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Extensions of different type parameterized inequalities for generalized \((m,h)\)-preinvex mappings via k-fractional integrals

Abstract

The authors discover a general k-fractional integral identity with multi-parameters for twice differentiable functions. By using this integral equation, the authors derive some new bounds on Hermite–Hadamard’s and Simpson’s inequalities for generalized \((m,h)\)-preinvex functions through k-fractional integrals. By taking the special parameter values for various suitable choices of function h, some interesting results are also obtained.

Introduction

The subsequent inequalities are notable in the literature as Hermite–Hadamard’s inequality and Simpson’s inequality, respectively.

Theorem 1.1

Suppose that \(f:I\subseteq \mathbb{R}\rightarrow \mathbb{R}\) is a convex function defined on the interval I of real numbers and \(a,b\in I\) along with \(a< b\). The following double inequality holds:

$$ f \biggl(\frac{a+b}{2} \biggr)\leq \frac{1}{b-a} \int ^{b}_{a} f(x)\,\mathrm{d}x\leq \frac{f(a)+f(b)}{2}. $$

Theorem 1.2

Assume that \(f:[a,b]\rightarrow \mathbb{R}\) is a four times continuously differentiable mapping on \((a,b)\) and \(\Vert f^{(4)} \Vert _{\infty }=\mathrm{sup}_{x\in (a,b)} \vert f^{(4)}(x) \vert <\infty \). Then the following inequality holds:

$$ \biggl\vert \frac{1}{3} \biggl[ \frac{f(a)+f(b)}{2}+2f \biggl(\frac{a+b}{2} \biggr) \biggr]-\frac{1}{b-a} \int ^{b}_{a} f(x)\,\mathrm{d}x \biggr\vert \leq \frac{1}{2880} \bigl\Vert f^{(4)} \bigr\Vert _{\infty }(b-a)^{4}. $$

Hermite–Hadamard’s inequalities and Simpson’s inequalities have remained an area of great interest owing to their extensive applications in mathematics and other sciences. Many researchers generalized these inequalities. For recent results, for example, see [18] and the references mentioned in these papers.

In 2013, Sarikaya et al. established the subsequent interesting Hermite–Hadamard’s inequalities by utilizing Riemann–Liouville fractional integrals.

Theorem 1.3

([9])

Let \(f:[a,b]\rightarrow \mathbb{R}\) be a positive function along with \(0\leq a< b\), and let \(f\in L^{1}[a,b]\). Suppose that f is a convex function on \([a,b]\), then the following inequalities for fractional integrals hold:

$$ f \biggl(\frac{a+b}{2} \biggr) \leq \frac{\Gamma (\mu +1)}{2(b-a)^{\mu }}\bigl[J^{\mu }_{a^{+}}f(b)+J^{\mu }_{b^{-}}f(a) \bigr]\leq \frac{f(a)+f(b)}{2}, $$
(1.1)

where the symbols \(J^{\mu }_{a^{+}} f\) and \(J^{\mu }_{b^{-}} f\) denote respectively the left-sided and right-sided Riemann–Liouville fractional integrals of order \(\mu >0\) defined by

$$ J^{\mu }_{a^{+}}f(x) = \frac{1}{\Gamma (\mu )} \int ^{x}_{a}(x-t)^{\mu -1}f(t)\,\mathrm{d}t, \quad a< x, $$

and

$$ J^{\mu }_{b^{-}}f(x) = \frac{1}{\Gamma (\mu )} \int ^{b}_{x}(t-x)^{\mu -1}f(t)\,\mathrm{d}t, \quad x< b. $$

Here, \(\Gamma (\mu )\) is the gamma function and its definition is \(\Gamma (\mu )=\int _{0}^{\infty }e^{-t}t^{\mu -1}\,\mathrm{d}t\). It is to be noted that \(J^{0}_{a^{+}}f(x)=J^{0}_{b^{-}}f(x)=f(x)\).

In the case of \(\mu =1\), the fractional integral recaptures the classical integral.

Because of the extensive application of Riemann–Liouville fractional integrals, some authors extended their studies to fractional Hermite–Hadamard’s inequalities via mappings of different classes. For example, refer to [1012] for convex mappings, to [13] for s-convex mappings, to [14] for \((s,m)\)-convex mappings, to [15] for s-Godunova–Levin mappings, to [16] for harmonically convex mappings, to [17] for preinvex mappings, to [18] for MT m -preinvex mappings, to [19] for h-convex mappings, to [20] for r-convex mappings, and see the references cited therein.

In 2012, Mubeen and Habibullah introduced the following class of fractional integrals.

Definition 1.1

([21])

Let \(f\in L^{1}[a,b]\), then the Riemann–Liouville k-fractional integrals \({}_{k}J^{\mu }_{a^{+}}f(x)\) and \({}_{k}J^{\mu }_{b^{-}}f(x)\) of order \(\mu >0\) are given as

$$ {}_{k}J^{\mu }_{a^{+}}f(x) = \frac{1}{k\Gamma _{k}(\mu )} \int ^{x}_{a}(x-t)^{\frac{\mu }{k}-1}f(t)\,\mathrm{d}t \quad (0\leq a< x< b) $$

and

$$ {} _{k}J^{\mu }_{b^{-}}f(x) = \frac{1}{k\Gamma _{k}(\mu )} \int ^{b}_{x}(t-x)^{\frac{\mu }{k}-1}f(t)\,\mathrm{d}t\quad (0\leq a< x< b), $$

respectively, where \(k>0\) and \(\Gamma _{k}(\mu )\) is the k-gamma function defined by \(\Gamma _{k}(\mu )=\int ^{\infty }_{0} t^{\mu -1} e^{-\frac{t^{k}}{k}}\,\mathrm{d}t\). Furthermore, \(\Gamma _{k}(\mu +k)=\mu \Gamma _{k}(\mu )\) and \({}_{k}J^{0}_{a^{+}}f(x)={}_{k}J^{0}_{b^{-}}f(x)=f(x)\).

The concept of Riemann–Liouville k-fractional integral is an important generalization of Riemann–Liouville fractional integrals. We would like to stress here that for \(k\neq 1\) the properties of Riemann–Liouville k-fractional integrals are very dissimilar to those of classical Riemann–Liouville fractional integrals. Due to this, the Riemann–Liouville k-fractional integrals have aroused many researchers’ interest. Properties and estimations for the integral inequality related to this operator can be sought out in [2227] and the references cited therein.

The main purpose of the current paper is to establish some new bounds on Hermite–Hadamard’s and Simpson’s inequalities for mappings whose absolute values of second derivatives are generalized \((m,h)\)-preinvex. To do this, the authors derive a general k-fractional integral identity along with multi parameters for twice differentiable mappings. By using this integral identity, the authors derive some new inequalities of Simpson and Hermite–Hadamard type for these mappings.

To end this section, we restate some special functions and definitions as follows.

Let us consider the following special functions:

  1. (1)

    The beta function:

    $$ \begin{aligned} \beta (x,y)=\frac{\Gamma (x)\Gamma (y)}{\Gamma (x+y)}= \int ^{1}_{0}t^{x-1}(1-t)^{y-1}\,\mathrm{d}t, \quad x,y>0; \end{aligned} $$
  2. (2)

    The incomplete beta function:

    $$ \begin{aligned} \beta (a;x,y)= \int ^{a}_{0}t^{x-1}(1-t)^{y-1}\,\mathrm{d}t, \quad 0< a< 1,x,y>0; \end{aligned} $$
  3. (3)

    The hypergeometric function:

    $$ {}_{2}F_{1}(a,b;c;z)= \frac{1}{\beta (b,c-b)} \int ^{1}_{0} t^{b-1}(1-t)^{c-b-1}(1-zt)^{-a}\,\mathrm{d}t,\quad c>b>0,\vert z \vert < 1. $$

Definition 1.2

([28])

A function \(f : [0,\infty )\rightarrow \mathbb{R}\) is named s-convex in the second sense along with \(s\in (0,1]\) if

$$ \begin{aligned} f(\alpha x +\beta y)\leq \alpha ^{s} f(x)+ \beta ^{s} f(y) \end{aligned} $$

holds for all \(x,y\in [0,\infty )\) and \(\alpha ,\beta \geq 0\) along with \(\alpha +\beta =1\).

Definition 1.3

([29])

A function \(f :A \subseteq \mathbb{R}\rightarrow \mathbb{R}\) is called s-Godunova–Levin function of the second kind along with \(s\in [0,1]\) if

$$ \begin{aligned} f \bigl(tx+(1-t)y \bigr)\leq \frac{f(x)}{t^{s}}+ \frac{f(y)}{(1-t)^{s}} \end{aligned} $$

holds for all \(x,y\in A\) and \(t\in (0,1)\).

Definition 1.4

([30])

A function \(f:A\subseteq \mathbb{R}\rightarrow \mathbb{R}\) is named \(tgs\)-convex on A if f is non-negative and

$$ f \bigl(tx+ (1-t)y \bigr)\leq t(1-t)\bigl[f(x)+f(y) \bigr] $$

holds for all \(x, y \in A \) and \(t\in (0,1)\).

Definition 1.5

([31])

A function \(f: A\subseteq \mathbb{R}\rightarrow \mathbb{R}\) is called MT-convex if f is non-negative and

$$ f \bigl(tx+(1-t)y \bigr)\leq \frac{\sqrt{t}}{2\sqrt{1-t}}f(x)+\frac{\sqrt{1-t}}{2\sqrt{t}}f(y) $$

holds for all \(x,y\in A\) and \(t\in (0, 1)\).

Definition 1.6

([32])

A set \(A\subseteq \mathbb{R}^{n}\) is called m-invex with respect to the mapping \(\eta :A \times A\times (0, 1]\rightarrow \mathbb{R}^{n}\) for some fixed \(m\in (0, 1]\) if \(mx+\lambda \eta (y, x, m) \in A\) holds for all \(x, y \in A\) and \(\lambda \in [0, 1]\).

Definition 1.7

([33])

Let \(A \subseteq \mathbb{R}\) be an open m-invex subset with respect to \(\eta : A \times A \times (0,1]\rightarrow \mathbb{R}\), and let \(h_{1}, h_{2} :[0,1]\rightarrow \mathbb{R}_{0}\). A function \(f:A\rightarrow \mathbb{R}\) is said to be generalized \((m, h_{1}, h_{2})\)-preinvex if

$$ f \bigl(m x+ t\eta (y, x, m) \bigr)\leq mh_{1}(t) f(x)+ h_{2}(t) f(y) $$
(1.2)

is valid for all \(x, y \in A\) and \(t \in [0, 1]\). If inequality (1.2) reverses, then f is said to be generalized \((m, h_{1}, h_{2})\)-preincave on A.

Clearly, if we take \(h_{1}(t)=h(1-t)\), \(h_{2}(t)=h(t)\) in Definition 1.7, then f becomes generalized \((m,h)\)-preinvex functions as follows.

Definition 1.8

Let \(A\subseteq \mathbb{R}\) be an open m-invex subset with respect to \(\eta : A\times A\times (0,1] \rightarrow \mathbb{R}\), and let \(h :[0,1]\rightarrow \mathbb{R}_{0}\). A function \(f:A\rightarrow \mathbb{R}\) is called generalized \((m,h)\)-preinvex if

$$ f \bigl(mx+t\eta (y,x,m) \bigr)\leq h(t)f(y)+mh(1-t)f(x) $$
(1.3)

is valid for all \(x, y \in A\) and \(t \in [0,1]\).

Remark 1.1

Let us discuss some special cases of Definition 1.8 as follows:

  1. (i)

    choosing \(h(t)=1\), we obtain the definition of generalized \((m,P)\)-preinvex functions;

  2. (ii)

    choosing \(h(t)=t^{s}\) for \(s\in (0,1]\), we obtain the definition of generalized \((m,s)\)-Breckner-preinvex functions;

  3. (iii)

    choosing \(h(t)=t^{-s}\) for \(s\in (0,1)\), we obtain the definition of generalized \((m,s)\)-Godunova–Levin–Dragomir-preinvex functions;

  4. (iv)

    choosing \(h(t)=t(1-t)\), we obtain the definition of generalized \((m, tgs)\)-preinvex functions;

  5. (v)

    choosing \(h(t)=\frac{\sqrt{t}}{2\sqrt{1-t}}\), we obtain the definition of generalized m-MT-preinvex functions.

It is worth mentioning here that, as far as we know, all the special cases considered above are new in the literature.

Main results

In order to derive our main results, we need the subsequent identity.

Lemma 2.1

Let \(A\subseteq \mathbb{R}\) be an open m-invex subset with respect to \(\eta : A\times A\times (0,1] \rightarrow \mathbb{R}\setminus \{0\}\) for some fixed \(m \in (0,1]\), and let \(a, b \in A\), \(a< b\) with \(\eta (b,a,m)>0\). Assume that \(f:A\rightarrow \mathbb{R}\) is a twice differentiable function on A such that \(f''\) is integrable on \([ma,ma+\eta (b,a,m) ]\). Then the following identity for Riemann–Liouville k-fractional integrals along with \(x \in [a, b]\), \(\lambda \in [0, 1]\), \(\mu > 0\), and \(k>0\) exists:

$$\begin{aligned} &I_{f, \eta }(\mu ,k;x,\lambda ,m,a,b) \\ &\quad =\frac{\eta ^{\frac{\mu }{k}+2}(x,a,m)}{(\frac{\mu }{k}+1)\eta (b,a,m)} \int ^{1}_{0}t \biggl[ \biggl(\frac{\mu }{k}+1 \biggr)\lambda -t^{\frac{\mu }{k}} \biggr] f'' \bigl(ma+t \eta (x,a,m) \bigr)\,\mathrm{d}t \\ &\quad \quad {} +\frac{(-1)^{\frac{\mu }{k}+2}\eta ^{\frac{\mu }{k}+2}(x,b,m)}{(\frac{\mu }{k}+1)\eta (b,a,m)} \int ^{1}_{0}t \biggl[ \biggl(\frac{\mu }{k}+1 \biggr)\lambda -t^{\frac{\mu }{k}} \biggr] f'' \bigl(mb+t \eta (x,b,m) \bigr)\,\mathrm{d}t, \end{aligned}$$
(2.1)

where

$$\begin{aligned} &I_{f, \eta }(\mu ,k;x,\lambda ,m,a,b) \\ &\quad:=\frac{1-\lambda }{\eta (b,a,m)} \bigl[\eta ^{\frac{\mu }{k}}(x,a,m)f \bigl(ma+\eta (x,a,m) \bigr) \\ &\quad\quad {} +(-1)^{\frac{\mu }{k}}\eta ^{\frac{\mu }{k}}(x,b,m)f \bigl(mb+\eta (x,b,m) \bigr) \bigr] \\ &\quad\quad {} +\frac{\lambda }{\eta (b,a,m)} \bigl[\eta ^{\frac{\mu }{k}}(x,a,m)f(ma) +(-1)^{\frac{\mu }{k}}\eta ^{\frac{\mu }{k}}(x,b,m)f(mb) \bigr] \\ &\quad\quad {} +\frac{\frac{1}{\frac{\mu }{k}+1}-\lambda }{\eta (b,a,m)} \bigl[(-1)^{\frac{\mu }{k}+1}\eta ^{\frac{\mu }{k}+1}(x,b,m)f' \bigl(mb+\eta (x,b,m) \bigr) \\ &\quad\quad {} -\eta ^{\frac{\mu }{k}+1}(x,a,m)f' \bigl(ma+\eta (x,a,m) \bigr) \bigr] \\ &\quad \quad -\frac{\Gamma _{k}(\mu +k)}{\eta (b,a,m)} \bigl[{}_{k}J^{\mu }_{(ma+\eta (x,a,m))^{-}} f(ma)+{}_{k}J^{\mu }_{(mb+\eta (x,b,m))^{+}}f(mb) \bigr] \end{aligned}$$

and \(\Gamma _{k}\) is the k-gamma function.

Proof

By integration by parts and replacing the variable, we can state

$$\begin{aligned} &\int ^{1}_{0}t \biggl[ \biggl(\frac{\mu }{k}+1 \biggr)\lambda -t^{\frac{\mu }{k}} \biggr]f'' \bigl(ma+t \eta (x,a,m) \bigr)\,\mathrm{d}t \\ &\quad =t \biggl[ \biggl(\frac{\mu }{k}+1 \biggr)\lambda -t^{\frac{\mu }{k}} \biggr] \frac{f' (ma+t\eta (x,a,m) )}{\eta (x,a,m)} \bigg| _{0}^{1} \\ &\quad \quad {} -\frac{\frac{\mu }{k}+1}{\eta (x,a,m)} \int ^{1}_{0} \bigl(\lambda -t^{\frac{\mu }{k}} \bigr)f' \bigl(ma+t\eta (x,a,m) \bigr)\,\mathrm{d}t \\ &\quad = \biggl[ \biggl(\frac{\mu }{k}+1 \biggr)\lambda -1 \biggr] \frac{f' (ma+\eta (x,a,m) )}{\eta (x,a,m)} \\ &\quad \quad {} -\frac{\frac{\mu }{k}+1}{\eta (x,a,m)} \biggl[ \bigl(\lambda -t^{\frac{\mu }{k}} \bigr) \frac{f (ma+t\eta (x,a,m) )}{\eta (x,a,m)} \bigg| _{0}^{1} \\ &\quad \quad {} + \int ^{1}_{0}\frac{\mu }{k}t^{\frac{\mu }{k}-1} \frac{f (ma+t\eta (x,a,m) )}{\eta (x,a,m)}\,\mathrm{d}t \biggr] \\ &\quad = \biggl[ \biggl(\frac{\mu }{k}+1 \biggr)\lambda -1 \biggr] \frac{f' (ma+\eta (x,a,m) )}{\eta (x,a,m)} \\ &\quad \quad {} +\frac{\frac{\mu }{k}+1}{\eta ^{2}(x,a,m)} \biggl[(1-\lambda )f \bigl(ma+\eta (x,a,m) \bigr)+ \lambda f(ma) \\ &\quad \quad {} -\frac{\frac{\mu }{k}}{\eta ^{\frac{\mu }{k}}(x,a,m)}\int ^{ma+\eta (x,a,m)}_{ma}(u-ma)^{\frac{\mu }{k}-1}f(u)\,\mathrm{d}u \biggr] \\ &\quad = \biggl[ \biggl(\frac{\mu }{k}+1 \biggr)\lambda -1 \biggr]\frac{f' (ma+\eta (x,a,m) )}{\eta (x,a,m)} \\ &\quad \quad {} +\frac{\frac{\mu }{k}+1}{\eta ^{2}(x,a,m)} \biggl[(1-\lambda )f \bigl(ma+\eta (x,a,m) \bigr)+ \lambda f(ma) \\ &\quad \quad {} -\frac{\Gamma _{k}(\mu +k)}{\eta ^{\frac{\mu }{k}}(x,a,m)}{}_{k}J^{\mu }_{(ma+\eta (x,a,m))^{-}}f(ma)\biggr]. \end{aligned}$$
(2.2)

Similarly, we get

$$\begin{aligned} &\int ^{1}_{0}t \biggl[ \biggl(\frac{\mu }{k}+1 \biggr)\lambda -t^{\frac{\mu }{k}} \biggr]f'' \bigl(mb+t \eta (x,b,m) \bigr)\,\mathrm{d}t \\ &\quad = \biggl[ \biggl(\frac{\mu }{k}+1 \biggr)\lambda -1 \biggr] \frac{f' (mb+\eta (x,b,m) )}{\eta (x,b,m)} \\ &\quad \quad {} +\frac{\frac{\mu }{k}+1}{\eta ^{2}(x,b,m)} \biggl[(1-\lambda )f \bigl(mb+\eta (x,b,m) \bigr)+ \lambda f(mb) \\ &\quad \quad {} -\frac{\Gamma _{k}(\mu +k)}{\eta ^{\frac{\mu }{k}}(x,b,m)}{}_{k}J^{\mu }_{(ma+\eta (x,b,m))^{-}}f(mb)\biggr]. \end{aligned}$$
(2.3)

Multiplying both sides of (2.2) and (2.3) by \(\frac{\eta ^{\frac{\mu }{k}+2}(x,a,m)}{(\frac{\mu }{k}+1)\eta (b,a,m)}\) and \(\frac{(-1)^{\frac{\mu }{k}+2}\eta ^{\frac{\mu }{k}+2}(x,b,m)}{(\frac{\mu }{k}+1)\eta (b,a,m)}\), respectively, and adding the resulting identities together, we obtain the desired result. □

Remark 2.1

In Lemma 2.1, if we put \(k=1\) and \(\eta (b,a,m)=b-ma\) along with \(m=1\), then we get the following identity:

$$\begin{aligned} &(1-\lambda ) \biggl[\frac{(x-a)^{\mu }+(b-x)^{\mu }}{b-a} \biggr]f(x)+ \lambda \biggl[\frac{(x-a)^{\mu }f(a)+(b-x)^{\mu }f(b)}{b-a} \biggr] \\ &\quad \quad {} + \biggl(\frac{1}{\mu +1}-\lambda \biggr) \biggl[\frac{(b-x)^{\mu +1}-(x-a)^{\mu +1}}{b-a} \biggr]f'(x)-\frac{\Gamma (\mu +1)}{b-a} \bigl[J^{\mu }_{x^{-}}f(a)+J^{\mu }_{x^{+}}f(b) \bigr] \\ &\quad =\frac{(x-a)^{\mu +2}}{(\mu +1)(b-a)} \int ^{1}_{0}t \bigl[ (\mu +1 )\lambda -t^{\mu } \bigr]f'' \bigl(tx+(1-t)a \bigr)\,\mathrm{d}t \\ &\quad \quad {} +\frac{(b-x)^{\mu +2}}{(\mu +1)(b-a)} \int ^{1}_{0}t \bigl[ (\mu +1 )\lambda -t^{\mu } \bigr]f'' \bigl(tx+(1-t)b \bigr)\,\mathrm{d}t, \end{aligned}$$

which is proved by İşcan in [34]. Further, if we put \(\mu =1, \lambda =\frac{1}{2}\), and \(x=a\) or \(x=b\), then the above identity recaptures Lemma 1 in [35].

Using Lemma 2.1, we now state the following theorem.

Theorem 2.1

Let \(A\subseteq \mathbb{R}\) be an open m-invex subset with respect to \(\eta : A\times A\times (0,1] \rightarrow \mathbb{R}\setminus \{0\}\) for some fixed \(m \in (0,1]\), and let \(a, b \in A\), \(a< b\) with \(\eta (b,a,m)>0\). Assume that \(f:A\rightarrow \mathbb{R}\) is a twice differentiable function on A such that \(f''\) is integrable on \([ma,ma+\eta (b,a,m) ]\). If \(\vert f'' \vert ^{q}\) for \(q\geq 1\) is a generalized \((m,h)\)-preinvex function with respect to η and \(h:[0,1]\rightarrow \mathbb{R}_{0}\), then the following inequality for k-fractional integrals with \(x \in [a,b]\), \(\lambda \in [0, 1]\), \(\mu > 0\), \(k>0\) exists:

$$\begin{aligned} & \bigl\vert I_{f,\eta }(\mu,k;x,\lambda ,m,a,b) \bigr\vert \\ &\quad \leq C_{0}^{1-\frac{1}{q}}(k,\mu ,\lambda ) \biggl\{ \biggl\vert \frac{\eta ^{\frac{\mu }{k}+2}(x,a,m)}{(\frac{\mu }{k}+1)\eta (b,a,m)} \biggr\vert \bigl[ \bigl\vert f''(x) \bigr\vert ^{q}C_{1}(k,\mu ,\lambda ;h) \\ &\quad \quad {} +m \bigl\vert f''(a) \bigr\vert ^{q}C_{2}(k,\mu ,\lambda ;h) \bigr]^{\frac{1}{q}} \\ &\quad \quad {} + \biggl\vert \frac{(-1)^{\frac{\mu }{k}+2}\eta ^{\frac{\mu }{k}+2}(x,b,m)}{(\frac{\mu }{k}+1)\eta (b,a,m)} \biggr\vert \bigl[ \bigl\vert f''(x) \bigr\vert ^{q}C_{1}(k, \mu ,\lambda ;h) \\ &\quad \quad {} +m \bigl\vert f''(b) \bigr\vert ^{q}C_{2}(k,\mu ,\lambda ;h) \bigr]^{\frac{1}{q}} \biggr\} , \end{aligned}$$
(2.4)

where

$$\begin{aligned} &C_{0}(k,\mu ,\lambda )= \int ^{1}_{0}t \biggl\vert \biggl( \frac{\mu }{k}+1 \biggr)\lambda -t^{\frac{\mu }{k}} \biggr\vert \,\mathrm{d}t \\ &\hphantom{C_{0}(k,\mu ,\lambda )}=\textstyle\begin{cases} \frac{\frac{\mu }{k}[(\frac{\mu }{k}+1)\lambda ]^{1+\frac{2k}{\mu }}}{\frac{\mu }{k}+2}-\frac{(\frac{\mu }{k}+1)\lambda }{2}+\frac{1}{\frac{\mu }{k}+2},&0\leq \lambda \leq \frac{1}{\frac{\mu }{k}+1}, \\ \frac{(\frac{\mu }{k}+1)\lambda }{2}-\frac{1}{\frac{\mu }{k}+2},&\frac{1}{\frac{\mu }{k}+1}< \lambda \leq 1, \end{cases}\displaystyle \end{aligned}$$
(2.5)
$$\begin{aligned} & C_{1}(k,\mu ,\lambda ;h)= \int ^{1}_{0}t \biggl\vert \biggl( \frac{\mu }{k}+1 \biggr)\lambda -t^{\frac{\mu }{k}} \biggr\vert h(t)\,\mathrm{d}t,\quad 0\leq \lambda \leq 1, \end{aligned}$$
(2.6)

and

$$ C_{2}(k,\mu ,\lambda ;h)= \int ^{1}_{0}t \biggl\vert \biggl( \frac{\mu }{k}+1 \biggr)\lambda -t^{\frac{\mu }{k}} \biggr\vert h(1-t)\,\mathrm{d}t,\quad 0\leq \lambda \leq 1. $$
(2.7)

Proof

Applying Lemma 2.1 and the power mean inequality, we have

$$\begin{aligned} & \bigl\vert I_{f,\eta }(\mu ,k;x,\lambda,m,a,b) \bigr\vert \\ &\quad\leq \biggl\vert \frac{\eta ^{\frac{\mu }{k}+2}(x,a,m)}{(\frac{\mu }{k}+1)\eta (b,a,m)} \biggr\vert \int ^{1}_{0}t \biggl\vert \biggl( \frac{\mu }{k}+1 \biggr)\lambda -t^{\frac{\mu }{k}} \biggr\vert \bigl\vert f'' \bigl(ma+t\eta (x,a,m) \bigr) \bigr\vert \,\mathrm{d}t \\ &\quad\quad {} + \biggl\vert \frac{(-1)^{\frac{\mu }{k}+2}\eta ^{\frac{\mu }{k}+2}(x,b,m)}{(\frac{\mu }{k}+1)\eta (b,a,m)} \biggr\vert \int ^{1}_{0}t \biggl\vert \biggl( \frac{\mu }{k}+1 \biggr)\lambda -t^{\frac{\mu }{k}} \biggr\vert \bigl\vert f'' \bigl(mb+t\eta (x,b,m) \bigr) \bigr\vert \,\mathrm{d}t \\ &\quad\leq \biggl\vert \frac{\eta ^{\frac{\mu }{k}+2}(x,a,m)}{(\frac{\mu }{k}+1)\eta (b,a,m)} \biggr\vert \biggl( \int ^{1}_{0}t \biggl\vert \biggl( \frac{\mu }{k}+1 \biggr)\lambda -t^{\frac{\mu }{k}} \biggr\vert \,\mathrm{d}t \biggr)^{1-\frac{1}{q}} \\ &\quad\quad {} \times \biggl(\int ^{1}_{0}t \biggl\vert \biggl(\frac{\mu }{k}+1 \biggr)\lambda -t^{\frac{\mu }{k}} \biggr\vert \bigl\vert f'' \bigl(ma+t\eta (x,a,m) \bigr) \bigr\vert ^{q}\,\mathrm{d}t \biggr)^{\frac{1}{q}} \\ &\quad\quad {} + \biggl\vert \frac{(-1)^{\frac{\mu }{k}+2}\eta ^{\frac{\mu }{k}+2}(x,b,m)}{(\frac{\mu }{k}+1)\eta (b,a,m)} \biggr\vert \biggl( \int ^{1}_{0}t \biggl\vert \biggl(\frac{\mu }{k}+1 \biggr)\lambda -t^{\frac{\mu }{k}} \biggr\vert \,\mathrm{d}t \biggr)^{1-\frac{1}{q}} \\ &\quad\quad {} \times \biggl(\int ^{1}_{0}t \biggl\vert \biggl( \frac{\mu }{k}+1 \biggr)\lambda -t^{\frac{\mu }{k}} \biggr\vert \bigl\vert f'' \bigl(mb+t\eta (x,b,m) \bigr) \bigr\vert ^{q}\,\mathrm{d}t \biggr)^{\frac{1}{q}} \\ &\quad =C_{0}^{1-\frac{1}{q}}(k,\mu,\lambda ) \\ &\quad \quad {} \times \biggl\{ \biggl\vert \frac{\eta ^{\frac{\mu }{k}+2}(x,a,m)}{(\frac{\mu }{k}+1)\eta (b,a,m)} \biggr\vert \biggl( \int ^{1}_{0}t \biggl\vert \biggl( \frac{\mu }{k}+1 \biggr)\lambda -t^{\frac{\mu }{k}} \biggr\vert \bigl\vert f'' \bigl(ma+t\eta (x,a,m) \bigr) \bigr\vert ^{q}\,\mathrm{d}t \biggr)^{\frac{1}{q}} \\ &\quad \quad {} + \biggl\vert \frac{(-1)^{\frac{\mu }{k}+2}\eta ^{\frac{\mu }{k}+2}(x,b,m)}{(\frac{\mu }{k}+1)\eta (b,a,m)} \biggr\vert \biggl( \int ^{1}_{0}t \biggl\vert \biggl( \frac{\mu }{k}+1 \biggr)\lambda \\ &\quad \quad {} -t^{\frac{\mu }{k}} \biggr\vert \bigl\vert f'' \bigl(mb+t\eta (x,b,m) \bigr) \bigr\vert ^{q}\,\mathrm{d}t \biggr)^{\frac{1}{q}} \biggr\} . \end{aligned}$$
(2.8)

Since \(\vert f'' \vert ^{q}\) is generalized \((m,h)\)-preinvex on \([ma,ma+\eta (b,a,m) ]\), we get

$$\begin{aligned} &\int ^{1}_{0}t \biggl\vert \biggl( \frac{\mu }{k}+1 \biggr)\lambda -t^{\frac{\mu }{k}} \biggr\vert \bigl\vert f'' \bigl(ma+t\eta (x,a,m) \bigr) \bigr\vert ^{q}\,\mathrm{d}t \\ &\quad \leq \int ^{1}_{0}t \biggl\vert \biggl( \frac{\mu }{k}+1 \biggr)\lambda -t^{\frac{\mu }{k}} \biggr\vert \bigl[h(t) \bigl\vert f''(x) \bigr\vert ^{q}+mh(1-t) \bigl\vert f''(a) \bigr\vert ^{q} \bigr]\,\mathrm{d}t \\ &\quad = \bigl\vert f''(x) \bigr\vert ^{q}C_{1}(k,\mu ,\lambda ;h)+m \bigl\vert f''(a) \bigr\vert ^{q}C_{2}(k, \mu ,\lambda ;h) \end{aligned}$$
(2.9)

and

$$\begin{aligned} & \int ^{1}_{0}t \biggl\vert \biggl( \frac{\mu }{k}+1 \biggr)\lambda -t^{\frac{\mu }{k}} \biggr\vert \bigl\vert f'' \bigl(mb+t\eta (x,b,m) \bigr) \bigr\vert ^{q}\,\mathrm{d}t \\ &\quad \leq \int ^{1}_{0}t \biggl\vert \biggl( \frac{\mu }{k}+1 \biggr)\lambda -t^{\frac{\mu }{k}} \biggr\vert \bigl[h(t) \bigl\vert f''(x) \bigr\vert ^{q}+mh(1-t) \bigl\vert f''(b) \bigr\vert ^{q} \bigr]\,\mathrm{d}t \\ &\quad = \bigl\vert f''(x) \bigr\vert ^{q}C_{1}(k,\mu ,\lambda ;h)+m \bigl\vert f''(b) \bigr\vert ^{q}C_{2}(k, \mu ,\lambda ;h), \end{aligned}$$
(2.10)

where \(C_{0}(k,\mu ,\lambda )\), \(C_{1}(k,\mu ,\lambda ;h)\), and \(C_{2}(k,\mu ,\lambda ;h)\) are defined by (2.5)–(2.7), respectively. Hence, if we use (2.9) and (2.10) in (2.8), we can get the desired result. This completes the proof. □

Let us point out some special cases of Theorem 2.1.

I. If \(h(t)=t^{s}\) in Theorem 2.1, then we have the following results.

Corollary 2.1

In Theorem 2.1, if \(\vert f'' \vert ^{q}\) for \(q\geq 1\) is generalized \((m,s)\)-Breckner-preinvex functions, then, for \(s\in (0,1]\) and \(m\in (0,1]\), we have

$$\begin{aligned} & \bigl\vert I_{f,\eta }(\mu ,k;x,\lambda ,m,a,b) \bigr\vert \\ &\quad\leq C_{0}^{1-\frac{1}{q}}(k,\mu ,\lambda ) \\ &\quad\quad {} \times \biggl\{ \biggl\vert \frac{\eta ^{\frac{\mu }{k}+2}(x,a,m)}{(\frac{\mu }{k}+1)\eta (b,a,m)} \biggr\vert \bigl[ \bigl\vert f''(x) \bigr\vert ^{q}T_{1}(k, \mu ,\lambda ;s)+m \bigl\vert f''(a) \bigr\vert ^{q}T_{2}(k,\mu ,\lambda ;s) \bigr]^{\frac{1}{q}} \\ &\quad\quad {} + \biggl\vert \frac{(-1)^{\frac{\mu }{k}+2}\eta ^{\frac{\mu }{k}+2}(x,b,m)}{(\frac{\mu }{k}+1)\eta (b,a,m)} \biggr\vert \bigl[ \bigl\vert f''(x) \bigr\vert ^{q}T_{1}(k, \mu ,\lambda ;s)+m \bigl\vert f''(b) \bigr\vert ^{q}T_{2}(k,\mu ,\lambda ;s) \bigr]^{\frac{1}{q}} \biggr\} , \end{aligned}$$

where we use the fact that

$$\begin{aligned}& T_{1}(k,\mu ,\lambda ;s)= \textstyle\begin{cases} \frac{\frac{2\mu }{k} [ (\frac{\mu }{k}+1 )\lambda ]^{1+\frac{k(s+2)}{\mu }}}{(s+2)(\frac{\mu }{k}+s+2)}-\frac{(\frac{\mu }{k}+1)\lambda }{s+2}+\frac{1}{\frac{\mu }{k}+s+2}, &0\leq \lambda \leq \frac{1}{\frac{\mu }{k}+1}, \\ \frac{(\frac{\mu }{k}+1)\lambda }{s+2}-\frac{1}{\frac{\mu }{k}+s+2}, &\frac{1}{\frac{\mu }{k}+1}< \lambda \leq 1, \end{cases}\displaystyle \\& T_{2}(k,\mu ,\lambda ;s)= \textstyle\begin{cases} \left [ \textstyle\begin{array}{@{}c@{}} \beta (\frac{\mu }{k}+2,s+1 )- (\frac{\mu }{k}+1 )\lambda \beta (2,s+1 )\\ {}+2 (\frac{\mu }{k}+1 )\lambda \beta ( [ (\frac{\mu }{k}+1 )\lambda ]^{\frac{k}{\mu }};2,s+1 )\\ {}-2\beta ( [ (\frac{\mu }{k}+1 )\lambda ]^{\frac{k}{\mu }};\frac{\mu }{k}+2,s+1 ) \end{array}\displaystyle \right ] , &0\leq \lambda \leq \frac{1}{\frac{\mu }{k}+1}, \\ (\frac{\mu }{k}+1 )\lambda \beta (2,s+1)-\beta (\frac{\mu }{k}+2,s+1 ), &\frac{1}{\frac{\mu }{k}+1}< \lambda \leq 1, \end{cases}\displaystyle \end{aligned}$$

and \(C_{0}(k,\mu ,\lambda )\) is defined by (2.5).

Corollary 2.2

In Theorem 2.1, if the mapping \(\eta (b,a,m)=b-ma\) along with \(m=1\), taking \(x=\frac{a+b}{2}\), then for \(s\in (0,1]\), we have the following inequality for s-convex functions:

$$\begin{aligned} & \biggl\vert \frac{2^{\frac{\mu }{k}-1}}{(b-a)^{\frac{\mu }{k}-1}}I_{f} \biggl(\mu ,k;\frac{a+b}{2},\lambda ,1,a,b \biggr) \biggr\vert \\ &\quad = \biggl\vert (1-\lambda )f \biggl(\frac{a+b}{2} \biggr)+\lambda \biggl[ \frac{f(a)+f(b)}{2} \biggr] \\ &\quad \quad {} -\frac{2^{\frac{\mu }{k}-1}\Gamma _{k}(\mu +k)}{(b-a)^{\frac{\mu }{k}}} \bigl[{}_{k}J^{\mu }_{(\frac{a+b}{2})^{-}} f(a)+{}_{k}J^{\mu }_{(\frac{a+b}{2})^{+}}f(b) \bigr] \biggr\vert \\ &\quad \leq \frac{(b-a)^{2}}{8 (\frac{\mu }{k}+1 )} C_{0}^{1-\frac{1}{q}}(k,\mu ,\lambda ) \biggl\{ \biggl[ \biggl\vert f'' \biggl(\frac{a+b}{2} \biggr) \biggr\vert ^{q}T_{1}(k,\mu ,\lambda ;s)+ \bigl\vert f''(a) \bigr\vert ^{q}T_{2}(k, \mu ,\lambda ;s) \biggr]^{\frac{1}{q}} \\ &\quad \quad {} + \biggl[ \biggl\vert f'' \biggl( \frac{a+b}{2} \biggr) \biggr\vert ^{q}T_{1}(k,\mu , \lambda ;s)+ \bigl\vert f''(b) \bigr\vert ^{q}T_{2}(k,\mu ,\lambda ;s) \biggr]^{\frac{1}{q}} \biggr\} . \end{aligned}$$

Remark 2.2

In Corollary 2.2,

  1. (i)

    taking \(\lambda =\frac{1}{2}\), we obtain

    $$\begin{aligned} & \biggl\vert \frac{2^{\frac{\mu }{k}-1}}{(b-a)^{\frac{\mu }{k}-1}}I_{f} \biggl(\mu ,k;\frac{a+b}{2},\frac{1}{2},1,a,b \biggr) \biggr\vert \\ &\quad = \biggl\vert \frac{1}{4} \biggl[f(a)+2f \biggl(\frac{a+b}{2} \biggr)+f(b) \biggr]-\frac{2^{\frac{\mu }{k}-1}\Gamma _{k}(\mu +k)}{(b-a)^{\frac{\mu }{k}}} \bigl[{}_{k}J^{\mu }_{(\frac{a+b}{2})^{-}}f(a)+{}_{k}J^{\mu }_{(\frac{a+b}{2})^{+}}f(b) \bigr] \biggr\vert \\ &\quad \leq \frac{(b-a)^{2}}{8 (\frac{\mu }{k}+1 )} C_{0}^{1-\frac{1}{q}} \biggl(k,\mu , \frac{1}{2} \biggr) \\ &\quad \quad {} \times \biggl\{ \biggl[ \biggl\vert f'' \biggl(\frac{a+b}{2} \biggr) \biggr\vert ^{q}T_{1} \biggl(k,\mu ,\frac{1}{2};s \biggr)+ \bigl\vert f''(a) \bigr\vert ^{q}T_{2} \biggl(k,\mu ,\frac{1}{2};s \biggr) \biggr]^{\frac{1}{q}} \\ &\quad \quad {} + \biggl[ \biggl\vert f'' \biggl( \frac{a+b}{2} \biggr) \biggr\vert ^{q}T_{1} \biggl(k, \mu ,\frac{1}{2};s \biggr)+ \bigl\vert f''(b) \bigr\vert ^{q}T_{2} \biggl(k,\mu ,\frac{1}{2};s \biggr) \biggr]^{\frac{1}{q}} \biggr\} ; \end{aligned}$$
  2. (ii)

    taking \(\lambda =\frac{1}{3}\), we obtain

    $$\begin{aligned} & \biggl\vert \frac{2^{\frac{\mu }{k}-1}}{(b-a)^{\frac{\mu }{k}-1}}I_{f} \biggl(\mu ,k;\frac{a+b}{2},\frac{1}{3},1,a,b \biggr) \biggr\vert \\ &\quad = \biggl\vert \frac{1}{6} \biggl[f(a)+4f \biggl(\frac{a+b}{2} \biggr)+f(b) \biggr] \\ &\quad \quad {}-\frac{2^{\frac{\mu }{k}-1}\Gamma _{k}(\mu +k)}{(b-a)^{\frac{\mu }{k}}} \bigl[{}_{k}J^{\mu }_{(\frac{a+b}{2})^{-}}f(a)+{}_{k}J^{\mu }_{(\frac{a+b}{2})^{+}}f(b) \bigr] \biggr\vert \\ &\quad \leq \frac{(b-a)^{2}}{8 (\frac{\mu }{k}+1 )} C_{0}^{1-\frac{1}{q}} \biggl(k,\mu ,\frac{1}{3} \biggr) \\ &\quad \quad {} \times \biggl\{ \biggl[ \biggl\vert f'' \biggl(\frac{a+b}{2} \biggr) \biggr\vert ^{q}T_{1} \biggl(k,\mu ,\frac{1}{3};s \biggr)+ \bigl\vert f''(a) \bigr\vert ^{q}T_{2} \biggl(k,\mu ,\frac{1}{3};s \biggr) \biggr]^{\frac{1}{q}} \\ &\quad \quad {} + \biggl[ \biggl\vert f'' \biggl( \frac{a+b}{2} \biggr) \biggr\vert ^{q}T_{1} \biggl(k, \mu ,\frac{1}{3};s \biggr)+ \bigl\vert f''(b) \bigr\vert ^{q}T_{2} \biggl(k,\mu ,\frac{1}{3};s \biggr) \biggr]^{\frac{1}{q}} \biggr\} . \end{aligned}$$

    Specially, if we put \(k=\mu =s=1\), then we obtain

    $$\begin{aligned} & \biggl\vert \frac{1}{6} \biggl[f(a)+4f \biggl(\frac{a+b}{2} \biggr)+f(b) \biggr]-\frac{1}{b-a} \int ^{b}_{a} f(x)\,\mathrm{d}x \biggr\vert \\ &\quad \leq \frac{(b-a)^{2}}{162} \biggl\{ \biggl[\frac{59}{96} \biggl\vert f'' \biggl(\frac{a+b}{2} \biggr) \biggr\vert ^{q}+\frac{37}{96} \bigl\vert f''(a) \bigr\vert ^{q} \biggr]^{\frac{1}{q}} \\ &\quad \quad {} + \biggl[\frac{59}{96} \biggl\vert f'' \biggl(\frac{a+b}{2} \biggr) \biggr\vert ^{q}+ \frac{37}{96} \bigl\vert f''(b) \bigr\vert ^{q} \biggr]^{\frac{1}{q}} \biggr\} \\ &\quad \leq \frac{(b-a)^{2}}{162} \biggl\{ \biggl(\frac{59\vert f''(a) \vert ^{q}+133\vert f''(b) \vert ^{q}}{192} \biggr)^{\frac{1}{q}} \\ &\quad \quad {} + \biggl(\frac{59\vert f''(b) \vert ^{q}+133\vert f''(a) \vert ^{q}}{192} \biggr)^{\frac{1}{q}} \biggr\} . \end{aligned}$$
    (2.11)

    It is noted that the result of the first inequality in (2.11) is proved by İşcan in [34], which is better than the result presented by Sarikaya et al. in [36, Theorem 6].

Remark 2.3

In Corollary 2.2, if we take \(k=1\) and \(\lambda =0,1\), we have the results (f) and (h) in [34, Corollary 2.3], respectively. Further, if we take \(\mu =1\), we have the results (g) and (i) in [34, Corollary 2.3], respectively.

II. If \(h(t)=t^{-s}\) in Theorem 2.1, then we have the following results.

Corollary 2.3

In Theorem 2.1, if \(\vert f'' \vert ^{q}\) for \(q\geq 1\) is generalized \((m,s)\)-Godunova–Levin–Dragomir-preinvex functions, then, for \(s\in (0,1)\) and \(m\in (0,1]\), we have

$$\begin{aligned} & \bigl\vert I_{f,\eta }(\mu ,k;x,\lambda ,m,a,b) \bigr\vert \\ &\quad \leq C_{0}^{1-\frac{1}{q}}(k,\mu ,\lambda ) \\ &\quad \quad {} \times \biggl\{ \biggl\vert \frac{\eta ^{\frac{\mu }{k}+2}(x,a,m)}{(\frac{\mu }{k}+1)\eta (b,a,m)} \biggr\vert \bigl[ \bigl\vert f''(x) \bigr\vert ^{q}U_{1}(k, \mu ,\lambda ;s)+m \bigl\vert f''(a) \bigr\vert ^{q}U_{2}(k,\mu ,\lambda ;s) \bigr]^{\frac{1}{q}} \\ &\quad \quad {} + \biggl\vert \frac{(-1)^{\frac{\mu }{k}+2}\eta ^{\frac{\mu }{k}+2}(x,b,m)}{(\frac{\mu }{k}+1)\eta (b,a,m)} \biggr\vert \bigl[ \bigl\vert f''(x) \bigr\vert ^{q}U_{1}(k, \mu ,\lambda ;s)+m \bigl\vert f''(b) \bigr\vert ^{q}U_{2}(k,\mu ,\lambda ;s) \bigr]^{\frac{1}{q}} \biggr\} , \end{aligned}$$

where we use the fact that

$$\begin{aligned}& U_{1}(k,\mu ,\lambda ;s)= \textstyle\begin{cases} \frac{\frac{2\mu }{k} [ (\frac{\mu }{k}+1 )\lambda ]^{1+\frac{k(2-s)}{\mu }}}{(2-s)(\frac{\mu }{k}-s+2)}-\frac{(\frac{\mu }{k}+1)\lambda }{2-s}+\frac{1}{\frac{\mu }{k}-s+2}, &0\leq \lambda \leq \frac{1}{\frac{\mu }{k}+1}, \\ \frac{(\frac{\mu }{k}+1)\lambda }{2-s}-\frac{1}{\frac{\mu }{k}-s+2}, &\frac{1}{\frac{\mu }{k}+1}< \lambda \leq 1, \end{cases}\displaystyle \\& U_{2}(k,\mu ,\lambda ;s)= \textstyle\begin{cases} \left [ \textstyle\begin{array}{@{}c@{}} \beta (\frac{\mu }{k}+2,1-s )- (\frac{\mu }{k}+1 )\lambda \beta (2,1-s )\\ {}+2 (\frac{\mu }{k}+1 )\lambda \beta ( [ (\frac{\mu }{k}+1 )\lambda ]^{\frac{k}{\mu }};2,1-s ) \\ {}-2\beta ( [ (\frac{\mu }{k}+1 )\lambda ]^{\frac{k}{\mu }};\frac{\mu }{k}+2,1-s ) \end{array}\displaystyle \right ] , &0\leq \lambda \leq \frac{1}{\frac{\mu }{k}+1}, \\ (\frac{\mu }{k}+1 )\lambda \beta (2,1-s)-\beta (\frac{\mu }{k}+2,1-s ), &\frac{1}{\frac{\mu }{k}+1}< \lambda \leq 1, \end{cases}\displaystyle \end{aligned}$$

and \(C_{0}(k,\mu ,\lambda )\) is defined by (2.5).

Corollary 2.4

In Theorem 2.1, if the mapping \(\eta (b,a,m)=b-ma\) together with \(m=1\), taking \(x=\frac{a+b}{2}\), for \(s\in (0,1)\), we have the following inequality for s-Godunova–Levin functions:

$$\begin{aligned} & \biggl\vert (1-\lambda )f \biggl(\frac{a+b}{2} \biggr)+\lambda \biggl[\frac{f(a)+f(b)}{2} \biggr] \\ &\quad \quad {} -\frac{2^{\frac{\mu }{k}-1}\Gamma _{k}(\mu +k)}{(b-a)^{\frac{\mu }{k}}} \bigl[{}_{k}J^{\mu }_{(\frac{a+b}{2})^{-}} f(a)+{}_{k}J^{\mu }_{(\frac{a+b}{2})^{+}}f(b) \bigr] \biggr\vert \\ &\quad \leq \frac{(b-a)^{2}}{8 (\frac{\mu }{k}+1 )} C_{0}^{1-\frac{1}{q}}(k,\mu ,\lambda ) \biggl\{ \biggl[ \biggl\vert f'' \biggl(\frac{a+b}{2} \biggr) \biggr\vert ^{q}U_{1}(k,\mu ,\lambda ;s)+ \bigl\vert f''(a) \bigr\vert ^{q}U_{2}(k, \mu ,\lambda ;s) \biggr]^{\frac{1}{q}} \\ &\quad \quad {} + \biggl[ \biggl\vert f'' \biggl( \frac{a+b}{2} \biggr) \biggr\vert ^{q}U_{1}(k,\mu , \lambda ;s)+ \bigl\vert f''(b) \bigr\vert ^{q}U_{2}(k,\mu ,\lambda ;s) \biggr]^{\frac{1}{q}} \biggr\} . \end{aligned}$$

Remark 2.4

In Corollary 2.4,

  1. (a)

    if \(\lambda =\frac{1}{3}\), then we obtain

    $$\begin{aligned} & \biggl\vert \frac{2^{\frac{\mu }{k}-1}}{(b-a)^{\frac{\mu }{k}-1}}I_{f} \biggl(\mu ,k;\frac{a+b}{2},\frac{1}{3},1,a,b \biggr) \biggr\vert \\ &\quad = \biggl\vert \frac{1}{6} \biggl[f(a)+4f \biggl(\frac{a+b}{2} \biggr)+f(b) \biggr]-\frac{2^{\frac{\mu }{k}-1}\Gamma _{k}(\mu +k)}{(b-a)^{\frac{\mu }{k}}} \bigl[{}_{k}J^{\mu }_{(\frac{a+b}{2})^{-}} f(a)+{}_{k}J^{\mu }_{(\frac{a+b}{2})^{+}}f(b) \bigr] \biggr\vert \\ &\quad \leq \frac{(b-a)^{2}}{8 (\frac{\mu }{k}+1 )} C_{0}^{1-\frac{1}{q}} \biggl(k,\mu , \frac{1}{3} \biggr) \\ &\quad \quad {} \times \biggl\{ \biggl[ \biggl\vert f'' \biggl(\frac{a+b}{2} \biggr) \biggr\vert ^{q}U_{1} \biggl(k,\mu ,\frac{1}{3};s \biggr)+ \bigl\vert f''(a) \bigr\vert ^{q}U_{2} \biggl(k,\mu ,\frac{1}{3};s \biggr) \biggr]^{\frac{1}{q}} \\ &\quad \quad {} + \biggl[ \biggl\vert f'' \biggl( \frac{a+b}{2} \biggr) \biggr\vert ^{q}U_{1} \biggl(k, \mu ,\frac{1}{3};s \biggr)+ \bigl\vert f''(b) \bigr\vert ^{q}U_{2} \biggl(k,\mu ,\frac{1}{3};s \biggr) \biggr]^{\frac{1}{q}} \biggr\} . \end{aligned}$$

    Specially, if we put \(k=1=\mu \), then we obtain a Simpson-type inequality:

    $$\begin{aligned} & \biggl\vert I_{f} \biggl(1,1; \frac{a+b}{2},\frac{1}{3},1,a,b \biggr) \biggr\vert \\ &\quad = \biggl\vert \frac{1}{6} \biggl[f(a)+4f \biggl(\frac{a+b}{2} \biggr)+f(b) \biggr]-\frac{1}{b-a} \int ^{b}_{a} f(x)\,\mathrm{d}x \biggr\vert \\ &\quad \leq \frac{(b-a)^{2}}{16} \biggl(\frac{8}{81} \biggr)^{1-\frac{1}{q}} \biggl\{ \biggl[ \biggl\vert f'' \biggl(\frac{a+b}{2}\biggr) \biggr\vert ^{q}U_{1} \biggl(1,1, \frac{1}{3};s \biggr) \\ &\quad \quad {}+ \bigl\vert f''(a) \bigr\vert ^{q}U_{2} \biggl(1,1,\frac{1}{3};s \biggr)\biggr]^{\frac{1}{q}} \\ &\quad \quad {} + \biggl[ \biggl\vert f'' \biggl(\frac{a+b}{2} \biggr) \biggr\vert ^{q}U_{1} \biggl(1,1,\frac{1}{3};s \biggr) + \bigl\vert f''(b)\bigr\vert ^{q}U_{2} \biggl(1,1,\frac{1}{3};s \biggr) \biggr]^{\frac{1}{q}} \biggr\} ; \end{aligned}$$
  2. (b)

    if \(\lambda =\frac{1}{2}\), then we obtain

    $$\begin{aligned} & \biggl\vert \frac{2^{\frac{\mu }{k}-1}}{(b-a)^{\frac{\mu }{k}-1}}I_{f} \biggl(\mu ,k;\frac{a+b}{2},\frac{1}{2},1,a,b \biggr) \biggr\vert \\ &\quad = \biggl\vert \frac{1}{4} \biggl[f(a)+2f \biggl(\frac{a+b}{2} \biggr)+f(b) \biggr]-\frac{2^{\frac{\mu }{k}-1}\Gamma _{k}(\mu +k)}{(b-a)^{\frac{\mu }{k}}} \bigl[{}_{k}J^{\mu }_{(\frac{a+b}{2})^{-}} f(a)+{}_{k}J^{\mu }_{(\frac{a+b}{2})^{+}}f(b) \bigr] \biggr\vert \\ &\quad \leq \frac{(b-a)^{2}}{8 (\frac{\mu }{k}+1 )} C_{0}^{1-\frac{1}{q}} \biggl(k,\mu , \frac{1}{2} \biggr) \\ &\quad \quad {} \times \biggl\{ \biggl[ \biggl\vert f'' \biggl(\frac{a+b}{2} \biggr) \biggr\vert ^{q}U_{1} \biggl(k,\mu ,\frac{1}{2};s \biggr)+ \bigl\vert f''(a) \bigr\vert ^{q}U_{2} \biggl(k,\mu ,\frac{1}{2};s \biggr) \biggr]^{\frac{1}{q}} \\ &\quad \quad {} + \biggl[ \biggl\vert f'' \biggl( \frac{a+b}{2} \biggr) \biggr\vert ^{q}U_{1} \biggl(k, \mu ,\frac{1}{2};s \biggr)+ \bigl\vert f''(b) \bigr\vert ^{q}U_{2} \biggl(k,\mu ,\frac{1}{2};s \biggr) \biggr]^{\frac{1}{q}} \biggr\} . \end{aligned}$$

    Specially, if we put \(k=1=\mu \), then we obtain an averaged midpoint-trapezoid-type inequality:

    $$\begin{aligned} & \biggl\vert I_{f} \biggl(1,1; \frac{a+b}{2},\frac{1}{2},1,a,b \biggr) \biggr\vert \\ &\quad = \biggl\vert \frac{1}{4} \biggl[f(a)+2f \biggl(\frac{a+b}{2} \biggr)+f(b) \biggr]-\frac{1}{b-a} \int ^{b}_{a} f(x)\,\mathrm{d}x \biggr\vert \\ &\quad \leq \frac{(b-a)^{2}}{16} \biggl(\frac{1}{6} \biggr)^{1-\frac{1}{q}} \biggl\{ \biggl[ \biggl\vert f'' \biggl(\frac{a+b}{2} \biggr) \biggr\vert ^{q}U_{1} \biggl(1,1, \frac{1}{2};s \biggr)+ \bigl\vert f''(a) \bigr\vert ^{q}U_{2} \biggl(1,1,\frac{1}{2};s \biggr) \biggr]^{\frac{1}{q}} \\ &\quad \quad {} + \biggl[ \biggl\vert f'' \biggl( \frac{a+b}{2} \biggr) \biggr\vert ^{q}U_{1} \biggl(1,1,\frac{1}{2};s \biggr)+ \bigl\vert f''(b) \bigr\vert ^{q}U_{2} \biggl(1,1,\frac{1}{2};s \biggr) \biggr]^{\frac{1}{q}} \biggr\} ; \end{aligned}$$
  3. (c)

    if \(\lambda =0\), then we obtain

    $$\begin{aligned} & \biggl\vert \frac{2^{\frac{\mu }{k}-1}}{(b-a)^{\frac{\mu }{k}-1}}I_{f} \biggl(\mu ,k;\frac{a+b}{2},0,1,a,b \biggr) \biggr\vert \\ &\quad = \biggl\vert f \biggl(\frac{a+b}{2} \biggr)-\frac{2^{\frac{\mu }{k}-1}\Gamma _{k}(\mu +k)}{(b-a)^{\frac{\mu }{k}}} \bigl[{}_{k}J^{\mu }_{(\frac{a+b}{2})^{-}} f(a)+{}_{k}J^{\mu }_{(\frac{a+b}{2})^{+}}f(b) \bigr] \biggr\vert \\ &\quad \leq \frac{(b-a)^{2}}{8 (\frac{\mu }{k}+1 )} \biggl(\frac{1}{\frac{\mu }{k}+2} \biggr)^{1-\frac{1}{q}} \biggl\{ \biggl[\frac{\vert f''(\frac{a+b}{2}) \vert ^{q}}{\frac{\mu }{k}-s+2}+ \bigl\vert f''(a) \bigr\vert ^{q}\beta \biggl(\frac{\mu }{k}+2,1-s \biggr) \biggr]^{\frac{1}{q}} \\ &\quad \quad {} + \biggl[\frac{\vert f''(\frac{a+b}{2}) \vert ^{q}}{\frac{\mu }{k}-s+2}+ \bigl\vert f''(b) \bigr\vert ^{q}\beta \biggl(\frac{\mu }{k}+2,1-s \biggr) \biggr]^{\frac{1}{q}} \biggr\} . \end{aligned}$$

    Specially, if we put \(k=1=\mu \), then we obtain a midpoint-type inequality:

    $$\begin{aligned} & \biggl\vert I_{f} \biggl(1,1; \frac{a+b}{2},0,1,a,b \biggr) \biggr\vert \\ &\quad = \biggl\vert f \biggl(\frac{a+b}{2} \biggr)-\frac{1}{b-a} \int ^{b}_{a} f(x)\,\mathrm{d}x \biggr\vert \\ &\quad \leq \frac{(b-a)^{2}}{16} \biggl(\frac{1}{3} \biggr)^{1-\frac{1}{q}} \biggl\{ \biggl[\frac{\vert f''(\frac{a+b}{2}) \vert ^{q}}{3-s}+ \bigl\vert f''(a) \bigr\vert ^{q}\beta (3,1-s ) \biggr]^{\frac{1}{q}} \\ &\quad \quad {}+ \biggl[\frac{\vert f''(\frac{a+b}{2}) \vert ^{q}}{3-s}+ \bigl\vert f''(b) \bigr\vert ^{q}\beta (3,1-s ) \biggr]^{\frac{1}{q}} \biggr\} ; \end{aligned}$$
  4. (d)

    if \(\lambda =1\), then we obtain

    $$\begin{aligned} & \biggl\vert \frac{2^{\frac{\mu }{k}-1}}{(b-a)^{\frac{\mu }{k}-1}}I_{f} \biggl(\mu ,k;\frac{a+b}{2},1,1,a,b \biggr) \biggr\vert \\ &\quad = \biggl\vert \frac{f(a)+f(b)}{2}-\frac{2^{\frac{\mu }{k}-1}\Gamma _{k}(\mu +k)}{(b-a)^{\frac{\mu }{k}}} \bigl[{}_{k}J^{\mu }_{(\frac{a+b}{2})^{-}} f(a)+{}_{k}J^{\mu }_{(\frac{a+b}{2})^{+}}f(b) \bigr] \biggr\vert \\ &\quad \leq \frac{(b-a)^{2}}{8 (\frac{\mu }{k}+1 )} \biggl[\frac{\frac{\mu }{k} (\frac{\mu }{k}+3 )}{2 (\frac{\mu }{k}+2 )} \biggr]^{1-\frac{1}{q}} \\ &\quad \quad {} \times \biggl\{ \biggl[\frac{\frac{\mu }{k} (\frac{\mu }{k}-s+3 )\vert f''(\frac{a+b}{2}) \vert ^{q}}{(2-s) (\frac{\mu }{k}-s+2 )} \\ &\quad \quad {} + \bigl\vert f''(a) \bigr\vert ^{q} \biggl( \biggl(\frac{\mu }{k}+1 \biggr)\beta (2,1-s)-\beta \biggl(\frac{\mu }{k}+2,1-s\biggr) \biggr) \biggr]^{\frac{1}{q}} \\ &\quad \quad {} + \biggl[\frac{\frac{\mu }{k} (\frac{\mu }{k}-s+3 )\vert f''(\frac{a+b}{2}) \vert ^{q}}{(2-s) (\frac{\mu }{k}-s+2 )} \\ &\quad \quad {} + \bigl\vert f''(b) \bigr\vert ^{q} \biggl( \biggl(\frac{\mu }{k}+1 \biggr)\beta (2,1-s)-\beta \biggl(\frac{\mu }{k}+2,1-s \biggr) \biggr) \biggr]^{\frac{1}{q}} \biggr\} . \end{aligned}$$

    Specially, if we put \(k=1=\mu \), then we obtain a trapezoid-type inequality:

    $$\begin{aligned} & \biggl\vert I_{f} \biggl(1,1; \frac{a+b}{2},1,1,a,b \biggr) \biggr\vert \\ &\quad = \biggl\vert \frac{f(a)+f(b)}{2}-\frac{1}{b-a} \int ^{b}_{a} f(x)\,\mathrm{d}x \biggr\vert \\ &\quad \leq \frac{(b-a)^{2}}{16} \biggl(\frac{2}{3} \biggr)^{1-\frac{1}{q}} \biggl\{ \biggl[\frac{(4-s)\vert f''(\frac{a+b}{2}) \vert ^{q}}{(2-s)(3-s)} + \bigl\vert f''(a) \bigr\vert ^{q} \bigl(2\beta (2,1-s)-\beta (3,1-s) \bigr) \biggr]^{\frac{1}{q}} \\ &\quad \quad {} + \biggl[\frac{(4-s)\vert f''(\frac{a+b}{2}) \vert ^{q}}{(2-s)(3-s)} + \bigl\vert f''(b) \bigr\vert ^{q} \bigl(2\beta (2,1-s)-\beta (3,1-s) \bigr) \biggr]^{\frac{1}{q}} \biggr\} . \end{aligned}$$

III. If \(h(t)=t(1-t)\) in Theorem 2.1, then we have the following results.

Corollary 2.5

In Theorem 2.1, if \(\vert f'' \vert ^{q}\) for \(q\geq 1\) is generalized \((m, tgs)\)-preinvex functions, then, for \(m\in (0,1]\), we have

$$ \begin{aligned} & \bigl\vert I_{f,\eta }(\mu ,k;x,\lambda ,m,a,b) \bigr\vert \\ &\quad \leq C_{0}^{1-\frac{1}{q}}(k,\mu ,\lambda )\theta ^{\frac{1}{q}}(k, \mu ,\lambda ) \biggl\{ \biggl\vert \frac{\eta ^{\frac{\mu }{k}+2}(x,a,m)}{(\frac{\mu }{k}+1)\eta (b,a,m)} \biggr\vert \bigl[ \bigl\vert f''(x) \bigr\vert ^{q}+m \bigl\vert f''(a) \bigr\vert ^{q} \bigr]^{\frac{1}{q}} \\ &\quad \quad {} + \biggl\vert \frac{(-1)^{\frac{\mu }{k}+2}\eta ^{\frac{\mu }{k}+2}(x,b,m)}{(\frac{\mu }{k}+1)\eta (b,a,m)} \biggr\vert \bigl[ \bigl\vert f''(x) \bigr\vert ^{q}+m \bigl\vert f''(b) \bigr\vert ^{q} \bigr]^{\frac{1}{q}} \biggr\} , \end{aligned} $$

where we use the fact that

$$ \theta (k,\mu ,\lambda )= \textstyle\begin{cases} \frac{\frac{2\mu }{k} [ (\frac{\mu }{k}+1 )\lambda ]^{\frac{3k}{\mu }+1}}{3(\frac{\mu }{k}+3)}-\frac{\frac{\mu }{k} [ (\frac{\mu }{k}+1 )\lambda ]^{\frac{4k}{\mu }+1}}{2(\frac{\mu }{k}+4)}-\frac{(\frac{\mu }{k}+1)\lambda }{12}+\frac{1}{(\frac{\mu }{k}+3)(\frac{\mu }{k}+4)}, &0\leq \lambda \leq \frac{1}{\frac{\mu }{k}+1}, \\ \frac{(\frac{\mu }{k}+1)\lambda }{12}-\frac{1}{(\frac{\mu }{k}+3)(\frac{\mu }{k}+4)}, &\frac{1}{\frac{\mu }{k}+1}< \lambda \leq 1\end{cases} $$

and \(C_{0}(k,\mu ,\lambda )\) is defined by (2.5).

Corollary 2.6

In Theorem 2.1, if the mapping \(\eta (b,a,m)=b-ma\) along with \(m=1\), taking \(x=\frac{a+b}{2}\), we get the following inequality for \(tgs\)-convex functions:

$$ \begin{aligned} & \biggl\vert (1-\lambda )f \biggl(\frac{a+b}{2} \biggr)+\lambda \biggl[\frac{f(a)+f(b)}{2} \biggr] \\ &\quad {} -\frac{2^{\frac{\mu }{k}-1}\Gamma _{k}(\mu +k)}{(b-a)^{\frac{\mu }{k}}} \bigl[{}_{k}J^{\mu }_{(\frac{a+b}{2})^{-}} f(a)+{}_{k}J^{\mu }_{(\frac{a+b}{2})^{+}}f(b) \bigr] \biggr\vert \\ &\leq \frac{(b-a)^{2}}{8 (\frac{\mu }{k}+1 )} C_{0}^{1-\frac{1}{q}}(k,\mu ,\lambda )\theta ^{\frac{1}{q}}(k,\mu ,\lambda ) \\ &\quad {} \times \biggl\{ \biggl[ \biggl\vert f'' \biggl(\frac{a+b}{2} \biggr) \biggr\vert ^{q}+ \bigl\vert f''(a) \bigr\vert ^{q} \biggr]^{\frac{1}{q}} + \biggl[ \biggl\vert f'' \biggl(\frac{a+b}{2} \biggr) \biggr\vert ^{q}+ \bigl\vert f''(b) \bigr\vert ^{q} \biggr]^{\frac{1}{q}} \biggr\} . \end{aligned} $$

Remark 2.5

In Corollary 2.6,

  1. (a)

    if \(\lambda =\frac{1}{3}\), then we obtain

    $$\begin{aligned} & \biggl\vert \frac{1}{6} \biggl[f(a)+4f \biggl(\frac{a+b}{2} \biggr)+f(b) \biggr]-\frac{2^{\frac{\mu }{k}-1}\Gamma _{k}(\mu +k)}{(b-a)^{\frac{\mu }{k}}} \bigl[{}_{k}J^{\mu }_{(\frac{a+b}{2})^{-}} f(a)+{}_{k}J^{\mu }_{(\frac{a+b}{2})^{+}}f(b) \bigr] \biggr\vert \\ &\quad \leq \frac{(b-a)^{2}}{8 (\frac{\mu }{k}+1 )}C_{0}^{1-\frac{1}{q}} \biggl(k,\mu , \frac{1}{3} \biggr)\theta ^{\frac{1}{q}} \biggl(k,\mu ,\frac{1}{3} \biggr) \\ &\quad \quad {} \times \biggl\{ \biggl[ \biggl\vert f'' \biggl(\frac{a+b}{2} \biggr) \biggr\vert ^{q}+ \bigl\vert f''(a) \bigr\vert ^{q} \biggr]^{\frac{1}{q}} + \biggl[ \biggl\vert f'' \biggl(\frac{a+b}{2} \biggr) \biggr\vert ^{q}+ \bigl\vert f''(b) \bigr\vert ^{q} \biggr]^{\frac{1}{q}} \biggr\} . \end{aligned}$$

    Specially, if we put \(k=1=\mu \), then we obtain a Simpson-type inequality:

    $$\begin{aligned} & \biggl\vert \frac{1}{6} \biggl[f(a)+4f \biggl(\frac{a+b}{2} \biggr)+f(b) \biggr]-\frac{1}{b-a} \int ^{b}_{a} f(x)\,\mathrm{d}x \biggr\vert \\ &\quad \leq \frac{(b-a)^{2}}{162}{ \biggl(\frac{23}{160} \biggr)}^{\frac{1}{q}} \\ &\quad \quad {} \times \biggl\{ \biggl[ \biggl\vert f'' \biggl(\frac{a+b}{2} \biggr) \biggr\vert ^{q}+ \bigl\vert f''(a) \bigr\vert ^{q} \biggr]^{\frac{1}{q}} + \biggl[ \biggl\vert f'' \biggl(\frac{a+b}{2} \biggr) \biggr\vert ^{q}+ \bigl\vert f''(b) \bigr\vert ^{q} \biggr]^{\frac{1}{q}} \biggr\} ; \end{aligned}$$
  2. (b)

    if \(\lambda =\frac{1}{2}\), then we obtain

    $$ \begin{aligned} & \biggl\vert \frac{1}{4} \biggl[f(a)+2f \biggl(\frac{a+b}{2} \biggr)+f(b) \biggr]-\frac{2^{\frac{\mu }{k}-1}\Gamma _{k}(\mu +k)}{(b-a)^{\frac{\mu }{k}}} \bigl[{}_{k}J^{\mu }_{(\frac{a+b}{2})^{-}}f(a)+{}_{k}J^{\mu }_{(\frac{a+b}{2})^{+}}f(b) \bigr] \biggr\vert \\ &\quad \leq \frac{(b-a)^{2}}{8 (\frac{\mu }{k}+1 )}C_{0}^{1-\frac{1}{q}} \biggl(k,\mu , \frac{1}{2} \biggr)\theta ^{\frac{1}{q}} \biggl(k,\mu ,\frac{1}{2} \biggr) \\ &\quad \quad {} \times \biggl\{ \biggl[ \biggl\vert f'' \biggl(\frac{a+b}{2} \biggr) \biggr\vert ^{q}+ \bigl\vert f''(a) \bigr\vert ^{q} \biggr]^{\frac{1}{q}} + \biggl[ \biggl\vert f'' \biggl(\frac{a+b}{2} \biggr) \biggr\vert ^{q}+ \bigl\vert f''(b) \bigr\vert ^{q} \biggr]^{\frac{1}{q}} \biggr\} . \end{aligned} $$

    Specially, if we put \(k=1=\mu \), then we derive an averaged midpoint-trapezoid-type inequality:

    $$\begin{aligned} & \biggl\vert \frac{1}{4} \biggl[f(a)+2f \biggl(\frac{a+b}{2} \biggr)+f(b) \biggr]-\frac{1}{b-a} \int ^{b}_{a} f(x)\,\mathrm{d}x \biggr\vert \\ &\quad \leq \frac{(b-a)^{2}}{96}{ \biggl(\frac{1}{5} \biggr)}^{\frac{1}{q}} \biggl\{ \biggl[ \biggl\vert f'' \biggl( \frac{a+b}{2} \biggr) \biggr\vert ^{q}+ \bigl\vert f''(a) \bigr\vert ^{q} \biggr]^{\frac{1}{q}} \\ &\quad \quad {}+ \biggl[ \biggl\vert f'' \biggl(\frac{a+b}{2} \biggr) \biggr\vert ^{q}+ \bigl\vert f''(b) \bigr\vert ^{q} \biggr]^{\frac{1}{q}} \biggr\} ; \end{aligned}$$
  3. (c)

    if \(\lambda =0\), then we obtain

    $$ \begin{aligned} & \biggl\vert f \biggl(\frac{a+b}{2} \biggr)- \frac{2^{\frac{\mu }{k}-1}\Gamma _{k}(\mu +k)}{(b-a)^{\frac{\mu }{k}}} \bigl[{}_{k}J^{\mu }_{(\frac{a+b}{2})^{-}}f(a)+{}_{k}J^{\mu }_{(\frac{a+b}{2})^{+}}f(b) \bigr] \biggr\vert \\ &\quad \leq \frac{(b-a)^{2}}{8 (\frac{\mu }{k}+1 )} \biggl(\frac{1}{\frac{\mu }{k}+2} \biggr)^{1-\frac{1}{q}} \biggl[\frac{1}{(\frac{\mu }{k}+3)(\frac{\mu }{k}+4)} \biggr]^{\frac{1}{q}} \\ &\quad \quad {} \times \biggl\{ \biggl[ \biggl\vert f'' \biggl(\frac{a+b}{2} \biggr) \biggr\vert ^{q}+ \bigl\vert f''(a) \bigr\vert ^{q} \biggr]^{\frac{1}{q}} + \biggl[ \biggl\vert f'' \biggl(\frac{a+b}{2} \biggr) \biggr\vert ^{q}+ \bigl\vert f''(b) \bigr\vert ^{q} \biggr]^{\frac{1}{q}} \biggr\} . \end{aligned} $$

    Specially, if we put \(k=1=\mu \), then we derive a midpoint-type inequality:

    $$\begin{aligned} & \biggl\vert f \biggl(\frac{a+b}{2} \biggr)- \frac{1}{b-a} \int ^{b}_{a} f(x)\,\mathrm{d}x \biggr\vert \\ &\quad \leq \frac{(b-a)^{2}}{48} \biggl(\frac{3}{20} \biggr)^{\frac{1}{q}} \biggl\{ \biggl[ \biggl\vert f'' \biggl(\frac{a+b}{2} \biggr) \biggr\vert ^{q}+ \bigl\vert f''(a) \bigr\vert ^{q} \biggr]^{\frac{1}{q}} \\ &\quad \quad {}+ \biggl[ \biggl\vert f'' \biggl(\frac{a+b}{2} \biggr) \biggr\vert ^{q}+ \bigl\vert f''(b) \bigr\vert ^{q} \biggr]^{\frac{1}{q}} \biggr\} ; \end{aligned}$$
  4. (d)

    if \(\lambda =1\), then we obtain

    $$\begin{aligned} & \biggl\vert \frac{f(a)+f(b)}{2}-\frac{2^{\frac{\mu }{k}-1}\Gamma _{k}(\mu +k)}{(b-a)^{\frac{\mu }{k}}} \bigl[{}_{k}J^{\mu }_{(\frac{a+b}{2})^{-}}f(a)+{}_{k}J^{\mu }_{(\frac{a+b}{2})^{+}}f(b) \bigr] \biggr\vert \\ &\quad \leq \frac{(b-a)^{2}}{8 (\frac{\mu }{k}+1 )} \biggl[\frac{\frac{\mu }{k} (\frac{\mu }{k}+3 )}{2 (\frac{\mu }{k}+2 )} \biggr]^{1-\frac{1}{q}} \biggl[\frac{\frac{\mu }{k} (\frac{\mu ^{2}}{k^{2}}+\frac{8\mu }{k}+19 )}{12 (\frac{\mu }{k}+3 ) (\frac{\mu }{k}+4 )} \biggr]^{\frac{1}{q}} \\ &\quad \quad {} \times \biggl\{ \biggl[ \biggl\vert f'' \biggl(\frac{a+b}{2} \biggr) \biggr\vert ^{q}+ \bigl\vert f''(a) \bigr\vert ^{q} \biggr]^{\frac{1}{q}} + \biggl[ \biggl\vert f'' \biggl(\frac{a+b}{2} \biggr) \biggr\vert ^{q}+ \bigl\vert f''(b) \bigr\vert ^{q} \biggr]^{\frac{1}{q}} \biggr\} . \end{aligned}$$

    Specially, if we put \(k=1=\mu \), then we obtain a trapezoid-type inequality:

    $$\begin{aligned} & \biggl\vert \frac{f(a)+f(b)}{2}-\frac{1}{b-a} \int ^{b}_{a} f(x)\,\mathrm{d}x \biggr\vert \\ &\quad \leq \frac{(b-a)^{2}}{24} \biggl(\frac{7}{40} \biggr)^{\frac{1}{q}} \biggl\{ \biggl[ \biggl\vert f'' \biggl(\frac{a+b}{2} \biggr) \biggr\vert ^{q}+ \bigl\vert f''(a) \bigr\vert ^{q} \biggr]^{\frac{1}{q}} \\ &\quad \quad {}+ \biggl[ \biggl\vert f'' \biggl(\frac{a+b}{2} \biggr) \biggr\vert ^{q}+ \bigl\vert f''(b) \bigr\vert ^{q} \biggr]^{\frac{1}{q}} \biggr\} . \end{aligned}$$

IV. If \(h(t)=\frac{\sqrt{t}}{2\sqrt{1-t}}\) in Theorem 2.1, then we have the following results.

Corollary 2.7

In Theorem 2.1, if \(\vert f'' \vert ^{q}\) for \(q\geq 1\) is generalized m-MT-preinvex functions, then, for \(m\in (0,1]\), we have

$$ \begin{aligned} & \bigl\vert I_{f,\eta }(\mu ,k;x,\lambda ,m,a,b) \bigr\vert \\ &\quad \leq C_{0}^{1-\frac{1}{q}}(k,\mu ,\lambda ) \biggl\{ \biggl\vert \frac{\eta ^{\frac{\mu }{k}+2}(x,a,m)}{(\frac{\mu }{k}+1)\eta (b,a,m)} \biggr\vert \bigl[ \bigl\vert f''(x) \bigr\vert ^{q}\Phi _{1}(k,\mu ,\lambda )+m \bigl\vert f''(a) \bigr\vert ^{q}\Phi _{2}(k,\mu ,\lambda ) \bigr]^{\frac{1}{q}} \\ &\quad \quad {} + \biggl\vert \frac{(-1)^{\frac{\mu }{k}+2}\eta ^{\frac{\mu }{k}+2}(x,b,m)}{(\frac{\mu }{k}+1)\eta (b,a,m)} \biggr\vert \bigl[ \bigl\vert f''(x) \bigr\vert ^{q}\Phi _{1}(k,\mu ,\lambda )+m \bigl\vert f''(b) \bigr\vert ^{q}\Phi _{2}(k,\mu ,\lambda ) \bigr]^{\frac{1}{q}} \biggr\} , \end{aligned} $$

where we use the fact that

$$\begin{aligned}& \Phi _{1}(k,\mu ,\lambda ) \\& \quad = \textstyle\begin{cases} \left [ \textstyle\begin{array}{@{}c@{}} \frac{1}{2}\beta (\frac{\mu }{k}+\frac{5}{2},\frac{1}{2} )-\frac{3\lambda \pi (\frac{\mu }{k}+1)}{16}+ (\frac{\mu }{k}+1 )\lambda \beta ( [ (\frac{\mu }{k}+1 )\lambda ]^{\frac{k}{\mu }};\frac{5}{2},\frac{1}{2} )\\ {}-\beta ( [ (\frac{\mu }{k}+1 )\lambda ]^{\frac{k}{\mu }};\frac{\mu }{k}+\frac{5}{2},\frac{1}{2} )\end{array}\displaystyle \right ] , &0\leq \lambda \leq \frac{1}{\frac{\mu }{k}+1}, \\ \frac{3\lambda \pi (\frac{\mu }{k}+1)}{16}-\frac{1}{2}\beta (\frac{\mu }{k}+\frac{5}{2},\frac{1}{2} ), &\frac{1}{\frac{\mu }{k}+1}< \lambda \leq 1, \end{cases}\displaystyle \\& \Phi _{2}(k,\mu ,\lambda ) \\& \quad = \textstyle\begin{cases} \left [ \textstyle\begin{array}{@{}c@{}} \frac{1}{2}\beta (\frac{\mu }{k}+\frac{3}{2},\frac{3}{2} )-\frac{\lambda \pi (\frac{\mu }{k}+1)}{16} + (\frac{\mu }{k}+1 )\lambda \beta ( [ (\frac{\mu }{k}+1 )\lambda ]^{\frac{k}{\mu }};\frac{3}{2},\frac{3}{2} )\\ {}-\beta ( [ (\frac{\mu }{k}+1 )\lambda ]^{\frac{k}{\mu }};\frac{\mu }{k}+\frac{3}{2},\frac{3}{2} ) \end{array}\displaystyle \right ] , &0\leq \lambda \leq \frac{1}{\frac{\mu }{k}+1}, \\ \frac{\lambda \pi (\frac{\mu }{k}+1)}{16}-\frac{1}{2}\beta (\frac{\mu }{k}+\frac{3}{2},\frac{3}{2} ), &\frac{1}{\frac{\mu }{k}+1}< \lambda \leq 1, \end{cases}\displaystyle \end{aligned}$$

and \(C_{0}(k,\mu ,\lambda )\) is defined by (2.5).

Corollary 2.8

In Theorem 2.1, if the mapping \(\eta (b,a,m)=b-ma\) together with \(m=1\), taking \(x=\frac{a+b}{2}\), we get the following inequality for MT-convex functions:

$$\begin{aligned} & \biggl\vert (1-\lambda )f \biggl(\frac{a+b}{2} \biggr)+\lambda \biggl[\frac{f(a)+f(b)}{2} \biggr] \\ &\quad \quad {} -\frac{2^{\frac{\mu }{k}-1}\Gamma _{k}(\mu +k)}{(b-a)^{\frac{\mu }{k}}} \bigl[{}_{k}J^{\mu }_{(\frac{a+b}{2})^{-}} f(a)+{}_{k}J^{\mu }_{(\frac{a+b}{2})^{+}}f(b) \bigr] \biggr\vert \\ &\quad \leq \frac{(b-a)^{2}}{8 (\frac{\mu }{k}+1 )} C_{0}^{1-\frac{1}{q}}(k,\mu ,\lambda ) \biggl\{ \biggl[ \biggl\vert f'' \biggl(\frac{a+b}{2} \biggr) \biggr\vert ^{q}\Phi _{1}(k,\mu ,\lambda )+ \bigl\vert f''(a) \bigr\vert ^{q}\Phi _{2}(k,\mu ,\lambda ) \biggr]^{\frac{1}{q}} \\ &\quad \quad {} + \biggl[ \biggl\vert f'' \biggl( \frac{a+b}{2} \biggr) \biggr\vert ^{q}\Phi _{1}(k,\mu ,\lambda )+ \bigl\vert f''(b) \bigr\vert ^{q}\Phi _{2}(k,\mu ,\lambda ) \biggr]^{\frac{1}{q}} \biggr\} . \end{aligned}$$

Remark 2.6

In Corollary 2.8,

  1. (a)

    if \(\lambda =\frac{1}{3}\), then we obtain

    $$\begin{aligned} & \biggl\vert \frac{1}{6} \biggl[f(a)+4f \biggl(\frac{a+b}{2} \biggr)+f(b) \biggr]-\frac{2^{\frac{\mu }{k}-1}\Gamma _{k}(\mu +k)}{(b-a)^{\frac{\mu }{k}}} \bigl[{}_{k}J^{\mu }_{(\frac{a+b}{2})^{-}} f(a)+{}_{k}J^{\mu }_{(\frac{a+b}{2})^{+}}f(b) \bigr] \biggr\vert \\ &\quad \leq \frac{(b-a)^{2}}{8 (\frac{\mu }{k}+1 )}C_{0}^{1-\frac{1}{q}} \biggl(k,\mu ,\frac{1}{3} \biggr) \biggl\{ \biggl[ \biggl\vert f'' \biggl(\frac{a+b}{2} \biggr) \biggr\vert ^{q}\Phi _{1} \biggl(k,\mu ,\frac{1}{3} \biggr) \\ &\quad \quad {}+ \bigl\vert f''(a) \bigr\vert ^{q}\Phi _{2} \biggl(k,\mu ,\frac{1}{3} \biggr) \biggr]^{\frac{1}{q}} \\ &\quad \quad {} + \biggl[ \biggl\vert f'' \biggl( \frac{a+b}{2} \biggr) \biggr\vert ^{q}\Phi _{1} \biggl(k,\mu ,\frac{1}{3} \biggr)+ \bigl\vert f''(b) \bigr\vert ^{q}\Phi _{2} \biggl(k,\mu ,\frac{1}{3} \biggr) \biggr]^{\frac{1}{q}} \biggr\} . \end{aligned}$$

    Specially, if we put \(k=1=\mu \), then we derive a Simpson-type inequality:

    $$ \begin{aligned} & \biggl\vert \frac{1}{6} \biggl[f(a)+4f \biggl(\frac{a+b}{2} \biggr)+f(b) \biggr]-\frac{1}{b-a} \int ^{b}_{a} f(x)\,\mathrm{d}x \biggr\vert \\ &\quad \leq \frac{(b-a)^{2}}{16} { \biggl(\frac{8}{81} \biggr)}^{1-\frac{1}{q}} \biggl\{ \biggl[ \biggl\vert f'' \biggl( \frac{a+b}{2} \biggr) \biggr\vert ^{q}\Phi _{1} \biggl(1,1,\frac{1}{3} \biggr)+ \bigl\vert f''(a) \bigr\vert ^{q}\Phi _{2} \biggl(1,1,\frac{1}{3} \biggr) \biggr]^{\frac{1}{q}} \\ &\quad \quad {} + \biggl[ \biggl\vert f'' \biggl( \frac{a+b}{2} \biggr) \biggr\vert ^{q}\Phi _{1} \biggl(1,1,\frac{1}{3} \biggr)+ \bigl\vert f''(b) \bigr\vert ^{q}\Phi _{2} \biggl(1,1,\frac{1}{3} \biggr) \biggr]^{\frac{1}{q}} \biggr\} , \end{aligned} $$

    where

    $$ \Phi _{1} \biggl(1,1,\frac{1}{3} \biggr)=\frac{71\sqrt{2}}{648}+\frac{1}{8}\arcsin {\frac{\sqrt{3}}{3}}- \frac{\pi }{32} $$

    and

    $$ \Phi _{2} \biggl(1,1,\frac{1}{3} \biggr)=\frac{19\sqrt{2}}{648}+\frac{1}{12}\arcsin {\frac{1}{3}}+ \frac{1}{16}\arctan {2\sqrt{2}}-\frac{\pi }{32}; $$
  2. (b)

    if \(\lambda =\frac{1}{2}\), then we obtain

    $$\begin{aligned} & \biggl\vert \frac{1}{4} \biggl[f(a)+2f \biggl(\frac{a+b}{2} \biggr)+f(b) \biggr]-\frac{2^{\frac{\mu }{k}-1}\Gamma _{k}(\mu +k)}{(b-a)^{\frac{\mu }{k}}} \bigl[{}_{k}J^{\mu }_{(\frac{a+b}{2})^{-}} f(a)+{}_{k}J^{\mu }_{(\frac{a+b}{2})^{+}}f(b) \bigr] \biggr\vert \\ &\quad \leq \frac{(b-a)^{2}}{8 (\frac{\mu }{k}+1 )} C_{0}^{1-\frac{1}{q}} \biggl(k,\mu , \frac{1}{2} \biggr) \biggl\{ \biggl[ \biggl\vert f''\biggl(\frac{a+b}{2} \biggr) \biggr\vert ^{q}\Phi _{1} \biggl(k,\mu ,\frac{1}{2} \biggr) \\ &\quad \quad {}+ \bigl\vert f''(a) \bigr\vert ^{q}\Phi _{2} \biggl(k,\mu ,\frac{1}{2} \biggr) \biggr]^{\frac{1}{q}} \\ &\quad \quad {} + \biggl[ \biggl\vert f'' \biggl( \frac{a+b}{2} \biggr) \biggr\vert ^{q}\Phi _{1} \biggl(k,\mu ,\frac{1}{2} \biggr)+ \bigl\vert f''(b)\bigr\vert ^{q}\Phi _{2} \biggl(k,\mu ,\frac{1}{2} \biggr) \biggr]^{\frac{1}{q}} \biggr\} . \end{aligned}$$

    Specially, if we put \(k=1=\mu \), then we derive an averaged midpoint-trapezoid-type inequality:

    $$\begin{aligned} & \biggl\vert \frac{1}{4} \biggl[f(a)+2f \biggl(\frac{a+b}{2} \biggr)+f(b) \biggr]-\frac{1}{b-a} \int ^{b}_{a} f(x)\,\mathrm{d}x \biggr\vert \\ &\quad \leq \frac{(b-a)^{2}}{96} \biggl(\frac{3\pi }{16} \biggr)^{\frac{1}{q}} \biggl\{ \biggl[ \biggl\vert f'' \biggl(\frac{a+b}{2} \biggr) \biggr\vert ^{q}+ \bigl\vert f''(a) \bigr\vert ^{q} \biggr]^{\frac{1}{q}} \\ &\quad \quad {}+ \biggl[ \biggl\vert f'' \biggl(\frac{a+b}{2} \biggr) \biggr\vert ^{q}+ \bigl\vert f''(b) \bigr\vert ^{q} \biggr]^{\frac{1}{q}} \biggr\} ; \end{aligned}$$
  3. (c)

    if \(\lambda =0\), then we obtain

    $$\begin{aligned} & \biggl\vert f \biggl(\frac{a+b}{2} \biggr)- \frac{2^{\frac{\mu }{k}-1}\Gamma _{k}(\mu +k)}{(b-a)^{\frac{\mu }{k}}} \bigl[{}_{k}J^{\mu }_{(\frac{a+b}{2})^{-}}f(a)+{}_{k}J^{\mu }_{(\frac{a+b}{2})^{+}}f(b) \bigr] \biggr\vert \\ &\quad \leq \frac{(b-a)^{2}}{8 (\frac{\mu }{k}+1 )} \biggl(\frac{1}{\frac{\mu }{k}+2} \biggr)^{1-\frac{1}{q}} \biggl(\frac{1}{2} \biggr)^{\frac{1}{q}} \\ &\quad \quad {} \times \biggl\{ \biggl[ \biggl\vert f'' \biggl(\frac{a+b}{2} \biggr) \biggr\vert ^{q}\beta \biggl( \frac{\mu }{k}+\frac{5}{2},\frac{1}{2} \biggr)+ \bigl\vert f''(a) \bigr\vert ^{q}\beta \biggl( \frac{\mu }{k}+\frac{3}{2},\frac{3}{2} \biggr) \biggr]^{\frac{1}{q}} \\ &\quad \quad {} + \biggl[ \biggl\vert f'' \biggl( \frac{a+b}{2} \biggr) \biggr\vert ^{q}\beta \biggl( \frac{\mu }{k}+\frac{5}{2},\frac{1}{2} \biggr)+ \bigl\vert f''(b) \bigr\vert ^{q}\beta \biggl( \frac{\mu }{k}+\frac{3}{2},\frac{3}{2} \biggr) \biggr]^{\frac{1}{q}} \biggr\} . \end{aligned}$$

    Specially, if we put \(k=1=\mu \), then we obtain a midpoint-type inequality:

    $$\begin{aligned} & \biggl\vert f \biggl(\frac{a+b}{2} \biggr)- \frac{1}{b-a} \int ^{b}_{a} f(x)\,\mathrm{d}x \biggr\vert \\ &\quad \leq \frac{(b-a)^{2}}{48} \biggl(\frac{3}{2} \biggr)^{\frac{1}{q}} \\ &\quad\quad {} \times \biggl\{ \biggl[\frac{5\pi }{16} \biggl\vert f'' \biggl(\frac{a+b}{2} \biggr) \biggr\vert ^{q}+\frac{\pi }{16} \bigl\vert f''(a) \bigr\vert ^{q} \biggr]^{\frac{1}{q}} \\ &\quad \quad {}+ \biggl[\frac{5\pi }{16} \biggl\vert f'' \biggl(\frac{a+b}{2} \biggr) \biggr\vert ^{q}+\frac{\pi }{16} \bigl\vert f''(b) \bigr\vert ^{q} \biggr]^{\frac{1}{q}} \biggr\} ; \end{aligned}$$
  4. (d)

    if \(\lambda =1\), then we obtain

    $$\begin{aligned} & \biggl\vert \frac{f(a)+f(b)}{2}-\frac{2^{\frac{\mu }{k}-1}\Gamma _{k}(\mu +k)}{(b-a)^{\frac{\mu }{k}}} \bigl[{}_{k}J^{\mu }_{(\frac{a+b}{2})^{-}}f(a)+{}_{k}J^{\mu }_{(\frac{a+b}{2})^{+}}f(b) \bigr] \biggr\vert \\ &\quad \leq \frac{(b-a)^{2}}{8 (\frac{\mu }{k}+1 )} \biggl(\frac{\frac{\mu }{k}(\frac{\mu }{k}+3)}{2(\frac{\mu }{k}+2)} \biggr)^{1-\frac{1}{q}} \biggl\{ \biggl[ \biggl\vert f'' \biggl(\frac{a+b}{2} \biggr) \biggr\vert ^{q} \biggl(\frac{3(\frac{\mu }{k}+1)\pi }{16}-\frac{1}{2} \beta \biggl(\frac{\mu }{k}+\frac{5}{2},\frac{1}{2} \biggr) \biggr) \\ &\quad \quad {} + \bigl\vert f''(a) \bigr\vert ^{q} \biggl(\frac{(\frac{\mu }{k}+1)\pi }{16}-\frac{1}{2}\beta \biggl( \frac{\mu }{k}+\frac{3}{2},\frac{3}{2} \biggr) \biggr) \biggr]^{\frac{1}{q}} \\ &\quad \quad {} + \biggl[ \biggl\vert f'' \biggl( \frac{a+b}{2} \biggr) \biggr\vert ^{q} \biggl( \frac{3(\frac{\mu }{k}+1)\pi }{16}-\frac{1}{2}\beta \biggl(\frac{\mu }{k}+ \frac{5}{2},\frac{1}{2} \biggr) \biggr) \\ &\quad \quad {} + \bigl\vert f''(b) \bigr\vert ^{q} \biggl(\frac{(\frac{\mu }{k}+1)\pi }{16}-\frac{1}{2}\beta \biggl( \frac{\mu }{k}+\frac{3}{2},\frac{3}{2} \biggr) \biggr) \biggr]^{\frac{1}{q}} \biggr\} . \end{aligned}$$

    Specially, if we put \(k=1=\mu \), then we obtain a trapezoid-type inequality:

    $$\begin{aligned} & \biggl\vert \frac{f(a)+f(b)}{2}-\frac{1}{b-a} \int ^{b}_{a} f(x)\,\mathrm{d}x \biggr\vert \\ &\quad \leq \frac{(b-a)^{2}}{16} \biggl(\frac{2}{3} \biggr)^{1-\frac{1}{q}} \\ &\quad \quad {} \times \biggl\{ \biggl[\frac{7\pi }{32} \biggl\vert f'' \biggl(\frac{a+b}{2} \biggr) \biggr\vert ^{q}+\frac{3\pi }{32} \bigl\vert f''(a) \bigr\vert ^{q} \biggr]^{\frac{1}{q}} \\ &\qquad {} + \biggl[\frac{7\pi }{32} \biggl\vert f'' \biggl(\frac{a+b}{2} \biggr) \biggr\vert ^{q}+\frac{3\pi }{32} \bigl\vert f''(b) \bigr\vert ^{q} \biggr]^{\frac{1}{q}} \biggr\} . \end{aligned}$$

Now, we get ready to state the second theorem as follows.

Theorem 2.2

Let \(A\subseteq \mathbb{R}\) be an open m-invex subset with respect to \(\eta : A\times A\times (0,1] \rightarrow \mathbb{R}\setminus \{0\}\) for some fixed \(m \in (0,1]\), and let \(a, b \in A\), \(a< b\) with \(\eta (b,a,m)>0\). Assume that \(f:A\rightarrow \mathbb{R}\) is a twice differentiable function on A such that \(f''\) is integrable on \([ma,ma+\eta (b,a,m) ]\). If \(\vert f'' \vert ^{q}\) for \(q>1\) is a generalized \((m,h)\)-preinvex function with respect to η and \(h:[0,1]\rightarrow \mathbb{R}_{0}\), then the following inequality for k-fractional integrals with \(x \in [a,b]\), \(\lambda \in [0, 1]\), \(\mu > 0\), \(k>0\) exists:

$$\begin{aligned} & \bigl\vert I_{f,\eta }(\mu ,k;x,\lambda ,m,a,b) \bigr\vert \\ &\quad \leq C_{3}^{\frac{1}{p}}(k,\mu ,\lambda ,p) \\ &\quad \quad {} \times \biggl\{ \biggl\vert \frac{\eta ^{\frac{\mu }{k}+2}(x,a,m)}{(\frac{\mu }{k}+1)\eta (b,a,m)} \biggr\vert \biggl[ \bigl\vert f''(x) \bigr\vert ^{q} \int ^{1}_{0}h(t)\,\mathrm{d}t+m \bigl\vert f''(a) \bigr\vert ^{q} \int ^{1}_{0}h(1-t)\,\mathrm{d}t \biggr]^{\frac{1}{q}} \\ &\quad \quad {} + \biggl\vert \frac{(-1)^{\frac{\mu }{k}+2}\eta ^{\frac{\mu }{k}+2}(x,b,m)}{(\frac{\mu }{k}+1)\eta (b,a,m)} \biggr\vert \\ &\quad \quad {}\times \biggl[ \bigl\vert f''(x) \bigr\vert ^{q} \int ^{1}_{0}h(t)\,\mathrm{d}t+m \bigl\vert f''(b) \bigr\vert ^{q} \int ^{1}_{0}h(1-t)\,\mathrm{d}t \biggr]^{\frac{1}{q}} \biggr\} , \end{aligned}$$
(2.12)

where \(p=\frac{q}{q-1}\) and

$$\begin{aligned} &C_{3}(k,\mu ,\lambda ,p) \\ &\quad = \textstyle\begin{cases} \frac{1}{p(\frac{\mu }{k}+1)+1},& \lambda =0, \\ \left [ \textstyle\begin{array}{@{}c@{}} \frac{k [ (\frac{\mu }{k}+1 )\lambda ]^{\frac{k+kp(\frac{\mu }{k}+1)}{\mu }}}{\mu }\beta (\frac{k(1+p)}{\mu },1+p )\\ {}+\frac{k [1- (\frac{\mu }{k}+1 )\lambda ]^{p+1}}{\mu (p+1)}{}_{2}F_{1} (1-\frac{k(1+p)}{\mu },1;p+2;1- (\frac{\mu }{k}+1 )\lambda ) \end{array}\displaystyle \right ] ,&0< \lambda \leq \frac{1}{\frac{\mu }{k}+1}, \\ \frac{k [ (\frac{\mu }{k}+1 )\lambda ]^{\frac{k+kp(\frac{\mu }{k}+1)}{\mu }}}{\mu }\beta (\frac{1}{(\frac{\mu }{k}+1)\lambda };\frac{k(1+p)}{\mu },1+p ), & \frac{1}{\frac{\mu }{k}+1}< \lambda \leq 1. \end{cases}\displaystyle \end{aligned}$$

Proof

Using Lemma 2.1 and Hölder’s inequality, we have

$$\begin{aligned} & \bigl\vert I_{f,\eta }(\mu ,k;x,\lambda ,m,a,b) \bigr\vert \\ &\quad \leq \biggl\vert \frac{\eta ^{\frac{\mu }{k}+2}(x,a,m)}{(\frac{\mu }{k}+1)\eta (b,a,m)} \biggr\vert \int ^{1}_{0}t \biggl\vert \biggl( \frac{\mu }{k}+1 \biggr)\lambda -t^{\frac{\mu }{k}} \biggr\vert \bigl\vert f'' \bigl(ma+t\eta (x,a,m) \bigr) \bigr\vert \,\mathrm{d}t \\ &\quad \quad {} + \biggl\vert \frac{(-1)^{\frac{\mu }{k}+2}\eta ^{\frac{\mu }{k}+2}(x,b,m)}{(\frac{\mu }{k}+1)\eta (b,a,m)} \biggr\vert \int ^{1}_{0}t \biggl\vert \biggl( \frac{\mu }{k}+1 \biggr)\lambda -t^{\frac{\mu }{k}} \biggr\vert \bigl\vert f'' \bigl(mb+t\eta (x,b,m) \bigr) \bigr\vert \,\mathrm{d}t \\ &\quad \leq \biggl\vert \frac{\eta ^{\frac{\mu }{k}+2}(x,a,m)}{(\frac{\mu }{k}+1)\eta (b,a,m)} \biggr\vert \biggl[ \int ^{1}_{0}t^{p} \biggl\vert \biggl( \frac{\mu }{k}+1 \biggr)\lambda -t^{\frac{\mu }{k}} \biggr\vert ^{p}\,\mathrm{d}t \biggr]^{\frac{1}{p}} \\ &\quad \quad {} \times \biggl[ \int ^{1}_{0} \bigl\vert f'' \bigl(ma+t\eta (x,a,m) \bigr) \bigr\vert ^{q}\,\mathrm{d}t \biggr]^{\frac{1}{q}} \\ &\quad \quad {} + \biggl\vert \frac{(-1)^{\frac{\mu }{k}+2}\eta ^{\frac{\mu }{k}+2}(x,b,m)}{(\frac{\mu }{k}+1)\eta (b,a,m)} \biggr\vert \biggl[ \int ^{1}_{0}t^{p} \biggl\vert \biggl( \frac{\mu }{k}+1 \biggr)\lambda -t^{\frac{\mu }{k}} \biggr\vert ^{p}\,\mathrm{d}t \biggr]^{\frac{1}{p}} \\ &\quad \quad {} \times \biggl[ \int ^{1}_{0} \bigl\vert f'' \bigl(mb+t\eta (x,b,m) \bigr) \bigr\vert ^{q}\,\mathrm{d}t \biggr]^{\frac{1}{q}}. \end{aligned}$$
(2.13)

Since \(\vert f'' \vert ^{q}\) is generalized \((m,h)\)-preinvex on \([ma,ma+\eta (b,a,m)]\), we get

$$\begin{aligned} \int ^{1}_{0} \bigl\vert f'' \bigl(ma+t\eta (x,a,m) \bigr) \bigr\vert ^{q}\,\mathrm{d}t &\leq \int ^{1}_{0} \bigl[h(t) \bigl\vert f''(x) \bigr\vert ^{q}+mh(1-t) \bigl\vert f''(a) \bigr\vert ^{q} \bigr]\,\mathrm{d}t \\ &= \bigl\vert f''(x) \bigr\vert ^{q} \int ^{1}_{0}h(t)\,\mathrm{d}t+m \bigl\vert f''(a) \bigr\vert ^{q} \int ^{1}_{0}h(1-t)\,\mathrm{d}t, \end{aligned}$$
(2.14)
$$\begin{aligned} \int ^{1}_{0} \bigl\vert f'' \bigl(mb+t\eta (x,b,m) \bigr) \bigr\vert ^{q}\,\mathrm{d}t &\leq \int ^{1}_{0} \bigl[h(t) \bigl\vert f''(x) \bigr\vert ^{q}+mh(1-t) \bigl\vert f''(b) \bigr\vert ^{q} \bigr]\,\mathrm{d}t \\ &= \bigl\vert f''(x) \bigr\vert ^{q} \int ^{1}_{0}h(t)\,\mathrm{d}t+m \bigl\vert f''(b) \bigr\vert ^{q} \int ^{1}_{0}h(1-t)\,\mathrm{d}t, \end{aligned}$$
(2.15)

and

$$\begin{aligned}& C_{3}(k,\mu ,\lambda ,p) \\& \quad = \int ^{1}_{0}t^{p} \biggl\vert \biggl( \frac{\mu }{k}+1 \biggr)\lambda -t^{\frac{\mu }{k}} \biggr\vert ^{p}\,\mathrm{d}t \\& \quad =\textstyle\begin{cases} \int ^{1}_{0}t^{(\frac{\mu }{k}+1)p}\,\mathrm{d}t,& \lambda =0,\\ \left [ \textstyle\begin{array}{cc} \int ^{[(\frac{\mu }{k}+1)\lambda ]^{\frac{k}{\mu }}}_{0}t^{p}[(\frac{\mu }{k}+1)\lambda -t^{\frac{\mu }{k}}]^{p}\,\mathrm{d}t \\ +\int ^{1}_{[(\frac{\mu }{k}+1)\lambda ]^{\frac{k}{\mu }}}t^{p}[t^{\frac{\mu }{k}}-(\frac{\mu }{k}+1)\lambda ]^{p}\,\mathrm{d}t \end{array}\displaystyle \right ] , &0< \lambda \leq \frac{1}{\frac{\mu }{k}+1}, \\ \int ^{1}_{0}t^{p}[(\frac{\mu }{k}+1)\lambda -t^{\frac{\mu }{k}}]^{p}\,\mathrm{d}t, &\frac{1}{\frac{\mu }{k}+1}< \lambda \leq 1 \end{cases}\displaystyle \\& \quad = \textstyle\begin{cases} \frac{1}{p(\frac{\mu }{k}+1)+1},& \lambda =0, \\ \left [ \textstyle\begin{array}{@{}c@{}} \frac{k [ (\frac{\mu }{k}+1 )\lambda ]^{\frac{k+kp(\frac{\mu }{k}+1)}{\mu }}}{\mu }\beta (\frac{k(1+p)}{\mu },1+p )\\ {}+\frac{k [1- (\frac{\mu }{k}+1 )\lambda ]^{p+1}}{\mu (p+1)}{}_{2}F_{1} (1-\frac{k(1+p)}{\mu },1;p+2;1- (\frac{\mu }{k}+1 )\lambda ) \end{array}\displaystyle \right ] ,&0< \lambda \leq \frac{1}{\frac{\mu }{k}+1}, \\ \frac{k [ (\frac{\mu }{k}+1 )\lambda ]^{\frac{k+kp(\frac{\mu }{k}+1)}{\mu }}}{\mu }\beta (\frac{1}{(\frac{\mu }{k}+1)\lambda };\frac{k(1+p)}{\mu },1+p ), &\frac{1}{\frac{\mu }{k}+1}< \lambda \leq 1. \end{cases}\displaystyle \end{aligned}$$
(2.16)

Hence, if we use (2.14)–(2.16) in (2.13), we can get the desired result. This completes the proof. □

Let us point out some special cases of Theorem 2.2.

I. If \(h(t)=t^{s}\) in Theorem 2.2, then we have the following results.

Corollary 2.9

In Theorem 2.2, if we use the generalized \((m,s)\)-Breckner-preinvexity of \(\vert f'' \vert ^{q}\) along with \(q>1\) and \(p=\frac{q}{q-1}\), then, for \(s\in (0,1]\) and \(m\in (0,1]\), we have the following inequality:

$$ \begin{aligned} & \bigl\vert I_{f,\eta }(\mu ,k;x,\lambda ,m,a,b) \bigr\vert \\ &\quad \leq C_{3}^{\frac{1}{p}}(k,\mu ,\lambda ,p) \biggl\{ \biggl\vert \frac{\eta ^{\frac{\mu }{k}+2}(x,a,m)}{(\frac{\mu }{k}+1)\eta (b,a,m)} \biggr\vert \biggl[\frac{\vert f''(x) \vert ^{q}+m\vert f''(a) \vert ^{q}}{s+1} \biggr]^{\frac{1}{q}} \\ &\quad \quad {} + \biggl\vert \frac{(-1)^{\frac{\mu }{k}+2}\eta ^{\frac{\mu }{k}+2}(x,b,m)}{(\frac{\mu }{k}+1)\eta (b,a,m)} \biggr\vert \biggl[ \frac{\vert f''(x) \vert ^{q}+m\vert f''(b) \vert ^{q}}{s+1} \biggr]^{\frac{1}{q}} \biggr\} . \end{aligned} $$

Corollary 2.10

In Theorem 2.2, if the mapping \(\eta (b,a,m)=b-ma\) together with \(m=1\), choosing \(x=\frac{a+b}{2}\), for \(s\in (0,1]\), we have the following inequality for s-convex functions:

$$ \begin{aligned} & \biggl\vert \frac{2^{\frac{\mu }{k}-1}}{(b-a)^{\frac{\mu }{k}-1}}I_{f} \biggl(\mu ,k;\frac{a+b}{2},\lambda ,1,a,b \biggr) \biggr\vert \\ &\quad \leq \frac{(b-a)^{2}}{8 (\frac{\mu }{k}+1 )}C_{3}^{\frac{1}{p}}(k,\mu ,\lambda ,p) \biggl\{ \biggl[\frac{\vert f''(x) \vert ^{q}+\vert f''(a) \vert ^{q}}{s+1} \biggr]^{\frac{1}{q}} + \biggl[ \frac{\vert f''(x) \vert ^{q}+\vert f''(b) \vert ^{q}}{s+1} \biggr]^{\frac{1}{q}} \biggr\} . \end{aligned} $$

Remark 2.7

In Corollary 2.10,

  1. (i)

    if \(\lambda =\frac{1}{2}\), then we obtain

    $$ \begin{aligned} & \biggl\vert \frac{2^{\frac{\mu }{k}-1}}{(b-a)^{\frac{\mu }{k}-1}}I_{f} \biggl(\mu ,k;\frac{a+b}{2},\frac{1}{2},1,a,b \biggr) \biggr\vert \\ &\quad \leq \frac{(b-a)^{2}}{8 (\frac{\mu }{k}+1 )}C_{3}^{\frac{1}{p}} \biggl(k,\mu , \frac{1}{2},p \biggr) \biggl\{ \biggl[\frac{\vert f''(x) \vert ^{q}+\vert f''(a) \vert ^{q}}{s+1} \biggr]^{\frac{1}{q}} + \biggl[\frac{\vert f''(x) \vert ^{q}+\vert f''(b) \vert ^{q}}{s+1} \biggr]^{\frac{1}{q}} \biggr\} ; \end{aligned} $$
  2. (ii)

    if \(k=1\) and \(\lambda =\frac{1}{3},1\), then we have the results (c) and (f) of Corollary 2.3 in [34], respectively. Further, if we choose \(\mu =1\), then we have the results (d) and (g) of Corollary 2.3 in [34], respectively.

II. If \(h(t)=t^{-s}\) in Theorem 2.2, then we have the following results.

Corollary 2.11

In Theorem 2.2, if we use the generalized \((m,s)\)-Godunova–Levin–Dragomir-preinvexity of \(\vert f'' \vert ^{q}\) along with \(q>1\) and \(p=\frac{q}{q-1}\), then, for \(s\in (0,1)\) and \(m\in (0,1]\), we have the following inequality:

$$\begin{aligned} & \bigl\vert I_{f,\eta }(\mu ,k;x,\lambda ,m,a,b) \bigr\vert \\ &\quad \leq C_{3}^{\frac{1}{p}}(k,\mu ,\lambda ,p)\frac{1}{{(1-s)}^{\frac{1}{q}}} \biggl\{ \biggl\vert \frac{\eta ^{\frac{\mu }{k}+2}(x,a,m)}{(\frac{\mu }{k}+1)\eta (b,a,m)} \biggr\vert \bigl[ \bigl\vert f''(x) \bigr\vert ^{q}+m \bigl\vert f''(a) \bigr\vert ^{q} \bigr]^{\frac{1}{q}} \\ &\quad \quad {} + \biggl\vert \frac{(-1)^{\frac{\mu }{k}+2}\eta ^{\frac{\mu }{k}+2}(x,b,m)}{(\frac{\mu }{k}+1)\eta (b,a,m)} \biggr\vert \bigl[ \bigl\vert f''(x) \bigr\vert ^{q}+m \bigl\vert f''(b) \bigr\vert ^{q} \bigr]^{\frac{1}{q}} \biggr\} . \end{aligned}$$

Corollary 2.12

In Theorem 2.2, if the mapping \(\eta (b,a,m)=b-ma\) along with \(m=1\), choosing \(x=\frac{a+b}{2}\), for \(s\in (0,1)\), we have the following inequality for s-Godunova–Levin functions:

$$\begin{aligned} & \biggl\vert (1-\lambda )f \biggl(\frac{a+b}{2} \biggr)+\lambda \biggl[\frac{f(a)+f(b)}{2} \biggr]-\frac{2^{\frac{\mu }{k}-1}\Gamma _{k}(\mu +k)}{(b-a)^{\frac{\mu }{k}}} \bigl[{}_{k}J^{\mu }_{(\frac{a+b}{2})^{-}} f(a)+{}_{k}J^{\mu }_{(\frac{a+b}{2})^{+}}f(b) \bigr] \biggr\vert \\ &\quad \leq \frac{(b-a)^{2}}{8(\frac{\mu }{k}+1)(1-s)^{\frac{1}{q}}}C_{3}^{\frac{1}{p}}(k,\mu ,\lambda ,p) \\ &\quad \quad {} \times \biggl\{ \biggl[ \biggl\vert f'' \biggl(\frac{a+b}{2} \biggr) \biggr\vert ^{q}+ \bigl\vert f''(a) \bigr\vert ^{q} \biggr]^{\frac{1}{q}} + \biggl[ \biggl\vert f'' \biggl(\frac{a+b}{2} \biggr) \biggr\vert ^{q}+ \bigl\vert f''(b) \bigr\vert ^{q} \biggr]^{\frac{1}{q}} \biggr\} . \end{aligned}$$

Remark 2.8

In Corollary 2.12,

  1. (a)

    if \(\lambda =\frac{1}{3}\), then we obtain

    $$\begin{aligned} & \biggl\vert \frac{1}{6} \biggl[f(a)+4f \biggl(\frac{a+b}{2} \biggr)+f(b) \biggr]-\frac{2^{\frac{\mu }{k}-1}\Gamma _{k}(\mu +k)}{(b-a)^{\frac{\mu }{k}}} \bigl[{}_{k}J^{\mu }_{(\frac{a+b}{2})^{-}}f(a)+{}_{k}J^{\mu }_{(\frac{a+b}{2})^{+}}f(b) \bigr] \biggr\vert \\ &\quad \leq \frac{(b-a)^{2}}{8(\frac{\mu }{k}+1)(1-s)^{\frac{1}{q}}}C_{3}^{\frac{1}{p}} \biggl(k,\mu , \frac{1}{3},p \biggr) \\ &\quad \quad {} \times \biggl\{ \biggl[ \biggl\vert f'' \biggl(\frac{a+b}{2} \biggr) \biggr\vert ^{q}+ \bigl\vert f''(a) \bigr\vert ^{q} \biggr]^{\frac{1}{q}} + \biggl[ \biggl\vert f'' \biggl(\frac{a+b}{2} \biggr) \biggr\vert ^{q}+ \bigl\vert f''(b) \bigr\vert ^{q} \biggr]^{\frac{1}{q}} \biggr\} . \end{aligned}$$

    Specially, if we put \(k=1=\mu \), then we derive a Simpson-type inequality:

    $$\begin{aligned} & \biggl\vert \frac{1}{6} \biggl[f(a)+4f \biggl(\frac{a+b}{2} \biggr)+f(b) \biggr]-\frac{1}{b-a} \int ^{b}_{a} f(x)\,\mathrm{d}x \biggr\vert \\ &\quad \leq \frac{(b-a)^{2}}{16(1-s)^{\frac{1}{q}}}C_{3}^{\frac{1}{p}} \biggl(1,1, \frac{1}{3},p \biggr) \\ &\quad \quad {} \times \biggl\{ \biggl[ \biggl\vert f'' \biggl(\frac{a+b}{2} \biggr) \biggr\vert ^{q}+ \bigl\vert f''(a) \bigr\vert ^{q} \biggr]^{\frac{1}{q}} + \biggl[ \biggl\vert f'' \biggl(\frac{a+b}{2} \biggr) \biggr\vert ^{q}+ \bigl\vert f''(b) \bigr\vert ^{q} \biggr]^{\frac{1}{q}} \biggr\} ; \end{aligned}$$
  2. (b)

    if \(\lambda =\frac{1}{2}\), then we obtain

    $$\begin{aligned} & \biggl\vert \frac{1}{4} \biggl[f(a)+2f \biggl(\frac{a+b}{2} \biggr)+f(b) \biggr]-\frac{2^{\frac{\mu }{k}-1}\Gamma _{k}(\mu +k)}{(b-a)^{\frac{\mu }{k}}} \bigl[{}_{k}J^{\mu }_{(\frac{a+b}{2})^{-}}f(a)+{}_{k}J^{\mu }_{(\frac{a+b}{2})^{+}}f(b) \bigr] \biggr\vert \\ &\quad \leq \frac{(b-a)^{2}}{8(\frac{\mu }{k}+1)(1-s)^{\frac{1}{q}}}C_{3}^{\frac{1}{p}} \biggl(k,\mu , \frac{1}{2},p \biggr) \\ &\quad \quad {} \times \biggl\{ \biggl[ \biggl\vert f'' \biggl(\frac{a+b}{2} \biggr) \biggr\vert ^{q}+ \bigl\vert f''(a) \bigr\vert ^{q} \biggr]^{\frac{1}{q}} + \biggl[ \biggl\vert f'' \biggl(\frac{a+b}{2} \biggr) \biggr\vert ^{q}+ \bigl\vert f''(b) \bigr\vert ^{q} \biggr]^{\frac{1}{q}} \biggr\} . \end{aligned}$$

    Specially, if we put \(k=1=\mu \), then we derive an averaged midpoint-trapezoid-type inequality:

    $$\begin{aligned} & \biggl\vert \frac{1}{4} \biggl[f(a)+2f \biggl(\frac{a+b}{2} \biggr)+f(b) \biggr]-\frac{1}{b-a} \int ^{b}_{a} f(x)\,\mathrm{d}x \biggr\vert \\ &\quad \leq \frac{(b-a)^{2}}{16(1-s)^{\frac{1}{q}}}{\beta ^{\frac{1}{p}}(1+p,1+p)} \\ &\quad \quad {} \times \biggl\{ \biggl[ \biggl\vert f'' \biggl(\frac{a+b}{2} \biggr) \biggr\vert ^{q}+ \bigl\vert f''(a) \bigr\vert ^{q} \biggr]^{\frac{1}{q}} + \biggl[ \biggl\vert f'' \biggl(\frac{a+b}{2} \biggr) \biggr\vert ^{q}+ \bigl\vert f''(b) \bigr\vert ^{q} \biggr]^{\frac{1}{q}} \biggr\} ; \end{aligned}$$
  3. (c)

    if \(\lambda =0\), then we obtain

    $$\begin{aligned} & \biggl\vert f \biggl(\frac{a+b}{2} \biggr)- \frac{2^{\frac{\mu }{k}-1}\Gamma _{k}(\mu +k)}{(b-a)^{\frac{\mu }{k}}} \bigl[{}_{k}J^{\mu }_{(\frac{a+b}{2})^{-}}f(a)+{}_{k}J^{\mu }_{(\frac{a+b}{2})^{+}}f(b) \bigr] \biggr\vert \\ &\quad \leq \frac{(b-a)^{2}}{8(\frac{\mu }{k}+1)(1-s)^{\frac{1}{q}}}{ \biggl[\frac{1}{p (\frac{\mu }{k}+1 )+1} \biggr]}^{\frac{1}{p}} \\ &\quad \quad {} \times \biggl\{ \biggl[ \biggl\vert f'' \biggl(\frac{a+b}{2} \biggr) \biggr\vert ^{q}+ \bigl\vert f''(a) \bigr\vert ^{q} \biggr]^{\frac{1}{q}} + \biggl[ \biggl\vert f'' \biggl(\frac{a+b}{2} \biggr) \biggr\vert ^{q}+ \bigl\vert f''(b) \bigr\vert ^{q} \biggr]^{\frac{1}{q}} \biggr\} . \end{aligned}$$

    Specially, if we put \(k=1=\mu \), then we derive a midpoint-type inequality:

    $$\begin{aligned} & \biggl\vert f \biggl(\frac{a+b}{2} \biggr)- \frac{1}{b-a} \int ^{b}_{a} f(x)\,\mathrm{d}x \biggr\vert \\ &\quad \leq \frac{(b-a)^{2}}{16(1-s)^{\frac{1}{q}}}{ \biggl(\frac{1}{2p+1} \biggr)}^{\frac{1}{p}} \\ &\quad \quad {} \times \biggl\{ \biggl[ \biggl\vert f'' \biggl(\frac{a+b}{2} \biggr) \biggr\vert ^{q}+ \bigl\vert f''(a) \bigr\vert ^{q} \biggr]^{\frac{1}{q}} + \biggl[ \biggl\vert f'' \biggl(\frac{a+b}{2} \biggr) \biggr\vert ^{q}+ \bigl\vert f''(b) \bigr\vert ^{q} \biggr]^{\frac{1}{q}} \biggr\} ; \end{aligned}$$
  4. (d)

    if \(\lambda =1\), then we obtain

    $$\begin{aligned} & \biggl\vert \frac{f(a)+f(b)}{2}-\frac{2^{\frac{\mu }{k}-1}\Gamma _{k}(\mu +k)}{(b-a)^{\frac{\mu }{k}}} \bigl[{}_{k}J^{\mu }_{(\frac{a+b}{2})^{-}}f(a)+{}_{k}J^{\mu }_{(\frac{a+b}{2})^{+}}f(b) \bigr] \biggr\vert \\ &\quad \leq \frac{(b-a)^{2}}{8(\frac{\mu }{k}+1)(1-s)^{\frac{1}{q}}}C_{3}^{\frac{1}{p}}(k,\mu ,1,p) \\ &\quad \quad {} \times \biggl\{ \biggl[ \biggl\vert f'' \biggl(\frac{a+b}{2} \biggr) \biggr\vert ^{q}+ \bigl\vert f''(a) \bigr\vert ^{q} \biggr]^{\frac{1}{q}} + \biggl[ \biggl\vert f'' \biggl(\frac{a+b}{2} \biggr) \biggr\vert ^{q}+ \bigl\vert f''(b) \bigr\vert ^{q} \biggr]^{\frac{1}{q}} \biggr\} . \end{aligned}$$

    Specially, if we put \(k=1=\mu \), then we derive a trapezoid-type inequality:

    $$\begin{aligned} & \biggl\vert \frac{f(a)+f(b)}{2}-\frac{1}{b-a} \int ^{b}_{a} f(x)\,\mathrm{d}x \biggr\vert \\ &\quad \leq \frac{(b-a)^{2}}{16(1-s)^{\frac{1}{q}}}{ \biggl[2^{1+2p}\beta \biggl(\frac{1}{2};1+p,1+p \biggr) \biggr]}^{\frac{1}{p}} \\ &\quad \quad {} \times \biggl\{ \biggl[ \biggl\vert f'' \biggl(\frac{a+b}{2} \biggr) \biggr\vert ^{q}+ \bigl\vert f''(a) \bigr\vert ^{q} \biggr]^{\frac{1}{q}} + \biggl[ \biggl\vert f'' \biggl(\frac{a+b}{2} \biggr) \biggr\vert ^{q}+ \bigl\vert f''(b) \bigr\vert ^{q} \biggr]^{\frac{1}{q}} \biggr\} . \end{aligned}$$

III. If \(h(t)=t(1-t)\) in Theorem 2.2, then we have the following results.

Corollary 2.13

In Theorem 2.2, if we use the generalized \((m,tgs)\)-preinvexity of \(\vert f'' \vert ^{q}\) along with \(q>1\) and \(p=\frac{q}{q-1}\), then, for \(m\in (0,1]\), we have the following inequality:

$$\begin{aligned} & \bigl\vert I_{f,\eta }(\mu ,k;x,\lambda ,m,a,b) \bigr\vert \\ &\quad \leq C_{3}^{\frac{1}{p}}(k,\mu ,\lambda ,p){ \biggl( \frac{1}{6} \biggr)}^{\frac{1}{q}} \biggl\{ \biggl\vert \frac{\eta ^{\frac{\mu }{k}+2}(x,a,m)}{(\frac{\mu }{k}+1)\eta (b,a,m)} \biggr\vert \bigl[ \bigl\vert f''(x) \bigr\vert ^{q}+m \bigl\vert f''(a) \bigr\vert ^{q} \bigr]^{\frac{1}{q}} \\ &\quad \quad {} + \biggl\vert \frac{(-1)^{\frac{\mu }{k}+2}\eta ^{\frac{\mu }{k}+2}(x,b,m)}{(\frac{\mu }{k}+1)\eta (b,a,m)} \biggr\vert \bigl[ \bigl\vert f''(x) \bigr\vert ^{q}+m \bigl\vert f''(b) \bigr\vert ^{q} \bigr]^{\frac{1}{q}} \biggr\} , \end{aligned}$$

where \(C_{3}(k,\mu ,\lambda ,p)\) is defined by (2.16).

Corollary 2.14

In Theorem 2.2, if the mapping \(\eta (b,a,m)=b-ma\) together with \(m=1\), choosing \(x=\frac{a+b}{2}\), we get the following inequality for \(tgs\)-convex functions:

$$\begin{aligned} & \biggl\vert (1-\lambda )f \biggl(\frac{a+b}{2} \biggr)+\lambda \biggl[\frac{f(a)+f(b)}{2} \biggr] -\frac{2^{\frac{\mu }{k}-1}\Gamma _{k}(\mu +k)}{(b-a)^{\frac{\mu }{k}}} \bigl[{}_{k}J^{\mu }_{(\frac{a+b}{2})^{-}} f(a)+{}_{k}J^{\mu }_{(\frac{a+b}{2})^{+}}f(b) \bigr] \biggr\vert \\ &\quad \leq \frac{(b-a)^{2}}{8(\frac{\mu }{k}+1)} \biggl(\frac{1}{6} \biggr)^{\frac{1}{q}}C_{3}^{\frac{1}{p}}(k, \mu ,\lambda ,p) \\ &\quad \quad {} \times \biggl\{ \biggl[ \biggl\vert f'' \biggl(\frac{a+b}{2} \biggr) \biggr\vert ^{q}+ \bigl\vert f''(a) \bigr\vert ^{q} \biggr]^{\frac{1}{q}} + \biggl[ \biggl\vert f'' \biggl(\frac{a+b}{2} \biggr) \biggr\vert ^{q}+ \bigl\vert f''(b) \bigr\vert ^{q} \biggr]^{\frac{1}{q}} \biggr\} . \end{aligned}$$

Remark 2.9

In Corollary 2.14,

  1. (a)

    if \(\lambda =\frac{1}{3}\), then we obtain

    $$\begin{aligned} & \biggl\vert \frac{1}{6} \biggl[f(a)+4f \biggl(\frac{a+b}{2} \biggr)+f(b) \biggr]-\frac{2^{\frac{\mu }{k}-1}\Gamma _{k}(\mu +k)}{(b-a)^{\frac{\mu }{k}}} \bigl[{}_{k}J^{\mu }_{(\frac{a+b}{2})^{-}} f(a)+{}_{k}J^{\mu }_{(\frac{a+b}{2})^{+}}f(b) \bigr] \biggr\vert \\ &\quad \leq \frac{(b-a)^{2}}{8(\frac{\mu }{k}+1)} \biggl(\frac{1}{6} \biggr)^{\frac{1}{q}}C_{3}^{\frac{1}{p}} \biggl(k,\mu ,\frac{1}{3},p \biggr) \\ &\quad \quad {} \times \biggl\{ \biggl[ \biggl\vert f'' \biggl(\frac{a+b}{2} \biggr) \biggr\vert ^{q}+ \bigl\vert f''(a) \bigr\vert ^{q} \biggr]^{\frac{1}{q}} + \biggl[ \biggl\vert f'' \biggl(\frac{a+b}{2} \biggr) \biggr\vert ^{q}+ \bigl\vert f''(b) \bigr\vert ^{q} \biggr]^{\frac{1}{q}} \biggr\} . \end{aligned}$$

    Specially, if we put \(k=1=\mu \), then we derive a Simpson-type inequality:

    $$\begin{aligned} & \biggl\vert \frac{1}{6} \biggl[f(a)+4f \biggl(\frac{a+b}{2} \biggr)+f(b) \biggr]-\frac{1}{b-a} \int ^{b}_{a} f(x)\,\mathrm{d}x \biggr\vert \\ &\quad \leq \frac{(b-a)^{2}}{16} \biggl(\frac{1}{6} \biggr)^{\frac{1}{q}}C_{3}^{\frac{1}{p}} \biggl(1,1, \frac{1}{3},p \biggr) \\ &\quad \quad {} \times \biggl\{ \biggl[ \biggl\vert f'' \biggl(\frac{a+b}{2} \biggr) \biggr\vert ^{q}+ \bigl\vert f''(a) \bigr\vert ^{q} \biggr]^{\frac{1}{q}} + \biggl[ \biggl\vert f'' \biggl(\frac{a+b}{2} \biggr) \biggr\vert ^{q}+ \bigl\vert f''(b) \bigr\vert ^{q} \biggr]^{\frac{1}{q}} \biggr\} ; \end{aligned}$$
  2. (b)

    if \(\lambda =\frac{1}{2}\), then we obtain

    $$ \begin{aligned} & \biggl\vert \frac{1}{4} \biggl[f(a)+2f \biggl(\frac{a+b}{2} \biggr)+f(b) \biggr]-\frac{2^{\frac{\mu }{k}-1}\Gamma _{k}(\mu +k)}{(b-a)^{\frac{\mu }{k}}} \bigl[{}_{k}J^{\mu }_{(\frac{a+b}{2})^{-}}f(a)+{}_{k}J^{\mu }_{(\frac{a+b}{2})^{+}}f(b) \bigr] \biggr\vert \\ &\quad \leq \frac{(b-a)^{2}}{8(\frac{\mu }{k}+1)} \biggl(\frac{1}{6} \biggr)^{\frac{1}{q}}C_{3}^{\frac{1}{p}} \biggl(k,\mu ,\frac{1}{2},p \biggr) \\ &\quad \quad {} \times \biggl\{ \biggl[ \biggl\vert f'' \biggl(\frac{a+b}{2} \biggr) \biggr\vert ^{q}+ \bigl\vert f''(a) \bigr\vert ^{q} \biggr]^{\frac{1}{q}} + \biggl[ \biggl\vert f'' \biggl(\frac{a+b}{2} \biggr) \biggr\vert ^{q}+ \bigl\vert f''(b) \bigr\vert ^{q} \biggr]^{\frac{1}{q}} \biggr\} . \end{aligned} $$

    Specially, if we put \(k=1=\mu \), then we derive an averaged midpoint-trapezoid-type inequality:

    $$ \begin{aligned} & \biggl\vert \frac{1}{4} \biggl[f(a)+2f \biggl(\frac{a+b}{2} \biggr)+f(b) \biggr]-\frac{1}{b-a} \int ^{b}_{a} f(x)\,\mathrm{d}x \biggr\vert \\ &\quad \leq \frac{(b-a)^{2}}{16} \biggl(\frac{1}{6} \biggr)^{\frac{1}{q}}\beta ^{\frac{1}{p}}(1+p,1+p) \\ &\quad \quad {} \times \biggl\{ \biggl[ \biggl\vert f'' \biggl(\frac{a+b}{2} \biggr) \biggr\vert ^{q}+ \bigl\vert f''(a) \bigr\vert ^{q} \biggr]^{\frac{1}{q}} + \biggl[ \biggl\vert f'' \biggl(\frac{a+b}{2} \biggr) \biggr\vert ^{q}+ \bigl\vert f''(b) \bigr\vert ^{q} \biggr]^{\frac{1}{q}} \biggr\} ; \end{aligned} $$
  3. (c)

    if \(\lambda =0\), then we obtain

    $$\begin{aligned} & \biggl\vert f \biggl(\frac{a+b}{2} \biggr)- \frac{2^{\frac{\mu }{k}-1}\Gamma _{k}(\mu +k)}{(b-a)^{\frac{\mu }{k}}} \bigl[{}_{k}J^{\mu }_{(\frac{a+b}{2})^{-}}f(a)+{}_{k}J^{\mu }_{(\frac{a+b}{2})^{+}}f(b) \bigr] \biggr\vert \\ &\quad \leq \frac{(b-a)^{2}}{8(\frac{\mu }{k}+1)} \biggl(\frac{1}{6} \biggr)^{\frac{1}{q}}{ \biggl[\frac{1}{p (\frac{\mu }{k}+1 )+1} \biggr]}^{\frac{1}{p}} \\ &\quad \quad {} \times \biggl\{ \biggl[ \biggl\vert f'' \biggl(\frac{a+b}{2} \biggr) \biggr\vert ^{q}+ \bigl\vert f''(a) \bigr\vert ^{q} \biggr]^{\frac{1}{q}} + \biggl[ \biggl\vert f'' \biggl(\frac{a+b}{2} \biggr) \biggr\vert ^{q}+ \bigl\vert f''(b) \bigr\vert ^{q} \biggr]^{\frac{1}{q}} \biggr\} . \end{aligned}$$

    Specially, if we put \(k=1=\mu \), then we derive a midpoint-type inequality:

    $$\begin{aligned} & \biggl\vert f \biggl(\frac{a+b}{2} \biggr)- \frac{1}{b-a} \int ^{b}_{a} f(x)\,\mathrm{d}x \biggr\vert \\ &\quad \leq \frac{(b-a)^{2}}{16} \biggl(\frac{1}{6} \biggr)^{\frac{1}{q}}{ \biggl(\frac{1}{2p+1} \biggr)}^{\frac{1}{p}} \\ &\quad \quad {} \times \biggl\{ \biggl[ \biggl\vert f'' \biggl(\frac{a+b}{2} \biggr) \biggr\vert ^{q}+ \bigl\vert f''(a) \bigr\vert ^{q} \biggr]^{\frac{1}{q}} + \biggl[ \biggl\vert f'' \biggl(\frac{a+b}{2} \biggr) \biggr\vert ^{q}+ \bigl\vert f''(b) \bigr\vert ^{q} \biggr]^{\frac{1}{q}} \biggr\} ; \end{aligned}$$
  4. (d)

    if \(\lambda =1\), then we obtain

    $$ \begin{aligned} & \biggl\vert \frac{f(a)+f(b)}{2}-\frac{2^{\frac{\mu }{k}-1}\Gamma _{k}(\mu +k)}{(b-a)^{\frac{\mu }{k}}} \bigl[{}_{k}J^{\mu }_{(\frac{a+b}{2})^{-}}f(a)+{}_{k}J^{\mu }_{(\frac{a+b}{2})^{+}}f(b) \bigr] \biggr\vert \\ &\quad \leq \frac{(b-a)^{2}}{8(\frac{\mu }{k}+1)} \biggl(\frac{1}{6} \biggr)^{\frac{1}{q}}C_{3}^{\frac{1}{p}}(k, \mu ,1,p) \\ &\quad \quad {} \times \biggl\{ \biggl[ \biggl\vert f'' \biggl(\frac{a+b}{2} \biggr) \biggr\vert ^{q}+ \bigl\vert f''(a) \bigr\vert ^{q} \biggr]^{\frac{1}{q}} + \biggl[ \biggl\vert f'' \biggl(\frac{a+b}{2} \biggr) \biggr\vert ^{q}+ \bigl\vert f''(b) \bigr\vert ^{q} \biggr]^{\frac{1}{q}} \biggr\} . \end{aligned} $$

    Specially, if we put \(k=1=\mu \), then we derive a trapezoid-type inequality:

    $$ \begin{aligned} & \biggl\vert \frac{f(a)+f(b)}{2}-\frac{1}{b-a} \int ^{b}_{a} f(x)\,\mathrm{d}x \biggr\vert \\ &\quad \leq \frac{(b-a)^{2}}{16} \biggl(\frac{1}{6} \biggr)^{\frac{1}{q}}{ \biggl[2^{1+2p}\beta \biggl(\frac{1}{2};1+p,1+p \biggr) \biggr]}^{\frac{1}{p}} \\ &\quad \quad {} \times \biggl\{ \biggl[ \biggl\vert f'' \biggl(\frac{a+b}{2} \biggr) \biggr\vert ^{q}+ \bigl\vert f''(a) \bigr\vert ^{q} \biggr]^{\frac{1}{q}} + \biggl[ \biggl\vert f'' \biggl(\frac{a+b}{2} \biggr) \biggr\vert ^{q}+ \bigl\vert f''(b) \bigr\vert ^{q} \biggr]^{\frac{1}{q}} \biggr\} . \end{aligned} $$

IV. If \(h(t)=\frac{\sqrt{t}}{2\sqrt{1-t}}\) in Theorem 2.2, then we have the following results.

Corollary 2.15

In Theorem 2.2, if we use the generalized m-MT-preinvexity of \(\vert f'' \vert ^{q}\) along with \(q>1\) and \(p=\frac{q}{q-1}\), then, for \(m\in (0,1]\), we have the following inequality:

$$ \begin{aligned} & \bigl\vert I_{f,\eta }(\mu ,k;x,\lambda ,m,a,b) \bigr\vert \\ &\quad \leq C_{3}^{\frac{1}{p}}(k,\mu ,\lambda ,p) \biggl( \frac{\pi }{4} \biggr)^{\frac{1}{q}} \biggl\{ \biggl\vert \frac{\eta ^{\frac{\mu }{k}+2}(x,a,m)}{(\frac{\mu }{k}+1)\eta (b,a,m)} \biggr\vert \bigl[ \bigl\vert f''(x) \bigr\vert ^{q}+m \bigl\vert f''(a) \bigr\vert ^{q} \bigr]^{\frac{1}{q}} \\ &\quad \quad {} + \biggl\vert \frac{(-1)^{\frac{\mu }{k}+2}\eta ^{\frac{\mu }{k}+2}(x,b,m)}{(\frac{\mu }{k}+1)\eta (b,a,m)} \biggr\vert \bigl[ \bigl\vert f''(x) \bigr\vert ^{q}+m \bigl\vert f''(b) \bigr\vert ^{q} \bigr]^{\frac{1}{q}} \biggr\} , \end{aligned} $$

where we use the fact that

$$ \begin{aligned} \int ^{1}_{0}\frac{\sqrt{t}}{2\sqrt{1-t}}\,\mathrm{d}t= \int ^{1}_{0}\frac{\sqrt{1-t}}{2\sqrt{t}}\,\mathrm{d}t= \frac{\pi }{4}. \end{aligned} $$

Corollary 2.16

In Theorem 2.2, if the mapping \(\eta (b,a,m)=b-ma\) together with \(m=1\), choosing \(x=\frac{a+b}{2}\), we get the following inequality for MT-convex functions:

$$\begin{aligned} & \biggl\vert (1-\lambda )f \biggl(\frac{a+b}{2} \biggr)+\lambda \biggl[\frac{f(a)+f(b)}{2} \biggr] \\ &\quad \quad {} -\frac{2^{\frac{\mu }{k}-1}\Gamma _{k}(\mu +k)}{(b-a)^{\frac{\mu }{k}}} \bigl[{}_{k}J^{\mu }_{(\frac{a+b}{2})^{-}} f(a)+{}_{k}J^{\mu }_{(\frac{a+b}{2})^{+}}f(b) \bigr] \biggr\vert \\ &\quad \leq \frac{(b-a)^{2}}{8(\frac{\mu }{k}+1)} \biggl(\frac{\pi }{4} \biggr)^{\frac{1}{q}}C_{3}^{\frac{1}{p}}(k,\mu ,\lambda ,p) \\ &\quad \quad {} \times \biggl\{ \biggl[ \biggl\vert f'' \biggl(\frac{a+b}{2} \biggr) \biggr\vert ^{q}+ \bigl\vert f''(a) \bigr\vert ^{q} \biggr]^{\frac{1}{q}} + \biggl[ \biggl\vert f'' \biggl(\frac{a+b}{2} \biggr) \biggr\vert ^{q}+ \bigl\vert f''(b) \bigr\vert ^{q} \biggr]^{\frac{1}{q}} \biggr\} . \end{aligned}$$

Remark 2.10

In Corollary 2.16,

  1. (a)

    if \(\lambda =\frac{1}{3}\), then we obtain

    $$\begin{aligned} & \biggl\vert \frac{1}{6} \biggl[f(a)+4f \biggl(\frac{a+b}{2} \biggr)+f(b) \biggr] \\ &\quad \quad {}-\frac{2^{\frac{\mu }{k}-1}\Gamma _{k}(\mu +k)}{(b-a)^{\frac{\mu }{k}}} \bigl[{}_{k}J^{\mu }_{(\frac{a+b}{2})^{-}}f(a)+{}_{k}J^{\mu }_{(\frac{a+b}{2})^{+}}f(b) \bigr] \biggr\vert \\ &\quad \leq \frac{(b-a)^{2}}{8(\frac{\mu }{k}+1)} \biggl(\frac{\pi }{4} \biggr)^{\frac{1}{q}}C_{3}^{\frac{1}{p}} \biggl(k,\mu ,\frac{1}{3},p \biggr) \\ &\quad \quad {} \times \biggl\{ \biggl[ \biggl\vert f'' \biggl(\frac{a+b}{2} \biggr) \biggr\vert ^{q}+ \bigl\vert f''(a) \bigr\vert ^{q} \biggr]^{\frac{1}{q}} + \biggl[ \biggl\vert f'' \biggl(\frac{a+b}{2} \biggr) \biggr\vert ^{q}+ \bigl\vert f''(b) \bigr\vert ^{q} \biggr]^{\frac{1}{q}} \biggr\} . \end{aligned}$$

    Specially, if we put \(k=1=\mu \), then we derive a Simpson-type inequality:

    $$\begin{aligned} & \biggl\vert \frac{1}{6} \biggl[f(a)+4f \biggl(\frac{a+b}{2} \biggr)+f(b) \biggr]-\frac{1}{b-a} \int ^{b}_{a} f(x)\,\mathrm{d}x \biggr\vert \\ &\quad \leq \frac{(b-a)^{2}}{16} \biggl(\frac{\pi }{4} \biggr)^{\frac{1}{q}}C_{3}^{\frac{1}{p}} \biggl(1,1,\frac{1}{3},p \biggr) \\ &\quad \quad {} \times \biggl\{ \biggl[ \biggl\vert f'' \biggl( \frac{a+b}{2} \biggr) \biggr\vert ^{q}+ \bigl\vert f''(a) \bigr\vert ^{q} \biggr]^{\frac{1}{q}} + \biggl[ \biggl\vert f'' \biggl(\frac{a+b}{2} \biggr) \biggr\vert ^{q}+ \bigl\vert f''(b) \bigr\vert ^{q} \biggr]^{\frac{1}{q}}\biggr\} ; \end{aligned}$$
  2. (b)

    if \(\lambda =\frac{1}{2}\), then we obtain

    $$\begin{aligned} & \biggl\vert \frac{1}{4} \biggl[f(a)+2f \biggl(\frac{a+b}{2} \biggr)+f(b) \biggr] \\ &\quad \quad {}-\frac{2^{\frac{\mu }{k}-1}\Gamma _{k}(\mu +k)}{(b-a)^{\frac{\mu }{k}}} \bigl[{}_{k}J^{\mu }_{(\frac{a+b}{2})^{-}}f(a)+{}_{k}J^{\mu }_{(\frac{a+b}{2})^{+}}f(b) \bigr] \biggr\vert \\ &\quad \leq \frac{(b-a)^{2}}{8(\frac{\mu }{k}+1)} \biggl(\frac{\pi }{4} \biggr)^{\frac{1}{q}}C_{3}^{\frac{1}{p}} \biggl(k,\mu ,\frac{1}{2},p \biggr) \\ &\quad \quad {} \times \biggl\{ \biggl[ \biggl\vert f'' \biggl(\frac{a+b}{2} \biggr) \biggr\vert ^{q}+ \bigl\vert f''(a) \bigr\vert ^{q} \biggr]^{\frac{1}{q}} + \biggl[ \biggl\vert f'' \biggl(\frac{a+b}{2} \biggr) \biggr\vert ^{q}+ \bigl\vert f''(b) \bigr\vert ^{q} \biggr]^{\frac{1}{q}} \biggr\} . \end{aligned}$$

    Specially, if we put \(k=1=\mu \), then we derive an averaged midpoint-trapezoid-type inequality:

    $$\begin{aligned} & \biggl\vert \frac{1}{4} \biggl[f(a)+2f \biggl(\frac{a+b}{2} \biggr)+f(b) \biggr]-\frac{1}{b-a} \int ^{b}_{a} f(x)\,\mathrm{d}x \biggr\vert \\ &\quad \leq \frac{(b-a)^{2}}{16} \biggl(\frac{\pi }{4} \biggr)^{\frac{1}{q}}\beta ^{\frac{1}{p}}(1+p,1+p) \\ &\quad \quad {} \times \biggl\{ \biggl[ \biggl\vert f'' \biggl(\frac{a+b}{2} \biggr) \biggr\vert ^{q}+ \bigl\vert f''(a) \bigr\vert ^{q} \biggr]^{\frac{1}{q}} + \biggl[ \biggl\vert f'' \biggl(\frac{a+b}{2} \biggr) \biggr\vert ^{q}+ \bigl\vert f''(b) \bigr\vert ^{q} \biggr]^{\frac{1}{q}} \biggr\} ; \end{aligned}$$
  3. (c)

    if \(\lambda =0\), then we obtain

    $$\begin{aligned} & \biggl\vert f \biggl(\frac{a+b}{2} \biggr)- \frac{2^{\frac{\mu }{k}-1}\Gamma _{k}(\mu +k)}{(b-a)^{\frac{\mu }{k}}} \bigl[{}_{k}J^{\mu }_{(\frac{a+b}{2})^{-}}f(a)+{}_{k}J^{\mu }_{(\frac{a+b}{2})^{+}}f(b) \bigr] \biggr\vert \\ &\quad \leq \frac{(b-a)^{2}}{8(\frac{\mu }{k}+1)} \biggl(\frac{\pi }{4} \biggr)^{\frac{1}{q}}{ \biggl[\frac{1}{p (\frac{\mu }{k}+1 )+1} \biggr]}^{\frac{1}{p}} \\ &\quad\quad {} \times \biggl\{ \biggl[ \biggl\vert f'' \biggl(\frac{a+b}{2} \biggr) \biggr\vert ^{q}+ \bigl\vert f''(a) \bigr\vert ^{q} \biggr]^{\frac{1}{q}} + \biggl[ \biggl\vert f'' \biggl(\frac{a+b}{2} \biggr) \biggr\vert ^{q}+ \bigl\vert f''(b) \bigr\vert ^{q} \biggr]^{\frac{1}{q}} \biggr\} . \end{aligned}$$

    Specially, if we put \(k=1=\mu \), then we derive a midpoint-type inequality:

    $$ \begin{aligned} & \biggl\vert f \biggl(\frac{a+b}{2} \biggr)- \frac{1}{b-a} \int ^{b}_{a} f(x)\,\mathrm{d}x \biggr\vert \\ &\quad \leq \frac{(b-a)^{2}}{16} \biggl(\frac{\pi }{4} \biggr)^{\frac{1}{q}}{ \biggl(\frac{1}{2p+1} \biggr)}^{\frac{1}{p}} \\ &\quad \quad {} \times \biggl\{ \biggl[ \biggl\vert f'' \biggl(\frac{a+b}{2} \biggr) \biggr\vert ^{q}+ \bigl\vert f''(a) \bigr\vert ^{q} \biggr]^{\frac{1}{q}} + \biggl[ \biggl\vert f'' \biggl(\frac{a+b}{2} \biggr) \biggr\vert ^{q}+ \bigl\vert f''(b) \bigr\vert ^{q} \biggr]^{\frac{1}{q}} \biggr\} ; \end{aligned} $$
  4. (d)

    if \(\lambda =1\), then we obtain

    $$ \begin{aligned} & \biggl\vert \frac{f(a)+f(b)}{2}-\frac{2^{\frac{\mu }{k}-1}\Gamma _{k}(\mu +k)}{(b-a)^{\frac{\mu }{k}}} \bigl[{}_{k}J^{\mu }_{(\frac{a+b}{2})^{-}}f(a)+{}_{k}J^{\mu }_{(\frac{a+b}{2})^{+}}f(b) \bigr] \biggr\vert \\ &\quad \leq \frac{(b-a)^{2}}{8(\frac{\mu }{k}+1)} \biggl(\frac{\pi }{4} \biggr)^{\frac{1}{q}}C_{3}^{\frac{1}{p}}(k, \mu ,1,p) \\ &\quad \quad {} \times \biggl\{ \biggl[ \biggl\vert f'' \biggl(\frac{a+b}{2} \biggr) \biggr\vert ^{q}+ \bigl\vert f''(a) \bigr\vert ^{q} \biggr]^{\frac{1}{q}} + \biggl[ \biggl\vert f'' \biggl(\frac{a+b}{2} \biggr) \biggr\vert ^{q}+ \bigl\vert f''(b) \bigr\vert ^{q} \biggr]^{\frac{1}{q}} \biggr\} . \end{aligned} $$

    Specially, if we put \(k=1=\mu \), then we derive a trapezoid-type inequality:

    $$\begin{aligned} & \biggl\vert \frac{f(a)+f(b)}{2}-\frac{1}{b-a} \int ^{b}_{a} f(x)\,\mathrm{d}x \biggr\vert \\ &\quad \leq \frac{(b-a)^{2}}{16} \biggl(\frac{\pi }{4} \biggr)^{\frac{1}{q}}{ \biggl[2^{1+2p}\beta \biggl(\frac{1}{2};1+p,1+p \biggr) \biggr]}^{\frac{1}{p}} \\ &\quad \quad {} \times \biggl\{ \biggl[ \biggl\vert f'' \biggl(\frac{a+b}{2} \biggr) \biggr\vert ^{q}+ \bigl\vert f''(a) \bigr\vert ^{q} \biggr]^{\frac{1}{q}} + \biggl[ \biggl\vert f'' \biggl(\frac{a+b}{2} \biggr) \biggr\vert ^{q}+ \bigl\vert f''(b) \bigr\vert ^{q} \biggr]^{\frac{1}{q}} \biggr\} . \end{aligned}$$

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Acknowledgements

This work was partially supported by the National Natural Science Foundation of China under Grant No. 61374028.

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Zhang, Y., Du, T., Wang, H. et al. Extensions of different type parameterized inequalities for generalized \((m,h)\)-preinvex mappings via k-fractional integrals. J Inequal Appl 2018, 49 (2018). https://doi.org/10.1186/s13660-018-1639-5

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MSC

  • 26A33
  • 26A51
  • 26D07
  • 26D20
  • 41A55

Keywords

  • Hermite–Hadamard’s inequality
  • Simpson’s inequality
  • Generalized \((m,h)\)-preinvex functions
  • k-fractional integrals