert \,dt \\ &\quad \leq \frac{ (e^{i \varphi }\eta (b,a) )^{2}}{2(\alpha +1)} \biggl( \int _{0}^{1} \bigl[1-(1-t)^{\alpha +1}-t^{\alpha +1} \bigr] ^{p} \,dt \biggr)^{\frac{1}{p}}\\ &\qquad {}\times \biggl( \int _{0}^{1} \bigl\vert f'' \bigl(a+(1-t)e ^{i \varphi }\eta (b,a) \bigr) \bigr\vert ^{\frac{p}{p-1}}\,dt \biggr) ^{\frac{p-1}{p}} \\ &\quad =\frac{ (e^{i \varphi }\eta (b,a) )^{2}}{2(\alpha +1)} \biggl(1-\frac{1}{2^{\alpha }} \biggr) \biggl( \int _{0}^{1} \bigl\vert f'' \bigl(a+(1-t)e^{i \varphi }\eta (b,a) \bigr) \bigr\vert ^{\frac{p}{p-1}}\,dt \biggr)^{\frac{p-1}{p}} \end{aligned}$$
or, equivalently,
$$\begin{aligned} & \biggl(\frac{2^{\alpha +1}(\alpha +1)}{ (2^{\alpha }-1 ) (e^{i \varphi }\eta (b,a) )^{2}} \biggr)^{\frac{p}{p-1}}\biggl\vert \frac{f(a)+f (a+e^{i \varphi } \eta (b,a) )}{2} \\ &\qquad{} -\frac{ \varGamma (\alpha +1)}{2 (e^{i \varphi }\eta (b,a) )^{\alpha }} \bigl[J_{a^{+}}^{\alpha }f \bigl(a+e^{i \varphi }\eta (b,a) \bigr) +J_{ (a+e^{i \varphi }\eta (b,a) )^{-}}^{\alpha }f(a) \bigr] \biggr\vert ^{\frac{p}{p-1}} \\ &\quad \leq \int _{0}^{1} \bigl\vert f'' \bigl(a+(1-t)e^{i \varphi }\eta (b,a) \bigr) \bigr\vert ^{\frac{p}{p-1}}\,dt. \end{aligned}$$
Because \(T_{F,1}\) is nondecreasing with respect to the first variable, we get
$$\begin{aligned} &T_{F,1} \bigl(H_{1}(f,p), \bigl\vert f''(a) \bigr\vert ^{{\frac{p}{p-1}}}, \bigl\vert f''(b) \bigr\vert ^{{\frac{p}{p-1}}} \bigr)\leq 0. \end{aligned}$$
Thus, the proof is completed. □

Remark 14

If we choose \(\eta (b,a)=b-a\) and \(\varphi =0\) in Theorem 13, we get
$$\begin{aligned} &T_{F,1} \biggl( \biggl(\frac{2^{\alpha +1}(\alpha +1)}{ (2^{ \alpha }-1 ) (b-a)^{2}} \biggr)^{\frac{p}{p-1}} \biggl\vert \frac{f(a)+f(b)}{2} -\frac{ \varGamma (\alpha +1))}{2(b-a)^{\alpha }} \bigl[J_{a^{+}}^{\alpha }f(b)+J _{b^{-}}^{\alpha }f(a) \bigr] \biggr\vert , \\ &\quad \bigl\vert f''(a) \bigr\vert ,\bigl\vert f''(b) \bigr\vert \biggr) \leq 0. \end{aligned}$$

Corollary 13

Under the assumptions of Theorem 13, if \(|f''|^{ \frac{p}{p-1}}\) is ε-convex, then
$$\begin{aligned} & \biggl\vert \frac{f(a)+f (a+e^{i \varphi } \eta (b,a) )}{2}-\frac{ \varGamma (\alpha +1))}{2 (e^{i \varphi }\eta (b,a) )^{\alpha }} \bigl[J_{a^{+}}^{\alpha }f \bigl(a+e^{i \varphi }\eta (b,a) \bigr) +J_{ (a+e^{i \varphi }\eta (b,a) )^{-}}^{\alpha }f(a) \bigr] \biggr\vert \\ &\quad \leq \biggl(\frac{ (2^{\alpha }-1 ) (e^{i \varphi } \eta (b,a) )^{2}}{2^{\alpha +1}(\alpha +1)} \biggr) \biggl(\frac{ \vert f''(a) \vert ^{\frac{p}{p-1}}+ \vert f''(b) \vert ^{ \frac{p}{p-1}}}{2}+\varepsilon \biggr)^{\frac{p-1}{p}}. \end{aligned}$$

Proof

Using (12), (13), by Theorem 13, we have
$$\begin{aligned} 0 &\geq T_{F,1} \bigl(H_{1}(f,p), \bigl\vert f''(a) \bigr\vert ^{\frac{p}{p-1}}, \bigl\vert f''(b) \bigr\vert ^{\frac{p}{p-1}} \bigr) \\ &=H_{1}(f,p)-\frac{ \vert f''(a) \vert ^{\frac{p}{p-1}}+ \vert f''(b) \vert ^{\frac{p}{p-1}}}{2} -\varepsilon . \end{aligned}$$
This leads to
$$\begin{aligned} & \biggl\vert \frac{f(a)+f (a+e^{i \varphi } \eta (b,a) )}{2}-\frac{ \varGamma (\alpha +1))}{2 (e^{i \varphi }\eta (b,a) )^{\alpha }} \bigl[J_{a^{+}}^{\alpha }f \bigl(a+e^{i \varphi }\eta (b,a) \bigr) +J_{ (a+e^{i \varphi }\eta (b,a) )^{-}}^{\alpha }f(a) \bigr] \biggr\vert \\ &\quad \leq \biggl(\frac{ (2^{\alpha }-1 ) (e^{i \varphi } \eta (b,a) )^{2}}{2^{\alpha +1}(\alpha +1)} \biggr) \biggl(\frac{ \vert f''(a) \vert ^{\frac{p}{p-1}} + \vert f''(b) \vert ^{ \frac{p}{p-1}}}{2}+\varepsilon \biggr)^{\frac{p-1}{p}}. \end{aligned}$$
This completes the proof. □

Remark 15

In Corollary 13, if we choose
  1. (a)
    \(\eta (b,a)=b-a\) and \(\varphi =0\), we get
    $$\begin{aligned} & \biggl\vert \frac{f(a)+f(b)}{2}-\frac{\varGamma (\alpha +1))}{2(b-a)^{\alpha }} \bigl[J_{a^{+}}^{\alpha }f(b)+J_{b^{-}}^{\alpha }f(a) \bigr] \biggr\vert \\ &\quad \leq \biggl(\frac{ (2^{\alpha }-1 )(b-a)^{2}}{2^{\alpha +1}( \alpha +1)} \biggr) \biggl( \frac{ \vert f''(a) \vert ^{\frac{p}{p-1}} + \vert f''(b) \vert ^{\frac{p}{p-1}}}{2}+\varepsilon \biggr)^{ \frac{p-1}{p}}. \end{aligned}$$
     
  2. (b)
    \(\eta (b,a)=b-a\), \(\varphi =0\), and \(\varepsilon =0\), we get
    $$\begin{aligned} & \biggl\vert \frac{f(a)+f(b)}{2}-\frac{\varGamma (\alpha +1))}{2(b-a)^{\alpha }} \bigl[J_{a^{+}}^{\alpha }f(b)+J_{b^{-}}^{\alpha }f(a) \bigr] \biggr\vert \\ &\quad \leq \biggl(\frac{ (2^{\alpha }-1 )(b-a)^{2}}{2^{\alpha +1}( \alpha +1)} \biggr) \biggl( \frac{ \vert f''(a) \vert ^{\frac{p}{p-1}}+ \vert f''(b) \vert ^{\frac{p}{p-1}}}{2} \biggr)^{\frac{p-1}{p}}. \end{aligned}$$
     

Corollary 14

Under the assumptions of Theorem 13. If \(|f''|^{ \frac{p-1}{p}}\) is \(\lambda _{\varphi }\)-preinvex, we have
$$\begin{aligned} & \biggl\vert \frac{f(a)+f (a+e^{i \varphi } \eta (b,a) )}{2}-\frac{ \varGamma (\alpha +1))}{2 (e^{i \varphi }\eta (b,a) )^{\alpha }} \bigl[J_{a^{+}}^{\alpha }f \bigl(a+e^{i \varphi }\eta (b,a) \bigr) +J_{ (a+e^{i \varphi }\eta (b,a) )^{-}}^{\alpha }f(a) \bigr] \biggr\vert \\ &\quad \leq \biggl(\frac{ (2^{\alpha }-1 ) (e^{i \varphi } \eta (b,a) )^{2}}{2^{\alpha +1}(\alpha +1)} \biggr) \biggl[\frac{ \pi }{4} \biggl( \bigl\vert f''(a) \bigr\vert ^{\frac{p-1}{p}} + \frac{1-\lambda }{\lambda } \bigl\vert f''(b) \bigr\vert ^{\frac{p-1}{p}} \biggr) \biggr]^{ \frac{p-1}{p}}. \end{aligned}$$

Proof

Using (7), (14), by Theorem 13, we have
$$\begin{aligned} 0 &\geq T_{F,1} \bigl(H_{1}(f,p), \bigl\vert f''(a) \bigr\vert ^{\frac{p-1}{p}}, \bigl\vert f''(b) \bigr\vert ^{\frac{p-1}{p}} \bigr) \\ &= \biggl(\frac{2^{\alpha +1}(\alpha +1)}{ (2^{\alpha }-1 ) (e^{i \varphi }\eta (b,a) )^{2}} \biggr)^{\frac{p}{p-1}}\biggl\vert \frac{f(a)+f (a+e^{i \varphi } \eta (b,a) )}{2} \\ &\quad{} -\frac{ \varGamma (\alpha +1)}{2 (e^{i \varphi }\eta (b,a) )^{\alpha }} \bigl[J_{a^{+}}^{\alpha }f \bigl(a+e^{i \varphi }\eta (b,a) \bigr) +J_{ (a+e^{i \varphi }\eta (b,a) )^{-}}^{\alpha }f(a) \bigr] \biggr\vert ^{\frac{p}{p-1}} \\ &\quad{}-\frac{\pi }{4} \biggl( \bigl\vert f''(a) \bigr\vert ^{\frac{p-1}{p}} +\frac{1- \lambda }{\lambda } \bigl\vert f''(b) \bigr\vert ^{\frac{p-1}{p}} \biggr), \end{aligned}$$
that is,
$$\begin{aligned} & \biggl\vert \frac{f(a)+f (a+e^{i \varphi } \eta (b,a) )}{2}-\frac{ \varGamma (\alpha +1)}{2 (e^{i \varphi }\eta (b,a) )^{\alpha }} \bigl[J_{a^{+}}^{\alpha }f \bigl(a+e^{i \varphi }\eta (b,a) \bigr) +J_{ (a+e^{i \varphi }\eta (b,a) )^{-}}^{\alpha }f(a) \bigr] \biggr\vert \\ &\quad \leq \biggl(\frac{ (2^{\alpha }-1 ) (e^{i \varphi } \eta (b,a) )^{2}}{2^{\alpha +1}(\alpha +1)} \biggr) \biggl[\frac{ \pi }{4} \biggl( \bigl\vert f''(a) \bigr\vert ^{\frac{p-1}{p}} + \frac{1-\lambda }{\lambda } \bigl\vert f''(b) \bigr\vert ^{\frac{p-1}{p}} \biggr) \biggr]^{ \frac{p-1}{p}}. \end{aligned}$$
This proves Corollary 14. □

Remark 16

In Corollary 14, if we choose
  1. (a)
    \(\eta (b,a)=b-a\) and \(\varphi =0\), we get
    $$\begin{aligned} & \biggl\vert \frac{f(a)+f(b)}{2}-\frac{\varGamma (\alpha +1))}{2(b-a)^{\alpha }} \bigl[J_{a^{+}}^{\alpha }f(b)+J_{b^{-}}^{\alpha }f(a) \bigr] \biggr\vert \\ &\quad \leq \biggl(\frac{ (2^{\alpha }-1 )(b-a)^{2}}{2^{\alpha +1}( \alpha +1)} \biggr) \biggl[ \frac{\pi }{4} \biggl( \bigl\vert f''(a) \bigr\vert ^{\frac{p-1}{p}} +\frac{1-\lambda }{\lambda } \bigl\vert f''(b) \bigr\vert ^{\frac{p-1}{p}} \biggr) \biggr]^{\frac{p-1}{p}}. \end{aligned}$$
     
  2. (b)
    \(\eta (b,a)=b-a\), \(\varphi =0\), and \(\lambda =\frac{1}{2}\), we get
    $$\begin{aligned} & \biggl\vert \frac{f(a)+f(b)}{2}-\frac{\varGamma (\alpha +1))}{2(b-a)^{\alpha }} \bigl[J_{a^{+}}^{\alpha }f(b)+J_{b^{-}}^{\alpha }f(a) \bigr] \biggr\vert \\ &\quad \leq \biggl(\frac{ (2^{\alpha }-1 )(b-a)^{2}}{2^{\alpha +1}( \alpha +1)} \biggr) \biggl[ \frac{\pi \vert f''(a) \vert ^{ \frac{p-1}{p}} +\pi \vert f''(b) \vert ^{\frac{p-1}{p}}}{4} \biggr] ^{\frac{p-1}{p}}. \end{aligned}$$
     

Corollary 15

Under the assumptions of Theorem 13. If \(|f''|^{ \frac{p}{p-1}}\) is h-convex, then
$$\begin{aligned} & \biggl\vert \frac{f(a)+f (a+e^{i \varphi } \eta (b,a) )}{2}-\frac{ \varGamma (\alpha +1))}{2 (e^{i \varphi }\eta (b,a) )^{\alpha }} \bigl[J_{a^{+}}^{\alpha }f \bigl(a+e^{i \varphi }\eta (b,a) \bigr) +J_{ (a+e^{i \varphi }\eta (b,a) )^{-}}^{\alpha }f(a) \bigr] \biggr\vert \\ &\quad \leq \biggl(\frac{ (2^{\alpha }-1 ) (e^{i \varphi } \eta (b,a) )^{2}}{2^{\alpha +1}(\alpha +1)} \biggr) \biggl( \int _{0}^{1} h(t) \biggr)^{\frac{p-1}{p}} \biggl( \bigl\vert f''(a) \bigr\vert ^{ \frac{p-1}{p}} + \frac{1-\lambda }{\lambda } \bigl\vert f''(b) \bigr\vert ^{ \frac{p-1}{p}} \biggr). \end{aligned}$$

Proof

Using (9) and by Theorem 13, it can be proved easily. It is omitted. □

Theorem 14

Let \(f: [0,b]\to \mathbb{R}\) be a differentiable mapping and \(|f''|^{{\frac{p}{p-1}}}\) is measurable, decreasing, \(\lambda _{\varphi }\)-preinvex function on \([0,b]\) for \(\eta (b,a)>0\) and \(0\leq a< b\). Suppose that \(|f''|^{{\frac{p}{p-1}}}\) is F-convex on \([a,b]\), for some \(F\in \mathcal{F}\) and \(|f''|\in L^{{\frac{p}{p-1}}}(a,b)\), \(p>1\). Then we have
$$\begin{aligned} &T_{F,1} \bigl(H_{2}(f,p), \bigl\vert f''(a) \bigr\vert ^{\frac{p}{p-1}}, \bigl\vert f''(b) \bigr\vert ^{\frac{p}{p-1}} \bigr)+ \int _{0}^{1} L_{w(t)}\,dt \leq 0, \end{aligned}$$
(21)
where
$$\begin{aligned} H_{2}(f,p) & = \biggl(\frac{2(\alpha +1)}{ (e^{i \varphi }\eta (b,a) ) ^{2}} \biggr)^{\frac{p}{p-1}} \biggl(\frac{2^{\alpha }}{2^{\alpha }-1} \biggr) ^{\frac{1}{p-1}} \biggl\vert \frac{f(a)+f (a+e^{i \varphi } \eta (b,a) )}{2} \\ &\quad{}-\frac{ \varGamma (\alpha +1)}{2 (e^{i \varphi }\eta (b,a) )^{\alpha }} \bigl[J_{a^{+}}^{\alpha }f \bigl(a+e^{i \varphi }\eta (b,a) \bigr) +J_{ (a+e^{i \varphi }\eta (b,a) )^{-}}^{\alpha }f(a) \bigr] \biggr\vert ^{\frac{p}{p-1}} \end{aligned}$$
for \(w(t)=1-(1-t)^{\alpha +1}-t^{\alpha +1}\).

Proof

Since \(|f''|^{\frac{p}{p-1}}\) is F-convex, we have
$$\begin{aligned} F \bigl( \bigl\vert f'' \bigl(a+(1-t)e^{i \varphi } \eta (b,a) \bigr) \bigr\vert ^{\frac{p}{p-1}}, \bigl\vert f''(a) \bigr\vert ^{\frac{p}{p-1}}, \bigl\vert f''(b) \bigr\vert ^{\frac{p}{p-1}},t \bigr) \leq 0,\quad t\in [0,1]. \end{aligned}$$
Using (A3) with \(w(t)=1-(1-t)^{\alpha +1}-t^{\alpha +1}\), we obtain
$$\begin{aligned} &F \bigl(w(t) \bigl\vert f'' \bigl(a+(1-t)e^{i \varphi } \eta (b,a) \bigr) \bigr\vert ^{\frac{p}{p-1}}, w(t) \bigl\vert f''(a) \bigr\vert ^{\frac{p}{p-1}},w(t) \bigl\vert f''(b) \bigr\vert ^{\frac{p}{p-1}},t \bigr) +L_{w(t)}\leq 0,\\ &\quad t\in [0,1]. \end{aligned}$$
Integrating over \([0,1]\) and using axiom (A2), we obtain
$$\begin{aligned} &T_{F,w} \biggl( \int _{0}^{1} w(t) \bigl\vert f'' \bigl(a+(1-t)e^{i \varphi } \eta (b,a) \bigr) \bigr\vert ^{\frac{p}{p-1}}\,dt, \bigl\vert f''(a) \bigr\vert ^{\frac{p}{p-1}}, \bigl\vert f''(b) \bigr\vert ^{\frac{p}{p-1}} \biggr)+ \int _{0}^{1} L_{w(t)}\,dt\leq 0,\\ &\quad t\in [0,1]. \end{aligned}$$
Using Lemma 2 and the power mean inequality, we get
$$\begin{aligned} & \biggl\vert \frac{f(a)+f (a+e^{i \varphi } \eta (b,a) )}{2}-\frac{ \varGamma (\alpha +1)}{2 (e^{i \varphi }\eta (b,a) )^{\alpha }} \bigl[J_{a^{+}}^{\alpha }f \bigl(a+e^{i \varphi }\eta (b,a) \bigr) +J_{ (a+e^{i \varphi }\eta (b,a) )^{-}}^{\alpha }f(a) \bigr] \biggr\vert \\ &\quad =\frac{e^{i \varphi }\eta (b,a)}{2} \int _{0}^{1} \bigl[1-(1-t)^{ \alpha +1}-t^{\alpha +1} \bigr] \bigl\vert f'' \bigl(a+(1-t)e^{i \varphi } \eta (b,a) \bigr) \bigr\vert \,dt \\ &\quad \leq \frac{ (e^{i \varphi }\eta (b,a) )^{2}}{2(\alpha +1)} \biggl( \int _{0}^{1} \bigl[1-(1-t)^{\alpha +1}-t^{\alpha +1} \bigr]\,dt \biggr) ^{\frac{1}{p}}\\ &\qquad {}\times \biggl( \int _{0}^{1} w(t) \bigl\vert f' \bigl(a+(1-t)e^{i \varphi }\eta (b,a) \bigr) \bigr\vert ^{\frac{p}{p-1}}\,dt \biggr)^{ \frac{p-1}{p}} \\ &\quad =\frac{ (e^{i \varphi }\eta (b,a) )^{2}}{2(\alpha +1)} \biggl(1-\frac{1}{2^{\alpha }} \biggr)^{\frac{1}{p}} \biggl( \int _{0} ^{1} w(t) \bigl\vert f' \bigl(a+(1-t)e^{i \varphi }\eta (b,a) \bigr) \bigr\vert ^{\frac{p}{p-1}}\,dt \biggr) ^{\frac{p-1}{p}} \end{aligned}$$
or, equivalently,
$$\begin{aligned} & \biggl(\frac{2(\alpha +1)}{ (e^{i \varphi }\eta (b,a) ) ^{2}} \biggr)^{\frac{p}{p-1}} \biggl(\frac{2^{\alpha }}{2^{\alpha }-1} \biggr) ^{\frac{1}{p-1}} \biggl\vert \frac{f(a)+f (a+e^{i \varphi } \eta (b,a) )}{2} \\ &\qquad{}-\frac{ \varGamma (\alpha +1)}{2 (e^{i \varphi }\eta (b,a) )^{\alpha }} \bigl[J_{a^{+}}^{\alpha }f \bigl(a+e^{i \varphi }\eta (b,a) \bigr) +J_{ (a+e^{i \varphi }\eta (b,a) )^{-}}^{\alpha }f(a) \bigr] \biggr\vert ^{\frac{p}{p-1}} \\ &\quad \leq \int _{0}^{1} w(t) \bigl\vert f'' \bigl(a+(1-t)e^{i \varphi }\eta (b,a) \bigr) \bigr\vert ^{\frac{p}{p-1}}\,dt. \end{aligned}$$
Since \(T_{F,w}\) is nondecreasing with respect to the first variable, we have
$$\begin{aligned} &T_{F,w} \bigl(H_{2}(f,p), \bigl\vert f''(a) \bigr\vert ^{{\frac{p}{p-1}}}, \bigl\vert f''(b) \bigr\vert ^{{\frac{p}{p-1}}} \bigr)+ \int _{0}^{1} L_{w(t)}\,dt \leq 0. \end{aligned}$$
This completes the proof. □

Remark 17

Taking \(\eta (b,a)=b-a\) and \(\varphi =0\) in Theorem 14, we get
$$\begin{aligned} &T_{F,w} \biggl( \biggl(\frac{2(\alpha +1)}{ (e^{i \varphi }\eta (b,a) ) ^{2}} \biggr)^{\frac{p}{p-1}} \biggl(\frac{2^{\alpha }}{2^{\alpha }-1} \biggr) ^{\frac{1}{p-1}} \biggl\vert \frac{f(a)+f(b)}{2}- \frac{\varGamma (\alpha +1))}{2(b-a)^{ \alpha }} \bigl[J_{a^{+}}^{\alpha }f(b)+J_{b^{-}}^{\alpha }f(a) \bigr] \biggr\vert , \\ &\quad\bigl\vert f'(a) \bigr\vert , \bigl\vert f'(b) \bigr\vert \biggr)+ \int _{0}^{1} L_{w(t)}\,dt\leq 0. \end{aligned}$$

Corollary 16

Under the assumptions of Theorem 14, if \(|f''|^{ \frac{p}{p-1}}\) is ε-convex, we have
$$\begin{aligned} & \biggl\vert \frac{f(a)+f (a+e^{i \varphi } \eta (b,a) )}{2}-\frac{ \varGamma (\alpha +1))}{2 (e^{i \varphi }\eta (b,a) )^{\alpha }} \bigl[J_{a^{+}}^{\alpha }f \bigl(a+e^{i \varphi }\eta (b,a) \bigr) +J_{ (a+e^{i \varphi }\eta (b,a) )^{-}}^{\alpha }f(a) \bigr] \biggr\vert \\ &\quad \leq \biggl(\frac{ (e^{i \varphi }\eta (b,a) )^{2}}{2( \alpha +1)} \biggr) \biggl(\frac{2^{\alpha }-1}{2^{\alpha }} \biggr) ^{\frac{1}{p}} \biggl[\frac{\alpha }{2(\alpha +2)} \bigl( \bigl\vert f''(a) \bigr\vert ^{\frac{p}{p-1}} + \bigl\vert f''(b) \bigr\vert ^{\frac{p}{p-1}} \bigr)+2 \varepsilon \biggr]^{\frac{p-1}{p}}. \end{aligned}$$

Proof

Using (17), (18) and by Theorem 13, we obtain
$$\begin{aligned} & \biggl\vert \frac{f(a)+f (a+e^{i \varphi } \eta (b,a) )}{2}-\frac{ \varGamma (\alpha +1))}{2 (e^{i \varphi }\eta (b,a) )^{\alpha }} \bigl[J_{a^{+}}^{\alpha }f \bigl(a+e^{i \varphi }\eta (b,a) \bigr) +J_{ (a+e^{i \varphi }\eta (b,a) )^{-}}^{\alpha }f(a) \bigr] \biggr\vert \\ &\quad \leq \biggl(\frac{ (e^{i \varphi }\eta (b,a) )^{2}}{2( \alpha +1)} \biggr) \biggl(\frac{2^{\alpha }-1}{2^{\alpha }} \biggr) ^{\frac{1}{p}} \biggl[\frac{\alpha }{2(\alpha +2)} \bigl( \bigl\vert f''(a) \bigr\vert ^{\frac{p}{p-1}} + \bigl\vert f''(b) \bigr\vert ^{\frac{p}{p-1}} \bigr)+2 \varepsilon \biggr]^{\frac{p-1}{p}}. \end{aligned}$$
This completes the proof. □

Remark 18

In Corollary 16, if we choose
  1. (a)
    \(\eta (b,a)=b-a\) and \(\varphi =0\), we get
    $$\begin{aligned} & \biggl\vert \frac{f(a)+f(b)}{2}-\frac{\varGamma (\alpha +1))}{2(b-a)^{\alpha }} \bigl[J_{a^{+}}^{\alpha }f(b)+J_{b^{-}}^{\alpha }f(a) \bigr] \biggr\vert \\ &\quad \leq \biggl(\frac{(b-a)^{2}}{2(\alpha +1)} \biggr) \biggl(\frac{2^{ \alpha }-1}{2^{\alpha }} \biggr)^{\frac{1}{p}} \biggl[\frac{\alpha }{2( \alpha +2)} \bigl( \bigl\vert f''(a) \bigr\vert ^{\frac{p}{p-1}} + \bigl\vert f''(b) \bigr\vert ^{\frac{p}{p-1}} \bigr)+2\varepsilon \biggr]^{\frac{p-1}{p}}. \end{aligned}$$
     
  2. (b)
    \(\eta (b,a)=b-a\), \(\varphi =0\), and \(\varepsilon =0\), we get
    $$\begin{aligned} & \biggl\vert \frac{f(a)+f(b)}{2}-\frac{\varGamma (\alpha +1))}{2(b-a)^{\alpha }} \bigl[J_{a^{+}}^{\alpha }f(b)+J_{b^{-}}^{\alpha }f(a) \bigr] \biggr\vert \\ &\quad \leq \biggl(\frac{(b-a)^{2}}{2(\alpha +1)} \biggr) \biggl( \frac{2^{ \alpha }-1}{2^{\alpha }} \biggr)^{\frac{1}{p}} \biggl[\frac{\alpha }{2( \alpha +2)} \bigl( \bigl\vert f''(a) \bigr\vert ^{\frac{p}{p-1}} + \bigl\vert f''(b) \bigr\vert ^{\frac{p}{p-1}} \bigr) \biggr]^{\frac{p-1}{p}}. \end{aligned}$$
     

Corollary 17

Under the assumptions of Theorem 14. If \(|f''|^{ \frac{p-1}{p}}\) is \(\lambda _{\varphi }\)-preinvex, then
$$\begin{aligned} & \biggl\vert \frac{f(a)+f (a+e^{i \varphi } \eta (b,a) )}{2}-\frac{ \varGamma (\alpha +1))}{2 (e^{i \varphi }\eta (b,a) )^{\alpha }} \bigl[J_{a^{+}}^{\alpha }f \bigl(a+e^{i \varphi }\eta (b,a) \bigr) +J_{ (a+e^{i \varphi }\eta (b,a) )^{-}}^{\alpha }f(a) \bigr] \biggr\vert \\ &\quad \leq \biggl(\frac{ (e^{i \varphi }\eta (b,a) )^{2}}{2( \alpha +1)} \biggr) \biggl(\frac{2^{\alpha }-1}{2^{\alpha }} \biggr) ^{\frac{1}{p}}\\ &\qquad {}\times \biggl( \biggl[\frac{\pi }{2}-\beta \biggl(\frac{1}{2}, \alpha +\frac{5}{2} \biggr) -\beta \biggl(\frac{3}{2},\alpha + \frac{3}{2} \biggr) \biggr] \biggl( \bigl\vert f''(a) \bigr\vert ^{ \frac{p}{p-1}}+\frac{1-\lambda }{\lambda } \bigl\vert f''(b) \bigr\vert ^{ \frac{p}{p-1}} \biggr) \biggr)^{\frac{p-1}{p}}. \end{aligned}$$

Proof

Using (19), by Theorem 14, we have
$$\begin{aligned} 0 &\geq T_{F,w} \bigl(H_{2}(f,p), \bigl\vert f''(a) \bigr\vert ^{\frac{p-1}{p}}, \bigl\vert f''(b) \bigr\vert ^{\frac{p-1}{p}} \bigr) \\ &= \biggl(\frac{2(\alpha +1)}{ (e^{i \varphi }\eta (b,a) ) ^{2}} \biggr)^{\frac{p}{p-1}} \biggl(\frac{2^{\alpha }}{2^{\alpha }-1} \biggr) ^{\frac{1}{p-1}}\biggl\vert \frac{f(a)+f (a+e^{i \varphi } \eta (b,a) )}{2} \\ &\quad{}-\frac{ \varGamma (\alpha +1)}{2 (e^{i \varphi }\eta (b,a) )^{\alpha }} \bigl[J_{a^{+}}^{\alpha }f \bigl(a+e^{i \varphi }\eta (b,a) \bigr) +J_{ (a+e^{i \varphi }\eta (b,a) )^{-}}^{\alpha }f(a) \bigr] \biggr\vert ^{\frac{p}{p-1}} \\ &\quad{}- \biggl[\frac{\pi }{2}-\beta \biggl(\frac{1}{2},\alpha + \frac{5}{2} \biggr) - \beta \biggl(\frac{3}{2},\alpha + \frac{3}{2} \biggr) \biggr] \biggl( \bigl\vert f''(a) \bigr\vert ^{\frac{p-1}{p}} +\frac{1-\lambda }{\lambda } \bigl\vert f''(b) \bigr\vert ^{\frac{p-1}{p}} \biggr). \end{aligned}$$
This leads to
$$\begin{aligned} & \biggl\vert \frac{f(a)+f (a+e^{i \varphi } \eta (b,a) )}{2}-\frac{ \varGamma (\alpha +1)}{2 (e^{i \varphi }\eta (b,a) )^{\alpha }} \bigl[J_{a^{+}}^{\alpha }f \bigl(a+e^{i \varphi }\eta (b,a) \bigr) +J_{ (a+e^{i \varphi }\eta (b,a) )^{-}}^{\alpha }f(a) \bigr] \biggr\vert \\ &\quad \leq \biggl(\frac{ (e^{i \varphi }\eta (b,a) )^{2}}{2( \alpha +1)} \biggr) \biggl(\frac{2^{\alpha }-1}{2^{\alpha }} \biggr) ^{\frac{1}{p}}\\ &\qquad {}\times \biggl( \biggl[\frac{\pi }{2}-\beta \biggl(\frac{1}{2}, \alpha +\frac{5}{2} \biggr) -\beta \biggl(\frac{3}{2},\alpha + \frac{3}{2} \biggr) \biggr] \biggl( \bigl\vert f''(a) \bigr\vert ^{ \frac{p}{p-1}}+\frac{1-\lambda }{\lambda } \bigl\vert f''(b) \bigr\vert ^{ \frac{p}{p-1}} \biggr) \biggr)^{\frac{p-1}{p}}. \end{aligned}$$
This ends the proof. □

Remark 19

In Corollary 17, if we choose
  1. (a)
    \(\eta (b,a)=b-a\) and \(\varphi =0\), we get
    $$\begin{aligned} & \biggl\vert \frac{f(a)+f(b)}{2}-\frac{\varGamma (\alpha +1))}{2(b-a)^{\alpha }} \bigl[J_{a^{+}}^{\alpha }f(b)+J_{b^{-}}^{\alpha }f(a) \bigr] \biggr\vert \\ &\quad \leq \biggl(\frac{(b-a)^{2}}{2(\alpha +1)} \biggr) \biggl(\frac{2^{ \alpha }-1}{2^{\alpha }} \biggr)^{\frac{1}{p}} \biggl( \biggl[\frac{ \pi }{2}-\beta \biggl( \frac{1}{2},\alpha +\frac{5}{2} \biggr) -\beta \biggl( \frac{3}{2},\alpha +\frac{3}{2} \biggr) \biggr]\\ &\qquad {}\times \biggl( \bigl\vert f''(a) \bigr\vert ^{\frac{p}{p-1}}+\frac{1-\lambda }{\lambda } \bigl\vert f''(b) \bigr\vert ^{\frac{p}{p-1}} \biggr) \biggr)^{\frac{p-1}{p}}. \end{aligned}$$
     
  2. (b)
    \(\eta (b,a)=b-a\), \(\varphi =0\), and \(\lambda =\frac{1}{2}\), we get
    $$\begin{aligned} & \biggl\vert \frac{f(a)+f(b)}{2}-\frac{\varGamma (\alpha +1))}{2(b-a)^{\alpha }} \bigl[J_{a^{+}}^{\alpha }f(b)+J_{b^{-}}^{\alpha }f(a) \bigr] \biggr\vert \\ &\quad \leq \biggl(\frac{(b-a)^{2}}{2(\alpha +1)} \biggr) \biggl(\frac{2^{ \alpha }-1}{2^{\alpha }} \biggr)^{\frac{1}{p}} \biggl( \biggl[\frac{ \pi }{2}-\beta \biggl( \frac{1}{2},\alpha +\frac{5}{2} \biggr) -\beta \biggl( \frac{3}{2},\alpha +\frac{3}{2} \biggr) \biggr]\\ &\qquad {}\times \bigl( \bigl\vert f''(a) \bigr\vert ^{\frac{p}{p-1}}+ \bigl\vert f''(b) \bigr\vert ^{\frac{p}{p-1}} \bigr) \biggr) ^{\frac{p-1}{p}}. \end{aligned}$$
     

Corollary 18

Under the assumptions of Theorem 14. If \(|f''|^{ \frac{p}{p-1}}\) is h-convex, we have
$$\begin{aligned} & \biggl\vert \frac{f(a)+f (a+e^{i \varphi } \eta (b,a) )}{2}-\frac{ \varGamma (\alpha +1))}{2 (e^{i \varphi }\eta (b,a) )^{\alpha }} \bigl[J_{a^{+}}^{\alpha }f \bigl(a+e^{i \varphi }\eta (b,a) \bigr) +J_{ (a+e^{i \varphi }\eta (b,a) )^{-}}^{\alpha }f(a) \bigr] \biggr\vert \\ &\quad \leq \biggl(\frac{ (e^{i \varphi }\eta (b,a) )^{2}}{2( \alpha +1)} \biggr)\biggl(\frac{2^{\alpha }-1}{2^{\alpha }} \biggr) ^{\frac{1}{p}} \\ &\qquad {}\times \biggl( \int _{0}^{1} h(t) \bigl\vert (1-t)^{\alpha }-t^{ \alpha } \bigr\vert \,dt \biggr)^{\frac{p-1}{p}}\biggl( \bigl\vert f'(a) \bigr\vert ^{\frac{p}{p-1}}+\frac{1-\lambda }{\lambda } \bigl\vert f'(b) \bigr\vert ^{ \frac{p}{p-1}} \biggr)^{\frac{p-1}{p}}. \end{aligned}$$

Proof

Using (9), by Theorem 14, it can be proved easily. It is omitted. □

5 Conclusion

In the present paper, using the notion of F and F-convex function (see [17]), we construct some new inequalities of Hermite–Hadamard type for differentiable function via Riemann–Liouville fractional integral. We also established some trapezoid type inequalities for a function of whose second derivatives absolutely values are F-convex. Moreover, we obtained some new inequalities of Hermite–Hadamard type for Riemann–Liouville fractional integrals and via classical integrals. The results presented in this paper would provide generalizations and extension of those given in earlier work.

Declarations

Acknowledgements

The authors would like to express their thanks to the editor and the referees for their helpful comments.

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All authors contributed equally and significantly in writing this article. All authors read and approved the final manuscript.

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The authors declare that they have no competing interests.

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Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors’ Affiliations

(1)
Department of Mathematics, College of Education, University of Sulaimani, Sulaimani, Iraq
(2)
Department of Mathematics, Faculty of Science and Arts, Düzce University, Düzce, Turkey

References

  1. Niculescu, C., Persson, L.E.: Convex Functions and Their Application. Springer, Berlin (2004) Google Scholar
  2. Mohammed, P.O.: Inequalities of type Hermite–Hadamard for fractional integrals via differentiable convex functions. Turk. J. Anal. Number Theory 4(5), 135–139 (2016) Google Scholar
  3. Alomari, M., Darus, M., Kirmaci, U.S.: Refinements of Hadamard-type inequalities for quasi-convex functions with applications to trapezoidal formula and to special means. Comput. Math. Appl. 59, 225–232 (2010) MathSciNetView ArticleGoogle Scholar
  4. Noor, M.A., Noor, K.I., Awan, M.U.: Some quantum estimates for Hermite–Hadamard inequalities. Appl. Math. Comput. 251, 675–679 (2015) MathSciNetMATHGoogle Scholar
  5. İşcana, I., Turhan, S.: Generalized Hermite–Hadamard–Fejer type inequalities for GA-convex functions via fractional integral. Moroccan J. Pure Appl. Anal. 2(1), 34–46 (2016) Google Scholar
  6. Wu, Y., Qi, F.: On some Hermite–Hadamard type inequalities for \((s,QC)\)-convex functions. SpringerPlus 5, 49 (2016). https://doi.org/10.1186/s40064-016-1676-9 View ArticleGoogle Scholar
  7. Mohammed, P.O.: On new trapezoid type inequalities for h-convex functions via generalized fractional integral. Fract. Differ. Calc. 6(4), 125–128 (2018) Google Scholar
  8. Khan, M.A., Begum, S., Khurshid, Y., Chu, Y.-M.: Ostrowski type inequalities involving conformable fractional integrals. J. Inequal. Appl. 2018, 70 (2018) MathSciNetView ArticleGoogle Scholar
  9. Defnetti, B.: Sulla strati cazioni convesse. Ann. Mat. Pura Appl. 30, 173–183 (1949) MathSciNetView ArticleGoogle Scholar
  10. Mangasarian, O.L.: Pseudo-convex functions. SIAM J. Control 3, 281–290 (1965) MathSciNetMATHGoogle Scholar
  11. Mohammed, P.O.: Some new Hermite–Hadamard type inequalities for MT-convex functions on differentiable coordinates. J. King Saud Univ., Sci. 30, 258–262 (2018) View ArticleGoogle Scholar
  12. Polyak, B.T.: Existence theorems and convergence of minimizing sequences in extremum problems with restrictions. Sov. Math. Dokl. 7, 72–75 (1966) Google Scholar
  13. Hyers, D.H., Ulam, S.M.: Approximately convex functions. Proc. Am. Math. Soc. 3, 821–828 (1952) MathSciNetView ArticleGoogle Scholar
  14. Hudzik, H., Maligranda, L.: Some remarks on s-convex functions. Aequ. Math. 48, 100–111 (1994) MathSciNetView ArticleGoogle Scholar
  15. Varosanec, S.: On h-convexity. J. Math. Anal. Appl. 326(1), 303–311 (2007) MathSciNetView ArticleGoogle Scholar
  16. Ermeydan, S., Yildirim, H.: Riemann–Liouville fractional Hermite–Hadamard inequalities for differentiable \(\lambda _{\varphi }\)-preinvex functions. Malaya J. Mat. 4(3), 430–437 (2016) Google Scholar
  17. Samet, B.: On an implicit convexity concept and some integral inequalities. J. Inequal. Appl. 2016, 308 (2016) MathSciNetView ArticleGoogle Scholar
  18. Sarikaya, M.Z., Set, E., Yaldiz, H., Basak, N.: Hermite–Hadamard’s inequalities for fractional integrals and related fractional inequalities. Math. Comput. Model. 57, 2403–2407 (2013). https://doi.org/10.1016/j.mcm.2011.12.048 View ArticleMATHGoogle Scholar
  19. Budak, H., Sarikaya, M.Z.: On Ostrowski type inequalities for F-convex function. AIP Conf. Proc. 1833, 020057 (2017). https://doi.org/10.1063/1.4981705 View ArticleGoogle Scholar
  20. Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam (2006) MATHGoogle Scholar
  21. Budak, H., Sarikaya, M.Z., Yildiz, M.K.: Hermite–Hadamard type inequalities for F-convex function involving fractional integrals. Filomat (in press) Google Scholar
  22. Deng, J., Wang, J.: Fractional Hermite–Hadamard inequalities for \((\alpha ,m)\)-logarithmically convex functions. J. Inequal. Appl. 2013, Article ID 364 (2013) MathSciNetView ArticleGoogle Scholar

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