1 Introduction

Let \(h: I\subseteq \mathbb{R}\to \mathbb{R}\) be a convex function with \(a< b\) and \(a, b\in I\). Then

$$ h \biggl(\frac{a+b}{2} \biggr)\le \frac{1}{b-a} \int _{a}^{b} h(x)\,\operatorname {d}x \le \frac{h(a)+h(b)}{2}. $$
(1.1)

Inequality (1.1) is well known in the literature as the Hermite–Hadamard inequality. A number of mathematicians have devoted their efforts to generalize, refine, counterpart, and extend the Hermite–Hadamard inequality (1.1) for different classes of convex functions and mappings. For several recent results concerning inequality (1.1), we may refer the interested reader to [1, 10, 14, 27, 33, 34, 39].

Let us recall some definitions and known results concerning convexity.

Definition 1.1

([33])

A function \(h: I\subseteq \mathbb{R}\to \mathbb{R}\) is said to be convex on an interval I if the inequality

$$ h \bigl(\lambda x+(1-\lambda )y \bigr)\le \lambda h(x)+(1-\lambda )h(y) $$

holds for all \(x,y\in I\) and \(\lambda \in (0,1)\).

Definition 1.2

([1, 33])

A function: \(h: [0,b]\to \mathbb{R}\) is said to be m-convex if

$$ h \bigl(\lambda a+m(1-\lambda )b \bigr)\le \lambda h(a)+m(1-\lambda )h(b) $$

holds for all \(a,b\in [0,b]\) and \(\lambda \in [0,1]\) and for some \(m\in (0,1]\).

Definition 1.3

([1, 33])

Let \((\alpha ,m)\in (0,1]^{2}\). A function: \(h: [0,b]\to \mathbb{R}\) is said to be \((\alpha ,m)\)-convex if

$$ h \bigl(\lambda a+m(1-\lambda )b \bigr)\le \lambda ^{\alpha }h(a)+m \bigl(1- \lambda ^{\alpha } \bigr)h(b) $$

holds for all \(a,b\in [0,b]\) and \(\lambda \in [0,1]\) and for some \(m\in (0,1]\).

The Riemann–Liouville integrals \(J_{a+}^{\alpha }h(t)\) and \(J_{b-}^{\alpha }h(t)\) of order \(\alpha \ge 0\) are defined in [5] respectively by \(J_{a+}^{0} h(t)=J _{b-}^{0} h(t)=h(t)\),

$$ J_{a+}^{\alpha }h(t)=\frac{1}{\varGamma (\alpha )} \int _{a}^{t}(t-u)^{ \alpha -1}h(u)\,\operatorname {d}u, \quad t>a $$

and

$$ J_{b-}^{\alpha }h(t)=\frac{1}{\varGamma (\alpha )} \int _{t}^{b}(u-t)^{ \alpha -1}h(u)\,\operatorname {d}u, \quad t< b $$

for \(h \in L_{1}([a,b])\) and \(\alpha >0\), where Γ denotes the classical Euler gamma function which can be defined [17, 22] by

$$ \varGamma (w)=\lim_{n\to \infty }\frac{n!n^{w}}{\prod_{k=0}^{n}(w+k)}, \quad w\in \mathbb{C} \setminus \{0,-1,-2,\ldots \} $$

or by

$$ \varGamma (w)= \int ^{\infty }_{0}u^{w-1} e^{-u} \,\operatorname {d}u, \quad \Re (w)>0. $$

Recently, the following integral identity and the Riemann–Liouville fractional integral inequalities of the Hermite–Hadamard type for \((\alpha ,m)\)-convex functions were obtained.

Lemma 1.1

([26, Lemma 2.1])

Let \(h:[a,b]\subseteq \mathbb{R}\to \mathbb{R}\) be differentiable on an interval \((a,b)\) with \(a< b\) such that \(h'\in L_{1}([a,b])\). Then

$$\begin{aligned} Q_{\alpha }(a,b)&=\frac{b-a}{16} \biggl[ \int _{0}^{1} \bigl(1-u^{\alpha } \bigr)h' \biggl(\frac{3a+b}{4}u+\frac{a+b}{2}(1-u) \biggr) \,\operatorname {d}u \\ &\quad{} - \int _{0}^{1} u^{\alpha }h' \biggl(au+\frac{3a+b}{4}(1-u) \biggr)\,\operatorname {d}u \\ &\quad{}+ \int _{0} ^{1} \bigl(1-u^{\alpha } \bigr)h' \biggl(\frac{a+3b}{4}u+b(1-u) \biggr)\,\operatorname {d}u \\ &\quad{}- \int _{0}^{1} u^{\alpha }h' \biggl( \frac{a+b}{2}u+\frac{a+3b}{4}(1-u) \biggr)\,\operatorname {d}u \biggr] \end{aligned}$$
(1.2)

for \(\alpha >0\), where

$$\begin{aligned} Q_{\alpha }(a,b)&=\frac{1}{2} \biggl[\frac{h(a)+h(b)}{2}+h \biggl( \frac{a+b}{2} \biggr) \biggr] -\frac{4^{\alpha -1}\varGamma (\alpha +1)}{(b-a)^{ \alpha }} \biggl[J_{a+}^{\alpha }h \biggl(\frac{3a+b}{4} \biggr) \\ &\quad{} +J_{[(3a+b)/4]+}^{\alpha }h \biggl(\frac{a+b}{2} \biggr)+J_{[(a+b)/2]+} ^{\alpha }h \biggl(\frac{a+3b}{4} \biggr)+J_{[(a+3b)/4]+}^{\alpha }h(b) \biggr]. \end{aligned}$$

Theorem 1.1

([26, Theorem 3.1])

Let \(h: [0,\infty )\to \mathbb{R}\) be differentiable on \([0,\infty )\) and \(h'\in L_{1}([a,b])\) for \(0\le a< b\) and \(\alpha >0\). If \(\vert h'\vert ^{q}\) is \((\alpha _{1},m)\)-convex on \([0,\frac{b}{m} ]\) for some \((\alpha _{1},m)\in (0,1]^{2}\) and \(q\ge 1\), then

$$\begin{aligned} \bigl\vert Q_{\alpha }(a,b) \bigr\vert &\le \frac{b-a}{16(\alpha +1)} \biggl[ \frac{1}{( \alpha _{1}+1)(\alpha +\alpha _{1}+1)} \biggr]^{1/q} \biggl[ \biggl(( \alpha +1) (\alpha _{1}+1) \bigl\vert h'(a) \bigr\vert ^{q} \\ & \quad {} +m\alpha _{1}(\alpha _{1}+1) \biggl\vert h' \biggl(\frac{3a+b}{4m} \biggr) \biggr\vert ^{q} \biggr)^{1/q}+\alpha \biggl((\alpha +1) \biggl\vert h' \biggl(\frac{3a+b}{4} \biggr) \biggr\vert ^{q} \\ & \quad {} +m\alpha _{1}(\alpha _{1}+\alpha +2) \biggl\vert h' \biggl( \frac{a+b}{2m} \biggr) \biggr\vert ^{q} \biggr)^{1/q} \\ & \quad {} + \biggl((\alpha +1) (\alpha _{1}+1) \biggl\vert h' \biggl( \frac{a+b}{2} \biggr) \biggr\vert ^{q} +m\alpha _{1}( \alpha _{1}+1) \biggl\vert h' \biggl(\frac{a+3b}{4m} \biggr) \biggr\vert ^{q} \biggr)^{1/q} \\ & \quad {} +\alpha \biggl((\alpha +1) \biggl\vert h' \biggl( \frac{a+3b}{4} \biggr) \biggr\vert ^{q}+m\alpha _{1}( \alpha _{1}+\alpha +2) \biggl\vert h' \biggl( \frac{b}{m} \biggr) \biggr\vert ^{q} \biggr)^{1/q} \biggr]. \end{aligned}$$
(1.3)

Theorem 1.2

([26, Theorem 3.2])

Let \(h: [0,\infty )\to \mathbb{R}\) be differentiable on \([0,\infty )\) and \(h'\in L_{1}([a,b])\) for \(0\le a< b\) and \(\alpha >0\). If \(\vert h'\vert ^{q}\) is \((\alpha _{1},m)\)-convex on \([0,\frac{b}{m} ]\) for some \((\alpha _{1},m)\in (0,1]^{2}\) and for \(q>1\) and \(q\ge r\ge 0\), then

$$\begin{aligned} \bigl\vert Q_{\alpha }(a,b) \bigr\vert &\le \frac{b-a}{16} \biggl\{ \biggl(\frac{q-1}{ \alpha (q-r)+q-1} \biggr)^{1-1/q} \biggl[ \frac{1}{\alpha r+\alpha _{1}+1} \bigl\vert h'(a) \bigr\vert ^{q} \\ & \quad {} +\frac{m\alpha _{1}}{(\alpha r+1)(\alpha r+\alpha _{1}+1)} \biggl\vert h' \biggl( \frac{3a+b}{4m} \biggr) \biggr\vert ^{q} \biggr]^{1/q} \\ & \quad {} +\frac{1}{\alpha }B^{1-1/q} \biggl(\frac{2q-r-1}{q-1}, \frac{1}{\alpha } \biggr) \biggl[B \biggl(r+1,\frac{\alpha _{1}+1}{\alpha } \biggr) \biggl\vert h' \biggl(\frac{3a+b}{4} \biggr) \biggr\vert ^{q} \\ & \quad {} +m \biggl(B \biggl(r+1,\frac{1}{\alpha } \biggr) -B \biggl(r+1, \frac{ \alpha _{1}+1}{\alpha } \biggr) \biggr) \biggl\vert h' \biggl( \frac{a+b}{2m} \biggr) \biggr\vert ^{q} \biggr]^{1/q} \\ & \quad {} + \biggl(\frac{q-1}{\alpha (q-r)+q-1} \biggr)^{1-1/q} \biggl[ \frac{1}{ \alpha r+\alpha _{1}+1} \biggl\vert h' \biggl(\frac{a+b}{2} \biggr) \biggr\vert ^{q} \\ & \quad {} +\frac{m\alpha _{1}}{(\alpha r+1)(\alpha r+\alpha _{1}+1)} \biggl\vert h' \biggl( \frac{a+3b}{4m} \biggr) \biggr\vert ^{q} \biggr]^{1/q} \\ & \quad {} +\frac{1}{\alpha }B^{1-1/q} \biggl(\frac{2q-r-1}{q-1}, \frac{1}{\alpha } \biggr) \biggl[B \biggl(r+1,\frac{\alpha _{1}+1}{\alpha } \biggr) \biggl\vert h' \biggl(\frac{a+3b}{4} \biggr) \biggr\vert ^{q} \\ & \quad {} +m \biggl(B \biggl(r+1,\frac{1}{\alpha } \biggr)-B \biggl(r+1, \frac{ \alpha _{1}+1}{\alpha } \biggr) \biggr) \biggl\vert h' \biggl( \frac{b}{m} \biggr) \biggr\vert ^{q} \biggr]^{1/q} \biggr\} , \end{aligned}$$
(1.4)

where \(B(s,t)\) denotes the classical beta function which can be defined [18, 19] by

$$ B(s,t)= \int _{0}^{1}u^{s-1}(1-u)^{t-1} \,\operatorname {d}u,\quad s, t>0. $$

For more information about the Hermite–Hadamard type inequalities for \((\alpha ,m)\)-convex functions, please refer to the papers [2, 3, 6, 15, 21, 26, 28,29,30, 32, 35, 38] and closely related references therein.

2 A review for generalized fractional integral operators

Now we recall some necessary definitions and mathematical preliminaries of the generalized fractional integrals which are defined by Sarikaya and Ertuğral in [24].

Let \(\varphi : [0,\infty )\to [0,\infty )\) satisfy the condition \(\int _{0}^{1} \frac{\varphi (t)}{t}\,\operatorname {d}t<\infty \).

We now define the left-sided and right-sided generalized fractional integral operators \({}_{a^{+}}I_{\varphi }h(t)\) and \({}_{b^{-}}I_{ \varphi }h(t)\) by

$$ {}_{a^{+}}I_{\varphi }h(t)= \int _{a}^{t} \frac{\varphi (t-u)}{t-u}h(u)\,\operatorname {d}u, \quad t>a $$
(2.1)

and

$$ {}_{b^{-}}I_{\varphi }h(t)= \int _{t}^{b} \frac{\varphi (u-t)}{u-t}h(u)\,\operatorname {d}u, \quad t< b. $$
(2.2)

The most important feature of generalized fractional integrals is that they generalize some types of fractional integrals such as the Riemann–Liouville fractional integrals [25, 26, 31], the k-Riemann–Liouville fractional integrals [11, 36], the Katugampola fractional integrals [7, 8], conformable fractional integrals [23, 37], the Hadamard fractional integrals [16], and so on. These important special cases of the integral operators in (2.1) and (2.2) are mentioned below.

  1. 1.

    If we take \(\varphi (u)=u\), the operators in (2.1) and (2.2) reduce to the Riemann integrals

    $$ I_{a^{+}}h(t)= \int _{a}^{t} h(u)\,\operatorname {d}u, \quad t>a \quad \text{and} \quad I_{b^{-}}h(t)= \int _{t}^{b} h(u)\,\operatorname {d}u, \quad t< b. $$
  2. 2.

    If we take \(\varphi (u)=\frac{u^{\alpha }}{\varGamma (\alpha )}\), the operators in (2.1) and (2.2) become the Riemann–Liouville fractional integrals

    $$ I_{a^{+}}h(t)=\frac{1}{\varGamma (\alpha )} \int _{a}^{t} (t-u)^{\alpha -1}h(u)\,\operatorname {d}u, \quad t>a $$

    and

    $$ I_{b^{-}}h(t)=\frac{1}{\varGamma (\alpha )} \int _{t}^{b} (u-t)^{\alpha -1}h(u)\,\operatorname {d}u, \quad t< b. $$
  3. 3.

    If we take \(\varphi (u)=\frac{u^{\alpha /k}}{k\varGamma _{k}(\alpha )}\), the operators in (2.1) and (2.2) are the k-Riemann–Liouville fractional integrals

    $$ I_{a^{+},k}h(t)=\frac{1}{k\varGamma _{k}(\alpha )} \int _{a}^{t} (t-u)^{ \alpha /k-1}h(u)\,\operatorname {d}u, \quad t>a $$

    and

    $$ I_{b^{-},k}h(t)=\frac{1}{k\varGamma _{k}(\alpha )} \int _{t}^{b} (u-t)^{ \alpha /k-1}h(u)\,\operatorname {d}u, \quad t< b, $$

    where

    $$ \varGamma _{k}(\alpha )= \int _{0}^{\infty }u^{\alpha -1}e^{-u^{k}/k}\,\operatorname {d}u, \quad \mathbb{R}(\alpha )>0 $$

    and

    $$ \varGamma _{k}(\alpha )=k^{\alpha /k-1}\varGamma \biggl( \frac{\alpha }{k} \biggr),\quad \mathbb{R}(\alpha )>0, k>0 $$

    are given in [13, 20].

  4. 4.

    If we take \(\varphi (u)=\frac{u}{\alpha }\exp (-\frac{1-\alpha }{ \alpha }u )\), the operators in (2.1) and (2.2) reduce to the right-sided and left-sided fractional integral operators with exponential kernel for \(\alpha \in (0,1)\)

    $$ \mathcal{I}_{a^{+}}^{\alpha }h(t)=\frac{1}{\alpha } \int _{a}^{t} \exp \biggl(-\frac{1-\alpha }{\alpha }(t-u) \biggr)h(u)\,\operatorname {d}u, \quad t>a $$

    and

    $$ \mathcal{I}_{b^{-}}^{\alpha }h(t)=\frac{1}{\alpha } \int _{t}^{b} \exp \biggl(-\frac{1-\alpha }{\alpha }(u-t) \biggr)h(u)\,\operatorname {d}u, \quad t< b $$

    which are defined in [9].

Recently, Sarikaya and Ertuğral [24] established the following trapezoid inequalities for generalized fractional integrals.

Theorem 2.1

([24])

Let \(h: [a,b]\to \mathbb{R}\) be differentiable on \((a,b)\) with \(a< b\). If \(\vert h'\vert \) is convex on \([a,b]\), then

$$\begin{aligned}& \biggl\vert \frac{h(a)+h(b)}{2}-\frac{1}{2\varLambda (1)} \bigl[{}_{a^{+}}I _{\varphi }h(b)+{}_{b^{-}}I_{\varphi }h(a) \bigr] \biggr\vert \\& \quad \le \frac{ \vert h'(a) \vert + \vert h'(b) \vert }{2}\frac{b-a}{\varLambda (1)} \int _{0}^{1} u \bigl\vert \varLambda (1-u)- \varLambda (u) \bigr\vert \,\operatorname {d}u, \end{aligned}$$

where

$$ \varLambda (u)= \int _{0}^{u} \frac{\varphi ((b-a)t)}{t}\,\operatorname {d}t< \infty . $$

Theorem 2.2

([24])

Let \(h: [a,b]\to \mathbb{R}\) be differentiable on \((a,b)\) with \(a< b\). If \(\vert h'\vert ^{q}\) is convex on \([a,b]\) for \(p,q>1\) and \(\frac{1}{p}+ \frac{1}{q}=1\), then

$$\begin{aligned}& \biggl\vert \frac{h(a)+h(b)}{2}-\frac{{}_{a^{+}}I_{\varphi }h(b)+{}_{b ^{-}}I_{\varphi }h(a)}{2\varLambda (1)} \biggr\vert \\& \quad \le \frac{b-a}{2\varLambda (1)} \biggl[\frac{ \vert h'(a) \vert ^{q}+ \vert h'(b) \vert ^{q}}{2} \biggr]^{1/q} \biggl[ \int _{0}^{1} \bigl\vert \varLambda (1-u)- \varLambda (u) \bigr\vert ^{p}\,\operatorname {d}u \biggr]^{1/p}. \end{aligned}$$

In [4], Ertuğral and Sarikaya established the following trapezoid inequalities for generalized fractional integrals.

Theorem 2.3

([4])

Let \(h: [a,b]\to \mathbb{R}\) be absolutely continuous on \(I^{\circ }\) such that \(h'\in L_{1}([a,b])\) with \(a,b\in I^{\circ }\) with \(a< b\). If the mapping \(\vert h'\vert \) is convex on \([a,b]\), then

$$\begin{aligned}& \biggl\vert \frac{\nabla (0)h(b)+\Delta (0)h(a)}{b-a}-\frac{1}{b-a} \bigl[{}_{a^{+}}I_{\varphi }h(b)+{}_{b^{-}}I_{\varphi }h(a) \bigr] \biggr\vert \\& \quad \le \frac{b-x}{b-a} \bigl\vert h'(x) \bigr\vert \int _{0}^{1} \bigl\vert \nabla (u) \bigr\vert u \,\operatorname {d}u+ \frac{x-a}{b-a} \bigl\vert h'(x) \bigr\vert \int _{0}^{1} \bigl\vert \Delta (u) \bigr\vert u \,\operatorname {d}u \\ & \quad\quad{} +\frac{b-x}{b-a} \bigl\vert h'(b) \bigr\vert \int _{0}^{1} \bigl\vert \nabla (u) \bigr\vert (1-u)\,\operatorname {d}u+ \frac{x-a}{b-a} \bigl\vert h'(a) \bigr\vert \int _{0}^{1} \bigl\vert \Delta (u) \bigr\vert (1-u)\,\operatorname {d}u, \end{aligned}$$

where

$$ \Delta (u)= \int _{t}^{1} \frac{\varphi ((x-a)t)}{t}\,\operatorname {d}t< \infty \quad \textit{and}\quad \nabla (u)= \int _{t}^{1} \frac{\varphi ((b-x)t)}{t}\,\operatorname {d}t< \infty . $$

Theorem 2.4

([4])

Let \(h: [a,b]\to \mathbb{R}\) be differentiable on \((a,b)\) with \(a< b\). If \(\vert h'\vert ^{q}\) for \(q>1\) is convex on \([a,b]\), then

$$\begin{aligned}& \biggl\vert \frac{h(a)+h(b)}{2}-\frac{1}{2\varLambda (1)} \bigl[{}_{a^{+}}I _{\varphi }h(b)+{}_{b^{-}}I_{\varphi }h(a) \bigr] \biggr\vert \\ & \quad \le \frac{b-x}{b-a} \biggl[ \int _{0}^{1} \bigl\vert \nabla (u) \bigr\vert ^{p} \,\operatorname {d}u \biggr]^{1/p} \biggl[\frac{ \vert h'(a) \vert ^{q}+ \vert h'(b) \vert ^{q}}{2} \biggr]^{1/q} \\ & \quad\quad{} +\frac{b-x}{b-a} \biggl[ \int _{0}^{1} \bigl\vert \Delta (u) \bigr\vert ^{p} \,\operatorname {d}u \biggr]^{1/p} \biggl[\frac{ \vert h'(a) \vert ^{q}+ \vert h'(b) \vert ^{q}}{2} \biggr]^{1/q}, \end{aligned}$$

where \(\frac{1}{p}+\frac{1}{q}=1\).

Most recently, Mohammed and Sarikayain [12] established some generalized fractional integral inequalities of midpoint and trapezoid types for twice differential functions.

Theorem 2.5

([12])

Let \(h:I\subseteq \mathbb{R}\to \mathbb{R}\) be a twice differentiable function on \(I^{\circ }\) such that \(h''\in L_{1}([a,b])\) with \(a,b\in I^{\circ }\) and \(a< b\). If the function \(\vert h''\vert \) is convex on \([a,b]\), then

$$\begin{aligned}& \biggl\vert \bigl[{}_{ (\frac{a+b}{2} )^{+}}I_{\varphi }h(b)+ {}_{ (\frac{a+b}{2} )^{-}}I_{\varphi }h(a) \bigr] -2\nabla (1)h \biggl(\frac{a+b}{2} \biggr) \biggr\vert \\ & \quad \le \frac{(b-a)^{2}}{4} \bigl( \bigl\vert h''(a) \bigr\vert + \bigl\vert h''(b) \bigr\vert \bigr) \int _{0}^{1} \bigl\vert \Delta (t) \bigr\vert \,\operatorname {d}t, \end{aligned}$$

where

$$ \Delta (t)= \int _{0}^{t} \nabla (u)\,\operatorname {d}u< \infty \quad \textit{and} \quad \nabla (u)= \int _{0}^{u}\frac{\varphi ( (\frac{b-a}{2} )s )}{s}\,\operatorname {d}s< \infty . $$

Theorem 2.6

([12])

Let \(h:I\subseteq \mathbb{R}\to \mathbb{R}\) be twice differentiable on \(I^{\circ }\) such that \(h''\in L_{1}([a,b])\) with \(a,b\in I^{\circ }\) and \(a< b\). If \(\vert h''\vert ^{q}\) for \(q>1\) is convex on \([a,b]\), then

$$\begin{aligned} & \biggl\vert \bigl[{}_{ (\frac{a+b}{2} )^{+}}I_{\varphi }h(b)+ {}_{ (\frac{a+b}{2} )^{-}}I_{\varphi }h(a) \bigr] -2\nabla (1)h \biggl( \frac{a+b}{2} \biggr) \biggr\vert \\ &\quad \le \frac{(b-a)^{2}}{4} \biggl( \int _{0}^{1} \bigl\vert \Delta (u) \bigr\vert ^{p}\,\operatorname {d}u \biggr)^{1/p} \biggl\{ \biggl( \frac{ \vert h''(a) \vert ^{q}+3 \vert h''(b) \vert ^{q}}{4} \biggr)^{1/q} \\ &\quad \quad {} + \biggl(\frac{3 \vert h''(a) \vert ^{q}+ \vert h''(b) \vert ^{q}}{4} \biggr)^{1/q} \biggr\} \\ &\quad \le \frac{(b-a)^{2}}{2^{2/q}} \biggl( \int _{0}^{1} \bigl\vert \Delta (u) \bigr\vert ^{p}\,\operatorname {d}u \biggr)^{1/p} \bigl( \bigl\vert h''(a) \bigr\vert + \bigl\vert h''(b) \bigr\vert \bigr), \end{aligned}$$

where \(\frac{1}{p}+\frac{1}{q}=1\).

Theorem 2.7

([12])

Let \(h:I\subseteq \mathbb{R}\to \mathbb{R}\) be twice differentiable on \(I^{\circ }\) such that \(h''\in L_{1}([a,b])\) with \(a,b\in I^{\circ }\) and \(a< b\). If \(\vert h''\vert \) is convex on \([a,b]\), then

$$\begin{aligned}& \biggl\vert \frac{h(a)+h(b)}{2}-\frac{1}{2\varPhi (0)} \bigl[{}_{ ( \frac{a+b}{2} )^{+}}I_{\varphi }h(b) +{}_{ (\frac{a+b}{2} )^{-}}I_{\varphi }h(a) \bigr] \biggr\vert \\& \quad \le \frac{(b-a)^{2}}{8\varPhi (0)} \biggl[ \int _{0}^{1} \bigl\vert \Delta (u) \bigr\vert \,\operatorname {d}u \bigl( \bigl\vert h''(a) \bigr\vert + \bigl\vert h''(b) \bigr\vert \bigr) \biggr]. \end{aligned}$$

Theorem 2.8

([12])

Let \(h:I\subseteq \mathbb{R}\to \mathbb{R}\) be twice differentiable on \(I^{\circ }\) such that \(h''\in L_{1}([a,b])\) with \(a,b\in I^{\circ }\) and \(a< b\). If \(\vert h''\vert ^{q}\) for \(q>1\) is convex on \([a,b]\), then

$$\begin{aligned} & \biggl\vert \frac{h(a)+h(b)}{2}-\frac{1}{2\varPhi (0)} \bigl[{}_{ (\frac{a+b}{2} )^{+}}I_{\varphi }h(b) +{}_{ ( \frac{a+b}{2} )^{-}}I_{\varphi }h(a) \bigr] \biggr\vert \\ &\quad \le \frac{(b-a)^{2}}{8\varPhi (0)} \biggl( \int _{0}^{1} \bigl\vert \Delta (u) \bigr\vert ^{p}\,\operatorname {d}u \biggr)^{1/p} \biggl\{ \biggl( \frac{ \vert h''(a) \vert ^{q}+3 \vert h''(b) \vert ^{q}}{4} \biggr)^{1/q} \\ &\quad \quad {} + \biggl(\frac{3 \vert h''(a) \vert ^{q}+ \vert h''(b) \vert ^{q}}{4} \biggr)^{1/q} \biggr\} \\ &\quad \le \frac{(b-a)^{2}}{2^{3/q}\varPhi (0)} \biggl( \int _{0}^{1} \bigl\vert \Delta (u) \bigr\vert ^{p}\,\operatorname {d}u \biggr)^{1/p} \bigl( \bigl\vert h''(a) \bigr\vert + \bigl\vert h''(b) \bigr\vert \bigr), \end{aligned}$$

where \(\frac{1}{p}+\frac{1}{q}=1\).

3 A generalized fractional integral identity

Before stating and proving our main results, we formulate the following important fractional integral identity.

Lemma 3.1

Let \(f:[a,b]\to \mathbb{R}\) be a differentiable function on \((a,b)\) with \(a< b\) such that \(f\in L_{1}([a,b])\). Then

$$\begin{aligned}& \frac{f(a)+f(b)}{2}+2f \biggl(\frac{a+b}{2} \biggr)- \frac{2}{\Delta (1)} \biggl[{}_{a^{+}}I_{\varphi }f \biggl( \frac{3a+b}{4} \biggr) \\& \quad\quad {} +{}_{ (\frac{3a+b}{4} )^{+}}I_{\varphi }f \biggl( \frac{a+b}{2} \biggr) +{}_{ (\frac{a+b}{2} )^{+}}I_{\varphi }f \biggl(\frac{a+3b}{4} \biggr) +{}_{ (\frac{a+3b}{4} )^{-}}I _{\varphi }f(b) \biggr] \\& \quad =\frac{b-a}{8\Delta (1)} \biggl[ \int _{0}^{1}\nabla (t)f' \biggl( \frac{3a+b}{4}t+\frac{a+b}{2}(1-t) \biggr)\,\operatorname {d}t \\& \quad \quad {} - \int _{0} ^{1}\Delta (t)f' \biggl(at+ \frac{3a+b}{4}(1-t) \biggr)\,\operatorname {d}t + \int _{0}^{1}\nabla (t)f' \biggl( \frac{a+3b}{4}t+b(1-t) \biggr)\,\operatorname {d}t \\& \quad \quad {} - \int _{0}^{1}\Delta (t)f' \biggl( \frac{a+b}{2}t+\frac{a+3b}{4}(1-t) \biggr)\,\operatorname {d}t \biggr], \end{aligned}$$
(3.1)

where

$$ \Delta (t)= \int _{0}^{t}\frac{\varphi ( (\frac{b-a}{4} )u )}{u}\,\operatorname {d}u< \infty \quad \textit{and}\quad \nabla (t)= \int _{t} ^{1}\frac{\varphi ( (\frac{b-a}{4} )u )}{u}\,\operatorname {d}u< \infty . $$

Proof

Integrating by parts gives

$$\begin{aligned} I_{1}&=\frac{b-a}{8\Delta (1)} \int _{0}^{1}\nabla (t)f' \biggl( \frac{3a+b}{4}t+\frac{a+b}{2}(1-t) \biggr)\,\operatorname {d}t =\frac{1}{2 [ \Delta (1)+\nabla (0) ]} \\ &\quad{} \times \biggl[\nabla (0)f \biggl(\frac{a+b}{2} \biggr) - \int _{0}^{1} \frac{ \varphi ( (\frac{b-a}{4} )t )}{t} f \biggl( \frac{3a+b}{4}t+\frac{a+b}{2}(1-t) \biggr)\,\operatorname {d}t \biggr]. \end{aligned}$$

Changing the variable \(x=\frac{3a+b}{4}t+\frac{a+b}{2}(1-t)\) yields

$$\begin{aligned} I_{1} &=\frac{1}{2\Delta (1)} \biggl[\nabla (0)f \biggl(\frac{a+b}{2} \biggr) + \int _{\frac{a+b}{2}}^{\frac{3a+b}{4}}\frac{\varphi ( \frac{a+b}{2}-x )}{\frac{a+b}{2}-x}f(x)\,\operatorname {d}x \biggr] \\ &=\frac{1}{2\Delta (1)} \biggl[\nabla (0)f \biggl(\frac{a+b}{2} \biggr) -{}_{ (\frac{3a+b}{4} )^{+}}I_{\varphi }f \biggl(\frac{a+b}{2} \biggr) \biggr]. \end{aligned}$$

Similarly, we obtain

$$\begin{aligned}& \begin{aligned} I_{2} &=-\frac{b-a}{8\Delta (1)} \int _{0}^{1}\Delta (t)f' \biggl(at+ \frac{3a+b}{4}(1-t) \biggr)\,\operatorname {d}t \\ &=\frac{1}{2\Delta (1)} \biggl[\Delta (1)f(a)- \int _{0}^{1}\frac{ \varphi ( (\frac{b-a}{4} )t )}{t} f \biggl(at+ \frac{3a+b}{4}(1-t) \biggr)\,\operatorname {d}t \biggr] \\ &=\frac{1}{2\Delta (1)} \biggl[\Delta (1)f(a)-{}_{a^{+}}I_{\varphi }f \biggl(\frac{3a+b}{4} \biggr) \biggr], \end{aligned} \\& \begin{aligned} I_{3} &=\frac{b-a}{8\Delta (1)} \int _{0}^{1}\nabla (t)f' \biggl( \frac{a+3b}{4}t+b(1-t) \biggr)\,\operatorname {d}t \\ &=\frac{1}{2\Delta (1)} \biggl[\nabla (0)f(b)- \int _{0}^{1}\frac{ \varphi ( (\frac{b-a}{4} )t )}{t} f \biggl( \frac{a+3b}{4}t+b(1-t) \biggr)\,\operatorname {d}t \biggr] \\ &=\frac{1}{2\Delta (1)} \bigl[\nabla (0)f(b)-{}_{ ( \frac{a+3b}{4} )^{+}}I_{\varphi }f(b) \bigr] \end{aligned} \end{aligned}$$

and

$$\begin{aligned} I_{4} &=-\frac{b-a}{8\Delta (1)} \int _{0}^{1}\Delta (t)f' \biggl( \frac{3a+b}{4}t+\frac{a+b}{2}(1-t) \biggr)\,\operatorname {d}t \\ &=\frac{1}{2\Delta (1)} \biggl[\Delta (1)f \biggl(\frac{a+b}{2} \biggr)- \int _{0}^{1}\frac{\varphi ( (\frac{b-a}{4} )t )}{t} f \biggl( \frac{3a+b}{4}t+\frac{a+b}{2}(1-t) \biggr)\,\operatorname {d}t \biggr] \\ &=\frac{1}{2\Delta (1)} \biggl[\Delta (1)f \biggl(\frac{a+b}{2} \biggr) -{}_{ (\frac{a+b}{2} )^{+}}I_{\varphi }f \biggl(\frac{a+3b}{4} \biggr) \biggr], \end{aligned}$$

where we used the fact that \(\Delta (1)=\nabla (0)\). Adding \(I_{1}\), \(I_{2}\), \(I_{3}\), and \(I_{4}\) results in identity (3.1). The proof is thus completed. □

Remark 3.1

Since \(\Delta (1)=\nabla (0)\), we can write identity (3.1) in Lemma 3.1 as

$$\begin{aligned} & \frac{f(a)+f(b)}{2}+2f \biggl(\frac{a+b}{2} \biggr)-\frac{2}{ \nabla (0)} \biggl[{}_{a^{+}}I_{\varphi }f \biggl(\frac{3a+b}{4} \biggr) \\ &\quad \quad {} +{}_{ (\frac{3a+b}{4} )^{+}}I_{\varphi }f \biggl( \frac{a+b}{2} \biggr) +{}_{ (\frac{a+b}{2} )^{+}}I_{\varphi }f \biggl(\frac{a+3b}{4} \biggr) +{}_{ (\frac{a+3b}{4} )^{-}}I _{\varphi }f(b) \biggr] \\ &\quad =\frac{b-a}{8\nabla (0)} \biggl[ \int _{0}^{1}\nabla (t)f' \biggl( \frac{3a+b}{4}t+\frac{a+b}{2}(1-t) \biggr)\,\operatorname {d}t \\ &\quad\quad {} - \int _{0}^{1}\Delta (t)f' \biggl(at+ \frac{3a+b}{4}(1-t) \biggr)\,\operatorname {d}t + \int _{0}^{1}\nabla (t)f' \biggl( \frac{a+3b}{4}t \\ &\quad\quad {} +b(1-t) \biggr)\,\operatorname {d}t- \int _{0}^{1}\Delta (t)f' \biggl( \frac{a+b}{2}t+\frac{a+3b}{4}(1-t) \biggr)\,\operatorname {d}t \biggr]. \end{aligned}$$

Remark 3.2

Under assumptions of Lemma 3.1, if \(\varphi (t)=t\), then identity (3.1) reduces to

$$\begin{aligned} & \frac{1}{2} \biggl[\frac{f(a)+f(b)}{2}+f \biggl(\frac{a+b}{2} \biggr) \biggr]-\frac{1}{b-a} \int _{a}^{b} f(x)\,\operatorname {d}x \\ &\quad =\frac{b-a}{16} \biggl[ \int _{0}^{1}(1-t)f' \biggl( \frac{3a+b}{4}t+ \frac{a+b}{2}(1-t) \biggr)\,\operatorname {d}t \\ &\quad \quad {} - \int _{0}^{1}tf' \biggl(at+ \frac{3a+b}{4}(1-t) \biggr)\,\operatorname {d}t + \int _{0}^{1}(1-t)f' \biggl( \frac{a+3b}{4}t \\ &\quad \quad {} +b(1-t) \biggr)\,\operatorname {d}t- \int _{0}^{1}tf' \biggl( \frac{a+b}{2}t+ \frac{a+3b}{4}(1-t) \biggr)\,\operatorname {d}t \biggr], \end{aligned}$$

which has been proved in [26].

Remark 3.3

Under assumptions of Lemma 3.1, if \(\varphi (t)=\frac{t^{ \alpha }}{\varGamma (\alpha )}\), then identity (3.1) reduces to identity (1.2).

Remark 3.4

Under assumptions of Lemma 3.1, if \(\varphi (t)=\frac{t^{ \alpha /k}}{{k\varGamma _{k}(\alpha )}}\), then

$$\begin{aligned} &\frac{1}{2} \biggl[\frac{f(a)+f(b)}{2}+f \biggl(\frac{a+b}{2} \biggr) \biggr] -\frac{4^{\alpha /k-1}\varGamma _{k}(\alpha +1)}{(b-a)^{ \alpha /k}} \biggl[I^{\alpha }_{a^{+},k}f \biggl( \frac{3a+b}{4} \biggr) \\ &\quad \quad {} +I^{\alpha }_{ (\frac{3a+b}{4} )^{+},k}f \biggl( \frac{a+b}{2} \biggr) +I^{\alpha }_{ (\frac{a+b}{2} )^{+},k}f \biggl(\frac{a+3b}{4} \biggr) +I^{\alpha }_{ (\frac{a+3b}{4} )^{+},k}f(b) \biggr] \\ &\quad =\frac{b-a}{16} \biggl[ \int _{0}^{1} \bigl(1-t^{\alpha /k} \bigr)f' \biggl(\frac{3a+b}{4}t+\frac{a+b}{2}(1-t) \biggr) \,\operatorname {d}t \\ &\quad \quad {} - \int _{0}^{1} t^{\alpha /k} f' \biggl(at+\frac{3a+b}{4}(1-t) \biggr)\,\operatorname {d}t \\ &\quad \quad {} + \int _{0}^{1} \bigl(1-t^{\alpha /k} \bigr)f' \biggl( \frac{a+3b}{4}t+b(1-t) \biggr)\,\operatorname {d}t \\ &\quad \quad {} - \int _{0}^{1} t^{\alpha /k}f' \biggl( \frac{a+b}{2}t+ \frac{a+3b}{4}(1-t) \biggr)\,\operatorname {d}t \biggr]. \end{aligned}$$

Remark 3.5

Under assumptions of Lemma 3.1, applying \(\varphi (t)=\frac{t}{ \alpha }\exp (-\frac{1-\alpha }{\alpha }t )\) gives

$$\begin{aligned} &\frac{f(a)+f(b)}{2}+2f \biggl(\frac{a+b}{2} \biggr) - \frac{2(1- \alpha )}{1-\exp (-A)} \biggl[\mathcal{I}^{\alpha }{}_{a^{+}}f \biggl( \frac{3a+b}{4} \biggr) \\ &\quad \quad {} +\mathcal{I}^{\alpha }_{ (\frac{3a+b}{4} )^{+}}f \biggl( \frac{a+b}{2} \biggr) +\mathcal{I}^{\alpha }{}_{ ( \frac{a+b}{2} )^{+}}f \biggl( \frac{a+3b}{4} \biggr) +\mathcal{I} ^{\alpha }_{ (\frac{a+3b}{4} )^{-}}f(b) \biggr] \\ &\quad =\frac{b-a}{8[1-\exp (-A)]} \biggl[ \int _{0}^{1} \bigl[\exp (-At)-\exp (-A) \bigr]f' \biggl(\frac{3a+b}{4}t \\ &\quad \quad {} +\frac{a+b}{2}(1-t) \biggr)\,\operatorname {d}t - \int _{0}^{1} \bigl[1-\exp (-At) \bigr](t)f' \biggl(at+\frac{3a+b}{4}(1-t) \biggr)\,\operatorname {d}t \\ &\quad \quad {} + \int _{0}^{1} \bigl[\exp (-At)-\exp (-A) \bigr]f' \biggl(\frac{a+3b}{4}t+b(1-t) \biggr)\,\operatorname {d}t \\ &\quad \quad {} - \int _{0}^{1} \bigl[1-\exp (-At) \bigr]f' \biggl(\frac{a+b}{2}t+ \frac{a+3b}{4}(1-t) \biggr) \,\operatorname {d}t \biggr] \end{aligned}$$

for \(A=\frac{1-\alpha }{\alpha }\frac{b-a}{2}\).

4 Generalized fractional integral inequalities of Hermite–Hadamard type

Now we are in a position to state and prove our main results.

Theorem 4.1

Let \(f:[a,b]\to \mathbb{R}\) be a differentiable function on \((a,b)\) and \(f'\in L_{1}([a,b])\) for \(0\le a< b\) and \(\alpha >0\). If the mapping \(\vert f'\vert ^{q}\) for \(q\ge 1\) is \((\alpha _{1},m)\)-convex on \([0,\frac{b}{m} ]\) for some \((\alpha _{1},m)\in (0,1]^{2}\), then

$$\begin{aligned} & \biggl\vert \frac{f(a)+f(b)}{2}+2f \biggl(\frac{a+b}{2} \biggr) - \frac{2}{ \Delta (1)} \biggl[{}_{a^{+}}I_{\varphi }f \biggl( \frac{3a+b}{4} \biggr) \\ &\quad \quad {} +{}_{ (\frac{3a+b}{4} )^{+}}I_{\varphi }f \biggl( \frac{a+b}{2} \biggr) +{}_{ (\frac{a+b}{2} )^{+}}I_{\varphi }f \biggl(\frac{a+3b}{4} \biggr) +{}_{ (\frac{a+3b}{4} )^{-}}I _{\varphi }f(b) \biggr] \biggr\vert \\ &\quad \le \frac{b-a}{8\Delta (1)} \biggl[ \biggl( \int _{0}^{1} \bigl\vert \Delta (t) \bigr\vert \,\operatorname {d}t \biggr)^{1-1/q} \biggl(A_{1} \bigl\vert f'(a) \bigr\vert ^{q}+A_{2} \biggl\vert f' \biggl( \frac{3a+b}{4m} \biggr) \biggr\vert ^{q} \biggr)^{1/q} \\ &\quad \quad {} + \biggl( \int _{0}^{1} \bigl\vert \nabla (t) \bigr\vert \,\operatorname {d}t \biggr)^{1-1/q} \biggl(B_{1} \biggl\vert f' \biggl(\frac{3a+b}{4} \biggr) \biggr\vert ^{q} +B_{2} \biggl\vert f' \biggl(\frac{a+b}{4m} \biggr) \biggr\vert ^{q} \biggr)^{1/q} \\ &\quad \quad {} + \biggl( \int _{0}^{1} \bigl\vert \Delta (t) \bigr\vert \,\operatorname {d}t \biggr)^{1-1/q} \biggl(A_{1} \biggl\vert f' \biggl(\frac{a+b}{4} \biggr) \biggr\vert ^{q} +A_{2} \biggl\vert f' \biggl(\frac{a+3b}{4m} \biggr) \biggr\vert ^{q} \biggr)^{1/q} \\ &\quad \quad {} + \biggl( \int _{0}^{1} \bigl\vert \nabla (t) \bigr\vert \,\operatorname {d}t \biggr)^{1-1/q} \biggl(B_{1} \biggl\vert f' \biggl(\frac{a+3b}{4} \biggr) \biggr\vert ^{q}+B_{2} \biggl\vert f' \biggl( \frac{b}{m} \biggr) \biggr\vert ^{q} \biggr)^{1/q} \biggr], \end{aligned}$$

where the constants \(A_{1}\), \(A_{2}\), \(B_{1}\), and \(B_{2}\) are defined by

$$\begin{aligned}& A_{1} = \int _{0}^{1} t^{\alpha _{1}} \bigl\vert \Delta (t) \bigr\vert \,\operatorname {d}t, \qquad A_{2}= \int _{0}^{1} m \bigl(1-t^{\alpha _{1}} \bigr) \bigl\vert \Delta (t) \bigr\vert \,\operatorname {d}t, \\& B_{1} = \int _{0}^{1} t^{\alpha _{1}} \bigl\vert \nabla (t) \bigr\vert \,\operatorname {d}t, \qquad B_{2}= \int _{0}^{1} m \bigl(1-t^{\alpha _{1}} \bigr) \bigl\vert \nabla (t) \bigr\vert \,\operatorname {d}t. \end{aligned}$$

Proof

Using Lemma 3.1, the well-known power mean inequality, and the \((\alpha _{1},m)\)-convexity of \(\vert f'\vert ^{q}\) on \([0,\frac{b}{m} ]\) gives

$$\begin{aligned}& \biggl\vert \frac{f(a)+f(b)}{2}+2f \biggl( \frac{a+b}{2} \biggr) -\frac{2}{ \Delta (1)} \biggl[{}_{a^{+}}I_{\varphi }f \biggl( \frac{3a+b}{4} \biggr) \\& \quad \quad {} +{}_{ (\frac{3a+b}{4} )^{+}}I_{\varphi }f \biggl( \frac{a+b}{2} \biggr) +{}_{ (\frac{a+b}{2} )^{+}}I_{\varphi }f \biggl(\frac{a+3b}{4} \biggr) +{}_{ (\frac{a+3b}{4} )^{-}}I _{\varphi }f(b) \biggr] \biggr\vert \\& \quad \le \frac{b-a}{8\Delta (1)} \biggl[ \int _{0}^{1} \bigl\vert \Delta (t) \bigr\vert \biggl\vert f' \biggl(at+\frac{3a+b}{4}(1-t) \biggr) \biggr\vert \,\operatorname {d}t \\& \quad\quad {} + \int _{0}^{1} \bigl\vert \nabla (t) \bigr\vert \biggl\vert f' \biggl(\frac{3a+b}{4}t+\frac{a+b}{2}(1-t) \biggr) \biggr\vert \,\operatorname {d}t \\& \quad \quad {} + \int _{0}^{1} \bigl\vert \Delta (t) \bigr\vert \biggl\vert f' \biggl(\frac{a+b}{2}t+\frac{a+3b}{4}(1-t) \biggr) \biggr\vert \,\operatorname {d}t \\& \quad \quad {} + \int _{0}^{1} \bigl\vert \nabla (t) \bigr\vert \biggl\vert f' \biggl(\frac{a+3b}{4}t+b(1-t) \biggr) \biggr\vert \,\operatorname {d}t \biggr] \\& \quad \le \frac{b-a}{8\Delta (1)} \biggl\{ \biggl( \int _{0}^{1} \bigl\vert \Delta (t) \bigr\vert \,\operatorname {d}t \biggr)^{1-1/q} \biggl[ \int _{0}^{1} \bigl\vert \Delta (t) \bigr\vert \biggl(t^{\alpha _{1}} \bigl\vert f'(a) \bigr\vert ^{q} \\& \quad \quad {} +m \bigl(1-t^{\alpha _{1}} \bigr) \biggl\vert f' \biggl( \frac{3a+b}{4m} \biggr) \biggr\vert ^{q} \biggr)\,\operatorname {d}t \biggr]^{1/q} \\& \quad \quad {} + \biggl( \int _{0}^{1} \bigl\vert \nabla (t) \bigr\vert \,\operatorname {d}t \biggr)^{1-1/q} \biggl[ \int _{0}^{1} \bigl\vert \nabla (t) \bigr\vert \biggl(t^{\alpha _{1}} \biggl\vert f' \biggl( \frac{3a+b}{4} \biggr) \biggr\vert ^{q} \\& \quad \quad {} +m \bigl(1-t^{\alpha _{1}} \bigr) \biggl\vert f' \biggl( \frac{a+b}{2m} \biggr) \biggr\vert ^{q} \biggr)\,\operatorname {d}t \biggr]^{1/q} \\& \quad \quad {} + \biggl( \int _{0}^{1} \bigl\vert \Delta (t) \bigr\vert \,\operatorname {d}t \biggr)^{1-1/q} \biggl[ \int _{0}^{1} \bigl\vert \Delta (t) \bigr\vert \biggl(t^{\alpha _{1}} \biggl\vert f' \biggl(\frac{a+b}{2} \biggr) \biggr\vert ^{q} \\& \quad \quad {} +m \bigl(1-t ^{\alpha _{1}} \bigr) \biggl\vert f' \biggl( \frac{a+3b}{4m} \biggr) \biggr\vert ^{q} \biggr)\,\operatorname {d}t \biggr]^{1/q} \\& \quad\quad {} + \biggl( \int _{0}^{1} \bigl\vert \nabla (t) \bigr\vert \,\operatorname {d}t \biggr)^{1-1/q} \biggl[ \int _{0}^{1} \bigl\vert \nabla (t) \bigr\vert \biggl(t^{\alpha _{1}} \biggl\vert f' \biggl(\frac{a+3b}{4} \biggr) \biggr\vert ^{q} \\& \quad \quad {} +m \bigl(1-t ^{\alpha _{1}} \bigr) \biggl\vert f' \biggl( \frac{b}{m} \biggr) \biggr\vert ^{q} \biggr)\,\operatorname {d}t \biggr]^{1/q} \biggr\} \\& \quad =\frac{b-a}{8\Delta (1)} \biggl[ \biggl( \int _{0}^{1} \bigl\vert \Delta (t) \bigr\vert \,\operatorname {d}t \biggr)^{1-1/q} \biggl(A_{1} \bigl\vert f'(a) \bigr\vert ^{q}+A_{2} \biggl\vert f' \biggl( \frac{3a+b}{4m} \biggr) \biggr\vert ^{q} \biggr)^{1/q} \\& \quad \quad {} + \biggl( \int _{0}^{1} \bigl\vert \nabla (t) \bigr\vert \,\operatorname {d}t \biggr)^{1-1/q} \biggl(B_{1} \biggl\vert f' \biggl(\frac{3a+b}{4} \biggr) \biggr\vert ^{q}+B_{2} \biggl\vert f' \biggl( \frac{a+b}{4m} \biggr) \biggr\vert ^{q} \biggr)^{1/q} \\& \quad \quad {} + \biggl( \int _{0}^{1} \bigl\vert \Delta (t) \bigr\vert \,\operatorname {d}t \biggr)^{1-1/q} \biggl(A_{1} \biggl\vert f' \biggl(\frac{a+b}{4} \biggr) \biggr\vert ^{q}+A_{2} \biggl\vert f' \biggl( \frac{a+3b}{4m} \biggr) \biggr\vert ^{q} \biggr)^{1/q} \\& \quad \quad {} + \biggl( \int _{0}^{1} \bigl\vert \nabla (t) \bigr\vert \,\operatorname {d}t \biggr)^{1-1/q} \biggl(B_{1} \biggl\vert f' \biggl(\frac{a+3b}{4} \biggr) \biggr\vert ^{q}+B_{2} \biggl\vert f' \biggl( \frac{b}{m} \biggr) \biggr\vert ^{q} \biggr)^{1/q} \biggr]. \end{aligned}$$

This completes the proof. □

Remark 4.1

Under assumptions of Theorem 4.1, if \(\varphi (t)=t\) and \(m=\alpha _{1}=1\), then

$$\begin{aligned}& \biggl\vert \frac{1}{2} \biggl[\frac{f(a)+f(b)}{2}+f \biggl( \frac{a+b}{2} \biggr) \biggr]-\frac{1}{b-a} \int _{a}^{b} f(x)\,\operatorname {d}x \biggr\vert \\& \quad \le \frac{b-a}{32} \biggl(\frac{1}{3} \biggr)^{1/q} \biggl[ \biggl(2 \bigl\vert f'(a) \bigr\vert ^{q}+ \biggl\vert f' \biggl(\frac{3a+b}{4} \biggr) \biggr\vert ^{q} \biggr)^{1/q} \\& \quad\quad {} + \biggl( \biggl\vert f' \biggl(\frac{3a+b}{4} \biggr) \biggr\vert ^{q}+2 \biggl\vert f' \biggl( \frac{a+b}{2} \biggr) \biggr\vert ^{q} \biggr)^{1/q} \\& \quad\quad {} + \biggl(2 \biggl\vert f' \biggl(\frac{a+b}{2} \biggr) \biggr\vert ^{q}+ \biggl\vert f' \biggl( \frac{a+3b}{4} \biggr) \biggr\vert ^{q} \biggr)^{1/q} + \biggl( \biggl\vert f' \biggl(\frac{a+3b}{4} \biggr) \biggr\vert ^{q}+2 \bigl\vert f'(b) \bigr\vert ^{q} \biggr)^{1/q} \biggr], \end{aligned}$$

which was proved in [26].

Remark 4.2

Under assumptions of Theorem 4.1, if \(\varphi (t)=\frac{t^{ \alpha }}{\varGamma (\alpha )}\), then the inequality in Theorem 4.1 reduces to inequality (1.3).

Corollary 4.1

Under assumptions of Theorem 4.1, if \(\varphi (t)=\frac{t^{ \alpha /k}}{{k\varGamma _{k}(\alpha )}}\), then

$$\begin{aligned} & \biggl\vert \frac{1}{2} \biggl[\frac{f(a)+f(b)}{2}+f \biggl( \frac{a+b}{2} \biggr) \biggr] -\frac{4^{\alpha /k-1}\varGamma _{k}(\alpha +1)}{(b-a)^{ \alpha /k}} \biggl[I^{\alpha }_{a^{+},k}f \biggl(\frac{3a+b}{4} \biggr) \\ &\quad \quad {} +I^{\alpha }_{ (\frac{3a+b}{4} )^{+},k}f \biggl( \frac{a+b}{2} \biggr) +I^{\alpha }_{ (\frac{a+b}{2} )^{+},k}f \biggl(\frac{a+3b}{4} \biggr) +I^{\alpha }_{ (\frac{a+3b}{4} )^{+},k}f(b) \biggr] \biggr\vert \\ &\quad \le \frac{b-a}{16 (\frac{\alpha }{k}+1 )} \biggl[\frac{1}{( \alpha _{1}+1) (\frac{\alpha }{k}+\alpha _{1}+1 )} \biggr]^{1/q} \biggl\{ \biggl[ \biggl(\frac{\alpha }{k}+1 \biggr) (\alpha _{1}+1) \bigl\vert f'(a) \bigr\vert ^{q} \\ &\quad \quad {} +m\alpha _{1}(\alpha _{1}+1) \biggl\vert f' \biggl( \frac{3a+b}{4m} \biggr) \biggr\vert ^{q} \biggr]^{1/q} + \frac{\alpha }{k} \biggl[ \biggl(\frac{\alpha }{k}+1 \biggr) \biggl\vert f' \biggl(\frac{3a+b}{4} \biggr) \biggr\vert ^{q} \\ &\quad \quad {} +m\alpha _{1} \biggl(\alpha _{1}+ \frac{\alpha }{k}+2 \biggr) \biggl\vert f' \biggl( \frac{a+b}{2m} \biggr) \biggr\vert ^{q} \biggr]^{1/q} + \biggl[ \biggl( \frac{\alpha }{k}+1 \biggr) (\alpha _{1}+1) \\ &\quad \quad {} \times \biggl\vert f' \biggl(\frac{a+b}{2} \biggr) \biggr\vert ^{q}+m\alpha _{1}(\alpha _{1}+1) \biggl\vert f' \biggl(\frac{a+3b}{4m} \biggr) \biggr\vert ^{q} \biggr]^{1/q} \\ &\quad \quad {} +\alpha \biggl[ \biggl(\frac{\alpha }{k}+1 \biggr) \biggl\vert f' \biggl(\frac{a+b}{2} \biggr) \biggr\vert ^{q} +m \alpha _{1} \biggl( \alpha _{1}+ \frac{\alpha }{k}+2 \biggr) \biggl\vert f' \biggl( \frac{b}{m} \biggr) \biggr\vert ^{q} \biggr]^{1/q} \biggr\} . \end{aligned}$$

Corollary 4.2

Under assumptions of Theorem 4.1, if \(\alpha _{1}=1\) and \(\varphi (t)=\frac{t}{\alpha }\exp (-\frac{1-\alpha }{\alpha }t )\), then

$$\begin{aligned} & \biggl\vert \frac{f(a)+f(b)}{2}+2f \biggl(\frac{a+b}{2} \biggr) - \frac{2(1- \alpha )}{1-\exp (-A)} \biggl[\mathcal{I}^{\alpha }_{a^{+}}f \biggl( \frac{3a+b}{4} \biggr) \\ &\quad \quad {} +\mathcal{I}^{\alpha }_{ (\frac{3a+b}{4} )^{+}}f \biggl( \frac{a+b}{2} \biggr) +\mathcal{I}^{\alpha }_{ ( \frac{a+b}{2} )^{+}}f \biggl( \frac{a+3b}{4} \biggr) +\mathcal{I} ^{\alpha }_{ (\frac{a+3b}{4} )^{-}}f(b) \biggr] \biggr\vert \\ &\quad \le \frac{b-a}{8[1-\exp (-A)]} \biggl[ \biggl(\frac{A+\exp (-A)-1}{A} \biggr)^{1-1/q} \biggl(A_{3} \bigl\vert f'(a) \bigr\vert ^{q} \\ &\quad \quad {} +A_{4} \biggl\vert f' \biggl( \frac{3a+b}{4m} \biggr) \biggr\vert ^{q} \biggr)^{1/q} + \biggl( \frac{A\exp (-A)+\exp (-A)-1}{A} \biggr)^{1-1/q} \\ &\quad \quad {} \times \biggl(B_{3} \biggl\vert f' \biggl( \frac{3a+b}{4} \biggr) \biggr\vert ^{q}+B_{4} \biggl\vert f' \biggl(\frac{a+b}{4m} \biggr) \biggr\vert ^{q} \biggr)^{1/q} \\ &\quad \quad {} + \biggl(\frac{A+\exp (-A)-1}{A} \biggr)^{1-1/q} \biggl(A_{3} \biggl\vert f' \biggl(\frac{a+b}{4} \biggr) \biggr\vert ^{q} +A_{4} \biggl\vert f' \biggl( \frac{a+3b}{4m} \biggr) \biggr\vert ^{q} \biggr)^{1/q} \\ &\quad \quad {} + \biggl(\frac{A\exp (-A)+\exp (-A)-1}{A} \biggr)^{1-1/q} \\ &\quad \quad {} \times \biggl(B_{3} \biggl\vert f' \biggl( \frac{a+3b}{4} \biggr) \biggr\vert ^{q}+B_{4} \biggl\vert f' \biggl(\frac{b}{m} \biggr) \biggr\vert ^{q} \biggr)^{1/q} \biggr], \end{aligned}$$

where

$$\begin{aligned}& A=\frac{1-\alpha }{\alpha }\frac{b-a}{2}, \quad \quad A_{3}= \frac{A^{2}+2A \exp (-A)+2\exp (-A)-2}{2A^{2}}, \\& A_{4}=\frac{m(A+\exp (-A)-1)}{A^{2}}, \\& B_{3}=\frac{A^{2}\exp (-A)+2A\exp (-A)+2\exp (-A)-2}{2A^{2}}, \\& B_{4}=\frac{m (A^{2}+2A+2\exp (-A)-A^{2}\exp (-A)-2 )}{2A ^{2}}. \end{aligned}$$

Theorem 4.2

Let \(f:[a,b]\to \mathbb{R}\) be a differentiable function on \((a,b)\) and \(f'\in L_{1}([a,b])\) for \(0\le a< b\). If the mapping \(\vert f'\vert ^{q}\) is \((\alpha _{1},m)\)-convex on \([0,\frac{b}{m} ]\) for some \((\alpha _{1},m)\in (0,1]^{2}\), \(q\ge 1\), and \(q\ge r\ge 0\), then

$$ \begin{aligned}[b] & \biggl\vert \frac{f(a)+f(b)}{2}+2f \biggl(\frac{a+b}{2} \biggr) -\frac{2}{ \Delta (1)} \biggl[{}_{a^{+}}I_{\varphi }f \biggl(\frac{3a+b}{4} \biggr) \\ &\quad \quad {} +{}_{ (\frac{3a+b}{4} )^{+}}I_{\varphi }f \biggl( \frac{a+b}{2} \biggr) +{}_{ (\frac{a+b}{2} )^{+}}I_{\varphi }f \biggl(\frac{a+3b}{4} \biggr) +{}_{ (\frac{a+3b}{4} )^{-}}I _{\varphi }f(b) \biggr] \biggr\vert \\ &\quad \le \frac{b-a}{8\Delta (1)} \biggl[ \biggl( \int _{0}^{1} \bigl\vert \Delta (t) \bigr\vert ^{\frac{q-r}{q-1}}\,\operatorname {d}t \biggr) ^{1-1/q} \biggl(C_{1} \bigl\vert f'(a) \bigr\vert ^{q}+C_{2} \biggl\vert f' \biggl(\frac{3a+b}{4m} \biggr) \biggr\vert ^{q} \biggr)^{1/q} \\ &\quad \quad {} + \biggl( \int _{0}^{1} \bigl\vert \nabla (t) \bigr\vert ^{\frac{q-r}{q-1}}\,\operatorname {d}t \biggr)^{1-1/q} \biggl(D_{1} \biggl\vert f' \biggl(\frac{3a+b}{4} \biggr) \biggr\vert ^{q}+D_{2} \biggl\vert f' \biggl( \frac{a+b}{4m} \biggr) \biggr\vert ^{q} \biggr)^{1/q} \\ &\quad \quad {} + \biggl( \int _{0}^{1} \bigl\vert \Delta (t) \bigr\vert ^{\frac{q-r}{q-1}}\,\operatorname {d}t \biggr)^{1-1/q} \biggl(C_{1} \biggl\vert f' \biggl(\frac{a+b}{4} \biggr) \biggr\vert ^{q}+C_{2} \biggl\vert f' \biggl( \frac{a+3b}{4m} \biggr) \biggr\vert ^{q} \biggr)^{1/q} \\ &\quad \quad {} + \biggl( \int _{0}^{1} \bigl\vert \nabla (t) \bigr\vert ^{\frac{q-r}{q-1}}\,\operatorname {d}t \biggr)^{1-1/q} \biggl(D_{1} \biggl\vert f' \biggl(\frac{a+3b}{4} \biggr) \biggr\vert ^{q}+D_{2} \biggl\vert f' \biggl( \frac{b}{m} \biggr) \biggr\vert ^{q} \biggr)^{1/q} \biggr], \end{aligned} $$
(4.1)

where the constants \(C_{1}\), \(C_{2}\), \(D_{1}\), and \(D_{2}\) are defined by

$$\begin{aligned}& C_{1} = \int _{0}^{1} t^{\alpha _{1}} \bigl\vert \Delta (t) \bigr\vert ^{r}\,\operatorname {d}t, \quad\quad C_{2}= \int _{0}^{1} m \bigl(1-t^{\alpha _{1}} \bigr) \bigl\vert \Delta (t) \bigr\vert ^{r}\,\operatorname {d}t, \\& D_{1} = \int _{0}^{1} t^{\alpha _{1}} \bigl\vert \nabla (t) \bigr\vert ^{r}\,\operatorname {d}t, \quad\quad D_{2}= \int _{0}^{1} m \bigl(1-t^{\alpha _{1}} \bigr) \bigl\vert \nabla (t) \bigr\vert ^{r}\,\operatorname {d}t. \end{aligned}$$

Proof

By Lemma 3.1, the well-known Hölder inequality, and the \((\alpha _{1},m)\)-convexity of \(\vert f'\vert ^{q}\) on \([0,\frac{b}{m} ]\), we have

$$\begin{aligned} & \biggl\vert \frac{f(a)+f(b)}{2}+2f \biggl(\frac{a+b}{2} \biggr) - \frac{2}{ \Delta (1)} \biggl[{}_{a^{+}}I_{\varphi }f \biggl( \frac{3a+b}{4} \biggr) \\ &\quad \quad {} +{}_{ (\frac{3a+b}{4} )^{+}}I_{\varphi }f \biggl( \frac{a+b}{2} \biggr) +{}_{ (\frac{a+b}{2} )^{+}}I_{\varphi }f \biggl(\frac{a+3b}{4} \biggr) +{}_{ (\frac{a+3b}{4} )^{-}}I _{\varphi }f(b) \biggr] \biggr\vert \\ &\quad \le \frac{b-a}{8\Delta (1)} \biggl[ \int _{0}^{1} \bigl\vert \Delta (t) \bigr\vert \biggl\vert f' \biggl(at+\frac{3a+b}{4}(1-t) \biggr) \biggr\vert \,\operatorname {d}t + \int _{0}^{1} \bigl\vert \nabla (t) \bigr\vert \biggl\vert f' \biggl(\frac{3a+b}{4}t \\ &\quad \quad {} +\frac{a+b}{2}(1-t) \biggr) \biggr\vert \,\operatorname {d}t + \int _{0}^{1} \bigl\vert \Delta (t) \bigr\vert \biggl\vert f' \biggl(\frac{a+b}{2}t+\frac{a+3b}{4}(1-t) \biggr) \biggr\vert \,\operatorname {d}t \\ &\quad \quad {} + \int _{0}^{1} \bigl\vert \nabla (t) \bigr\vert \biggl\vert f' \biggl(\frac{a+3b}{4}t+b(1-t) \biggr) \biggr\vert \,\operatorname {d}t \biggr] \\ &\quad \le \frac{b-a}{8\Delta (1)} \biggl\{ \biggl( \int _{0}^{1} \bigl\vert \Delta (t) \bigr\vert ^{ \frac{q-r}{q-1}}\,\operatorname {d}t \biggr)^{1-1/q} \biggl[ \int _{0}^{1} \bigl\vert \Delta (t) \bigr\vert ^{r} \biggl(t^{\alpha _{1}} \bigl\vert f'(a) \bigr\vert ^{q} \\ &\quad \quad {} +m \bigl(1-t^{\alpha _{1}} \bigr) \biggl\vert f' \biggl( \frac{3a+b}{4m} \biggr) \biggr\vert ^{q} \biggr)\,\operatorname {d}t \biggr]^{1/q} + \biggl( \int _{0}^{1} \bigl\vert \nabla (t) \bigr\vert ^{\frac{q-r}{q-1}}\,\operatorname {d}t \biggr)^{1-1/q} \\ &\quad \quad {} \times \biggl[ \int _{0}^{1} \bigl\vert \nabla (t) \bigr\vert ^{r} \biggl(t^{\alpha _{1}} \biggl\vert f' \biggl( \frac{3a+b}{4} \biggr) \biggr\vert ^{q} +m \bigl(1-t^{ \alpha _{1}} \bigr) \biggl\vert f' \biggl(\frac{a+b}{2m} \biggr) \biggr\vert ^{q} \biggr)\,\operatorname {d}t \biggr]^{1/q} \\ &\quad \quad {} + \biggl( \int _{0}^{1} \bigl\vert \Delta (t) \bigr\vert ^{\frac{q-r}{q-1}}\,\operatorname {d}t \biggr)^{1-1/q} \biggl[ \int _{0}^{1} \bigl\vert \Delta (t) \bigr\vert ^{r} \biggl(t^{\alpha _{1}} \biggl\vert f' \biggl( \frac{a+b}{2} \biggr) \biggr\vert ^{q} \\ &\quad \quad {} +m \bigl(1-t^{\alpha _{1}} \bigr) \biggl\vert f' \biggl( \frac{a+3b}{4m} \biggr) \biggr\vert ^{q} \biggr)\,\operatorname {d}t \biggr]^{1/q}+ \biggl( \int _{0}^{1} \bigl\vert \nabla (t) \bigr\vert ^{\frac{q-r}{q-1}}\,\operatorname {d}t \biggr)^{1-1/q} \\ &\quad \quad {} \times \biggl[ \int _{0}^{1} \bigl\vert \nabla (t) \bigr\vert ^{r} \biggl(t^{\alpha _{1}} \biggl\vert f' \biggl( \frac{a+3b}{4} \biggr) \biggr\vert ^{q} +m \bigl(1-t^{ \alpha _{1}} \bigr) \biggl\vert f' \biggl(\frac{b}{m} \biggr) \biggr\vert ^{q} \biggr)\,\operatorname {d}t \biggr]^{1/q} \biggr\} \\ &\quad =\frac{b-a}{8\Delta (1)} \biggl[ \biggl( \int _{0}^{1} \bigl\vert \Delta (t) \bigr\vert ^{ \frac{q-r}{q-1}}\,\operatorname {d}t \biggr)^{1-1/q} \biggl(C_{1} \bigl\vert f'(a) \bigr\vert ^{q}+C _{2} \biggl\vert f' \biggl(\frac{3a+b}{4m} \biggr) \biggr\vert ^{q} \biggr)^{1/q} \\ &\quad \quad {} + \biggl( \int _{0}^{1} \bigl\vert \nabla (t) \bigr\vert ^{\frac{q-r}{q-1}}\,\operatorname {d}t \biggr)^{1-1/q} \biggl(D_{1} \biggl\vert f' \biggl(\frac{3a+b}{4} \biggr) \biggr\vert ^{q}+D_{2} \biggl\vert f' \biggl( \frac{a+b}{4m} \biggr) \biggr\vert ^{q} \biggr)^{1/q} \\ &\quad \quad {} + \biggl( \int _{0}^{1} \bigl\vert \Delta (t) \bigr\vert ^{\frac{q-r}{q-1}}\,\operatorname {d}t \biggr)^{1-1/q} \biggl(C_{1} \biggl\vert f' \biggl(\frac{a+b}{4} \biggr) \biggr\vert ^{q}+C_{2} \biggl\vert f' \biggl( \frac{a+3b}{4m} \biggr) \biggr\vert ^{q} \biggr)^{1/q} \\ &\quad \quad {} + \biggl( \int _{0}^{1} \bigl\vert \nabla (t) \bigr\vert ^{\frac{q-r}{q-1}}\,\operatorname {d}t \biggr)^{1-1/q} \biggl(D_{1} \biggl\vert f' \biggl(\frac{a+3b}{4} \biggr) \biggr\vert ^{q}+D_{2} \biggl\vert f' \biggl( \frac{b}{m} \biggr) \biggr\vert ^{q} \biggr)^{1/q} \biggr]. \end{aligned}$$

The required proof is complete. □

Remark 4.3

Under assumptions of Theorem 4.2, if \(\varphi (t)=t\), then inequality (4.1) reduces to

$$\begin{aligned} & \biggl\vert \frac{1}{2} \biggl[\frac{f(a)+f(b)}{2}+f \biggl( \frac{a+b}{2} \biggr) \biggr]-\frac{1}{b-a} \int _{a}^{b} f(x)\,\operatorname {d}x \biggr\vert \\ &\quad \le \frac{b-a}{16} \biggl(\frac{q-1}{2q-r-1} \biggr)^{1-1/q} \biggl[ \biggl(\frac{1}{r+\alpha _{1}+1} \bigl\vert f'(a) \bigr\vert ^{q} +\frac{m\alpha _{1}}{(r+1)(r+ \alpha _{1}+1)} \\ &\quad \quad {} \times \biggl\vert f' \biggl(\frac{3a+b}{4} \biggr) \biggr\vert ^{q} \biggr)^{1/q} + \biggl(B(r+1,\alpha _{1}+1) \biggl\vert f' \biggl(\frac{3a+b}{4} \biggr) \biggr\vert ^{q} +m \biggl(\frac{1}{r+1} \\ &\quad \quad {} -B(r+1,\alpha _{1}+1) \biggr) \biggl\vert f' \biggl(\frac{a+b}{2} \biggr) \biggr\vert ^{q} \biggr)^{1/q} + \biggl(\frac{1}{r+\alpha _{1}+1} \biggl\vert f' \biggl(\frac{a+b}{2} \biggr) \biggr\vert ^{q} \\ &\quad \quad {} +\frac{m\alpha _{1}}{(r+1)(r+\alpha _{1}+1)} \biggl\vert f' \biggl( \frac{a+3b}{4} \biggr) \biggr\vert ^{q} \biggr)^{1/q} +B(r+1,\alpha _{1}+1) \\ &\quad \quad {} \times \biggl( \biggl\vert f' \biggl( \frac{a+3b}{4} \biggr) \biggr\vert ^{q} +m \biggl( \frac{1}{r+1}-B(r+1,\alpha _{1}+1) \biggr) \biggl\vert f' \biggl( \frac{b}{m} \biggr) \biggr\vert ^{q} \biggr)^{1/q} \biggr], \end{aligned}$$

which was proved in [26].

Remark 4.4

Under assumptions of Theorem 4.2, if \(\varphi (t)=\frac{t^{ \alpha }}{\varGamma (\alpha )}\), then inequality (4.1) reduces to inequality (1.4).

Corollary 4.3

Under assumptions of Theorem 4.2, if \(\varphi (t)=\frac{t^{ \alpha /k}}{{k\varGamma _{k}(\alpha )}}\), then

$$\begin{aligned} & \biggl\vert \frac{1}{2} \biggl[\frac{f(a)+f(b)}{2}+f \biggl( \frac{a+b}{2} \biggr) \biggr] -\frac{4^{\alpha /k-1}\varGamma _{k}(\alpha +1)}{(b-a)^{ \alpha /k}} \biggl[I^{\alpha }_{a^{+},k}f \biggl(\frac{3a+b}{4} \biggr) \\ &\quad \quad {} +I^{\alpha }_{ (\frac{3a+b}{4} )^{+},k}f \biggl( \frac{a+b}{2} \biggr) +I^{\alpha }_{ (\frac{a+b}{2} )^{+},k}f \biggl(\frac{a+3b}{4} \biggr) +I^{\alpha }_{ (\frac{a+3b}{4} )^{+},k}f(b) \biggr] \biggr\vert \\ &\quad \le \frac{b-a}{16} \biggl\{ \biggl[ \frac{q-1}{\frac{\alpha }{k}(q-r)+q-1} \biggr]^{1-1/q} \biggl[\frac{1}{\frac{ \alpha r}{k}+\alpha _{1}+1} \bigl\vert f'(a) \bigr\vert ^{q} \\ &\quad \quad {} +\frac{m\alpha _{1}}{ (\frac{\alpha r}{k}+1 ) (\frac{ \alpha r}{k}+\alpha _{1}+1 )} \biggl\vert f \biggl(\frac{3a+b}{4m} \biggr) \biggr\vert ^{q} \biggr]^{1/q}+\frac{k}{\alpha } B^{1-1/q} \biggl(\frac{2q-r-1}{q-1},\frac{k}{\alpha } \biggr) \\ &\quad \quad {} \times \biggl[ \biggl(r+1,\frac{k(\alpha _{1}+1)}{\alpha } \biggr) \biggl\vert f \biggl(\frac{3a+b}{4} \biggr) \biggr\vert ^{q}+m \biggl[B \biggl(r+1,\frac{k}{\alpha } \biggr) -B \biggl(r+1, \\ &\quad \quad \frac{k(\alpha _{1}+1)}{\alpha } \biggr) \biggr] \biggl\vert f \biggl( \frac{a+b}{2m} \biggr) \biggr\vert ^{q} \biggr]^{1/q} + \biggl[\frac{q-1}{\frac{ \alpha }{k}(q-r)+q-1} \biggr]^{1-1/q} \biggl[\frac{1}{ \frac{\alpha r}{k} +\alpha _{1}+1} \\ &\quad \quad {} \times \bigl\vert f'(a) \bigr\vert ^{q}+ \frac{m\alpha _{1}}{ ( \frac{\alpha r}{k}+1 ) (\frac{\alpha r}{k}+\alpha _{1}+1 )} \biggl\vert f \biggl(\frac{a+3b}{4m} \biggr) \biggr\vert ^{q} \biggr]^{1/q} \\ &\quad \quad {} +\frac{k}{\alpha }B^{1-1/q} \biggl(\frac{2q-r-1}{q-1}, \frac{k}{ \alpha } \biggr) \biggl[ \biggl(r+1,\frac{k(\alpha _{1}+1)}{\alpha } \biggr) \biggl\vert f \biggl(\frac{a+3b}{4} \biggr) \biggr\vert ^{q} \\ &\quad \quad {} +m \biggl[B \biggl(r+1,\frac{k}{\alpha } \biggr)-B \biggl(r+1, \frac{k( \alpha _{1}+1)}{\alpha } \biggr) \biggr] \biggl\vert f \biggl( \frac{b}{m} \biggr) \biggr\vert ^{q} \biggr]^{1/q} \\ &\quad \quad {} + \biggl[\frac{q-1}{\frac{\alpha }{k}(q-r)+q-1} \biggr]^{1-1/q} \biggr\} . \end{aligned}$$

Corollary 4.4

Under assumptions of Theorem 4.2, if \(r=0\) and \(\varphi (t)=\frac{t}{ \alpha }\exp (-\frac{1-\alpha }{\alpha }t )\), then

$$\begin{aligned} & \biggl\vert \frac{f(a)+f(b)}{2}+2f \biggl(\frac{a+b}{2} \biggr) - \frac{2(1- \alpha )}{1-\exp (-A)} \biggl[\mathcal{I}^{\alpha }_{a^{+}}f \biggl( \frac{3a+b}{4} \biggr) \\ &\quad \quad {} +\mathcal{I}^{\alpha }_{ (\frac{3a+b}{4} )^{+}}f \biggl( \frac{a+b}{2} \biggr)+ \mathcal{I}^{\alpha }_{ ( \frac{a+b}{2} )^{+}}f \biggl( \frac{a+3b}{4} \biggr) +\mathcal{I} ^{\alpha }_{ (\frac{a+3b}{4} )^{-}}f(b) \biggr] \biggr\vert \\ &\quad \le \frac{b-a}{8[1-\exp (-A)]} \biggl[ \biggl( \int _{0}^{1} \bigl[1-\exp (-At) \bigr]^{p} \,\operatorname {d}t \biggr)^{1/p} \biggl( \bigl\vert f'(a) \bigr\vert ^{q} \\ &\quad \quad {} + \biggl\vert f' \biggl(\frac{3a+b}{4m} \biggr) \biggr\vert ^{q} \biggr)^{1/q}+ \biggl( \int _{0}^{1} \bigl[\exp (-At)-\exp (-A) \bigr]^{p}\,\operatorname {d}t \biggr)^{1/p} \\ &\quad \quad {} \times \biggl( \biggl\vert f' \biggl( \frac{3a+b}{4} \biggr) \biggr\vert ^{q} + \biggl\vert f' \biggl( \frac{a+b}{4m} \biggr) \biggr\vert ^{q} \biggr)^{1/q}+ \biggl( \int _{0}^{1} \bigl[1-\exp (-At) \bigr]^{p} \,\operatorname {d}t \biggr)^{1/p} \\ &\quad \quad {} \times \biggl( \biggl\vert f' \biggl( \frac{a+b}{4} \biggr) \biggr\vert ^{q} + \biggl\vert f' \biggl( \frac{a+3b}{4m} \biggr) \biggr\vert ^{q} \biggr)^{1/q}+ \biggl( \int _{0}^{1} \bigl[\exp (-At) \\ &\quad \quad {} -\exp (-A) \bigr]^{p}\,\operatorname {d}t \biggr)^{1/p} \biggl( \biggl\vert f' \biggl( \frac{a+3b}{4} \biggr) \biggr\vert ^{q}+ \biggl\vert f' \biggl(\frac{b}{m} \biggr) \biggr\vert ^{q} \biggr)^{1/q} \biggr], \end{aligned}$$

where \(\frac{1}{p}+\frac{1}{q}=1\) and \(A=\frac{1-\alpha }{\alpha } \frac{b-a}{2}\).

Remark 4.5

Under assumptions of Theorem 4.2, if \(A=\frac{1-\alpha }{ \alpha }\frac{b-a}{2}\), \(\alpha _{1}=r=1\), and \(\varphi (t)=\frac{t}{ \alpha }\exp (-\frac{1-\alpha }{\alpha }t )\), then Theorem 4.2 reduces to Corollary 4.2.

5 Conclusions

In this work, we establish generalized fractional integral inequalities, the Riemann–Liouville fractional integral inequalities, and some classical integral inequalities of the Hermite–Hadamard type for \((\alpha ,m)\)-convex functions. The results presented in this paper would provide generalizations and extensions of those given in earlier works.