Generalized fractional integral inequalities by means of quasiconvexity

Abstract

Using the newly introduced fractional integral operators in (Fasc. Math. 20(4):5-27, 2016) and (East Asian Math. J. 21(2):191-203, 2005), we establish some novel inequalities of the Hermite–Hadamard type for functions whose second derivatives in absolute value are η-quasiconvex. Results obtained herein give a broader generalization to some existing results in the literature by choosing appropriate values of the parameters under consideration. We apply our results to some special means such as the arithmetic, geometric, harmonic, logarithmic, generalized logarithmic, and identric means to obtain more results in this direction.

1 Introduction

In many areas of mathematics, convex functions play an important role. In the study of optimization problems, they are particularly important where they are distinguished by a number of convenient properties. For example, there is no more than one minimum for a (strictly) convex function on an open set. Even in infinite-dimensional spaces, convex functions remain in place under suitable additional hypotheses. Fractional integral inequalities established by means of (quasi)convexity have been a subject of immense investigation in recent times. In this paper, it is our objective to contribute to this subject area.

Let $\mathfrak{D}\subset \mathbb{R}$ be an interval. We start by collecting the following preliminaries:

Definition 1

A function $\mathcal{K}:\mathfrak{D}\to \mathbb{R}$ is called convex on $\mathfrak{D}$ if

$$ \mathcal{K}\bigl(\tau x+(1-\tau )y\bigr)\leq \tau \mathcal{K}(x)+(1-\tau ) \mathcal{K}(y) $$

for all $x,y\in \mathfrak{D}$ and $\tau \in [0,1]$.

Thanks to the following fractional integral operators introduced by Raina [10] and Agarwal et al. [1]: if $\mathcal{K}\in L([\mathfrak{\alpha },\beta ])$, $\rho , \lambda >0$, and $\sigma:\mathbb{N}\cup \{0\}\to (0, \infty )$ is a bounded sequence, then we define

$$ \bigl(\mathfrak{J}^{\sigma }_{\rho ,\lambda ,\mathfrak{\alpha }^{+}; \omega }\mathcal{K} \bigr) (x):= \int _{\mathfrak{\alpha }}^{x}(x- \tau )^{\lambda -1} \mathfrak{F}^{\sigma }_{\rho ,\lambda } \bigl[\omega (x- \tau )^{\rho } \bigr]\mathcal{K}(\tau ) \,d\tau \quad\text{$(x> \mathfrak{\alpha }>0)$} $$

(1)

and

$$ \bigl(\mathfrak{J}^{\sigma }_{\rho ,\lambda ,\beta ^{-};\omega } \mathcal{K} \bigr) (x):= \int _{x}^{\beta }(\tau -x)^{\lambda -1} \mathfrak{F}^{\sigma }_{\rho ,\lambda } \bigl[\omega (\tau -x)^{ \rho } \bigr]\mathcal{K}(\tau ) \,d\tau\quad \text{$(0< x< \beta )$}, $$

(2)

where

$$ \mathfrak{F}^{\sigma }_{\rho ,\lambda }(x)=\mathfrak{F}^{\sigma (0), \sigma (1),\ldots }_{\rho ,\lambda }(x)= \sum_{j=0}^{\infty }\frac{ \sigma (j)}{\varGamma (j\rho +\lambda )}x^{j}. $$

(3)

Set et al. [11] obtained, among other things, the following three consequences of their main results for the class of convex functions.

Theorem 2

([11])

Let $\mathfrak{\alpha },\beta ,\omega \in \mathbb{R}$ with $\mathfrak{\alpha }<\beta$, and $\rho , \lambda >0$. Suppose that $\mathcal{K}:[\mathfrak{\alpha },\beta ]\to \mathbb{R}$ is a twice differentiable function on $(\mathfrak{\alpha },\beta )$ and $\mathcal{K}''\in L([\mathfrak{\alpha },\beta ])$. If, in addition, $|\mathcal{K}''|$ is convex on $[\mathfrak{\alpha },\beta ]$, then the following inequality for generalized fractional integrals holds:

$$\begin{aligned} & \biggl\vert \mathfrak{F}^{\sigma }_{\rho ,\lambda +1} \bigl[\omega ( \beta -\mathfrak{\alpha })^{\rho } \bigr]\frac{\mathcal{K}( \mathfrak{\alpha })+\mathcal{K}(\beta )}{2}- \frac{1}{2(\beta - \mathfrak{\alpha })^{\lambda }} \bigl[ \bigl(\mathfrak{J}^{\sigma }_{ \rho ,\lambda ,\beta ^{-};\omega } \mathcal{K} \bigr) ( \mathfrak{\alpha })+ \bigl(\mathfrak{J}^{\sigma }_{\rho ,\lambda , \mathfrak{\alpha }^{+};\omega } \mathcal{K} \bigr) (\beta ) \bigr] \biggr\vert \\ &\quad\leq \frac{(\beta -\mathfrak{\alpha })^{2}}{4}\mathfrak{F}^{ \sigma _{1,1}}_{\rho ,\lambda +2} \bigl[ \vert \omega \vert (\beta - \mathfrak{\alpha })^{\rho } \bigr] \bigl( \bigl\vert \mathcal{K}''( \mathfrak{\alpha }) \bigr\vert + \bigl\vert \mathcal{K}(\beta ) \bigr\vert \bigr), \end{aligned}$$

(4)

where

$$ \sigma _{1,1}(j)=\sigma (j) \biggl[\frac{\lambda +j\rho }{2(\lambda +j \rho +2)} \biggr], \quad j\in \mathbb{N}\cup \{0\}. $$

Theorem 3

([11])

Let $\mathfrak{\alpha },\beta ,\omega \in \mathbb{R}$ with $\mathfrak{\alpha }<\beta$, and $\rho , \lambda >0$. Suppose that $\mathcal{K}:[\mathfrak{\alpha },\beta ]\to \mathbb{R}$ is a twice differentiable function on $(\mathfrak{\alpha },\beta )$, $\mathcal{K}''\in L([\mathfrak{\alpha },\beta ])$, and $p>1$ with $1/p+1/q=1$. If, in addition, $|\mathcal{K}''|^{q}$ is convex on $[\mathfrak{\alpha },\beta ]$, then the following inequality for generalized fractional integrals holds:

$$\begin{aligned} & \biggl\vert \mathfrak{F}^{\sigma }_{\rho ,\lambda +1} \bigl[\omega ( \beta -\mathfrak{\alpha })^{\rho } \bigr]\frac{\mathcal{K}( \mathfrak{\alpha })+\mathcal{K}(\beta )}{2}- \frac{1}{2(\beta - \mathfrak{\alpha })^{\lambda }} \bigl[ \bigl(\mathfrak{J}^{\sigma }_{ \rho ,\lambda ,\beta ^{-};\omega } \mathcal{K} \bigr) ( \mathfrak{\alpha })+ \bigl(\mathfrak{J}^{\sigma }_{\rho ,\lambda , \mathfrak{\alpha }^{+};\omega } \mathcal{K} \bigr) (\beta ) \bigr] \biggr\vert \\ &\quad\leq \frac{(\beta -\mathfrak{\alpha })^{2}}{2}\mathfrak{F}^{\sigma _{2}}_{\rho ,\lambda +2} \bigl[ \vert \omega \vert (\beta -\mathfrak{\alpha })^{ \rho } \bigr] \biggl[ \frac{ \vert \mathcal{K}''(\mathfrak{\alpha }) \vert ^{q}+ \vert \mathcal{K}(\beta ) \vert ^{q}}{2} \biggr]^{\frac{1}{q}}, \end{aligned}$$

(5)

where

$$ \sigma _{2}(j)=2\sigma (j) \biggl[\frac{1}{\lambda +j\rho }\mathcal{B} \biggl(\frac{p+1}{\lambda +j\rho },p+1 \biggr) \biggr]^{\frac{1}{p}},\quad j\in \mathbb{N}\cup \{0\} $$

and $\mathcal{B}(\cdot ,\cdot )$ is the Euler beta function defined as follows:

$$ \mathcal{B}(x,y)= \int _{0}^{1}\tau ^{x-1}(1-\tau )^{y-1} \,d\tau ,\quad x, y>0. $$

Theorem 4

([11])

Let $\mathfrak{\alpha },\beta ,\omega \in \mathbb{R}$ with $\mathfrak{\alpha }<\beta$, and $\rho , \lambda >0$. Suppose that $\mathcal{K}:[\mathfrak{\alpha },\beta ]\to \mathbb{R}$ is a twice differentiable function on $(\mathfrak{\alpha },\beta )$ and $\mathcal{K}''\in L([\mathfrak{\alpha },\beta ])$. If, in addition, $|\mathcal{K}''|^{q}$ is a convex function on $[\mathfrak{\alpha }, \beta ]$ for $q\geq 1$, then the following inequality for generalized fractional integrals holds:

$$\begin{aligned} & \biggl\vert \mathfrak{F}^{\sigma }_{\rho ,\lambda +1} \bigl[\omega ( \beta -\mathfrak{\alpha })^{\rho } \bigr]\frac{\mathcal{K}( \mathfrak{\alpha })+\mathcal{K}(\beta )}{2}- \frac{1}{2(\beta - \mathfrak{\alpha })^{\lambda }} \bigl[ \bigl(\mathfrak{J}^{\sigma }_{ \rho ,\lambda ,\beta ^{-};\omega } \mathcal{K} \bigr) ( \mathfrak{\alpha })+ \bigl(\mathfrak{J}^{\sigma }_{\rho ,\lambda , \mathfrak{\alpha }^{+};\omega } \mathcal{K} \bigr) (\beta ) \bigr] \biggr\vert \\ &\quad\leq \frac{(\beta -\mathfrak{\alpha })^{2}}{2}\mathfrak{F}^{ \sigma _{3,1}}_{\rho ,\lambda +2} \bigl[ \vert \omega \vert (\beta - \mathfrak{\alpha })^{\rho } \bigr], \end{aligned}$$

(6)

where, for each $j\in \mathbb{N}\cup \{0\}$,

$$\begin{aligned} \sigma _{3,1}(j)={}&\sigma (j) \biggl[\frac{\lambda +j\rho }{2(\lambda +j \rho +2)} \biggr]^{1-\frac{1}{q}} \\ &{} \times \biggl[ \biggl( \frac{ \lambda +j\rho }{3(\lambda +j\rho +3)} \bigl\vert \mathcal{K}''( \mathfrak{\alpha }) \bigr\vert ^{q}+ \frac{(\lambda +j\rho )(\lambda +j\rho +5)}{6( \lambda +j\rho +2)(\lambda +j\rho +3)} \bigl\vert \mathcal{K}''(\beta ) \bigr\vert ^{q} \biggr) ^{\frac{1}{q}} \\ & {} + \biggl( \frac{(\lambda +j\rho )(\lambda +j\rho +5)}{6(\lambda +j\rho +2)(\lambda +j\rho +3)} \bigl\vert \mathcal{K}''( \mathfrak{\alpha }) \bigr\vert ^{q}+ \frac{\lambda +j\rho }{3( \lambda +j\rho +3)} \bigl\vert \mathcal{K}''(\beta ) \bigr\vert ^{q} \biggr)^{\frac{1}{q}} \biggr]. \end{aligned}$$

Remark 5

It is easy to verify that $\mathfrak{J}^{\sigma }_{\rho ,\lambda , \mathfrak{\alpha }^{+};\omega }\mathcal{K}$ and $\mathfrak{J}^{\sigma }_{\rho ,\lambda ,\beta ^{-};\omega }\mathcal{K}$ are bounded integral operators. In addition, if $\omega =0$ and $\sigma (0)=1$, then (1) and (2) reduce to the left- and right- Riemann–Liouville fractional integral operators, respectively. That is,

$$ \bigl(\mathfrak{J}^{\sigma }_{\rho ,\lambda ,\mathfrak{\alpha }^{+};0} \mathcal{K} \bigr) (x)= \bigl( \mathfrak{J}^{1, \sigma (1), \sigma (2), \ldots }_{\rho ,\lambda ,\mathfrak{\alpha }^{+};0}\mathcal{K} \bigr) (x)= \frac{1}{ \varGamma (\lambda )} \int _{\mathfrak{\alpha }}^{x}(x-\tau )^{\lambda -1} \mathcal{K}( \tau ) \,d\tau =:\mathcal{J}^{\lambda }_{ \mathfrak{\alpha }^{+}}\mathcal{K}(x) $$

and

$$ \bigl(\mathfrak{J}^{\sigma }_{\rho ,\lambda ,\beta ^{-};0}\mathcal{K} \bigr) (x)= \bigl( \mathfrak{J}^{1, \sigma (1), \sigma (2),\ldots }_{\rho ,\lambda ,\beta ^{-};0}\mathcal{K} \bigr) (x)= \frac{1}{\varGamma (\lambda )} \int _{x}^{\beta }(\tau -x)^{\lambda -1}\mathcal{K}(\tau ) \,d\tau =: \mathcal{J}^{\lambda }_{\beta ^{-}}\mathcal{K}(x), $$

where the gamma function $\varGamma (\lambda )=\int _{0}^{\infty }e^{-v}v ^{\lambda -1} \,dv$.

The class of convex functions has been generalized as follows.

Definition 6

A function $\mathcal{K}:\mathfrak{D}\to \mathbb{R}$ is called quasiconvex on $\mathfrak{D}$ if

$$ \mathcal{K}\bigl(\tau x+(1-\tau )y\bigr)\leq \max \bigl\{ \mathcal{K}(x), \mathcal{K}(y) \bigr\} $$

for all $x,y\in \mathfrak{D}$ and $\tau \in [0,1]$.

Recently, Gordji et al. further generalized the class of quasiconvex functions in the following manner.

Definition 7

([4])

A function $\mathcal{K}:\mathfrak{D}\to \mathbb{R}$ is called η-quasiconvex on $\mathfrak{D}$ with respect to $\eta: \mathbb{R}\times \mathbb{R}\to \mathbb{R}$ if

$$ \mathcal{K}\bigl(\tau x+(1-\tau )y\bigr)\leq \max \bigl\{ \mathcal{K}(y), \mathcal{K}(y)+\eta \bigl(\mathcal{K}(x),\mathcal{K}(y)\bigr) \bigr\} $$

for all $x,y\in \mathfrak{D}$ and $\tau \in [0,1]$.

Some inequalities via η-(quasi)convex functions can be found in [3,4,5, 9].

Remark 8

We summarize Definitions 1, 6, and 7 as follows:

To see that quasiconvexity does not always imply convexity, consider the following example: let $\mathcal{K}:[-2,2]\to \mathbb{R}$ be defined by

$$ \mathcal{K}(v):= \textstyle\begin{cases} 1,& v\in [-2,-1], \\ v^{2},& v\in (-1,2]. \end{cases} $$

The function $\mathcal{K}$ is quasiconvex on $[-2,2]$ but not convex on $[-2,2]$.

Theorems 2, 3, and 4 cannot be applied to functions whose second derivatives in absolute value, raise to some powers, are not convex. For this reason, it is our primary objective in this paper to extend Theorems 2, 3, and 4 to a more general class of functions—the class of η-quasiconvex functions. More precisely, we obtain inequalities akin to (4), (5), and (6) in the case when $\mathcal{K}:[ \mathfrak{\alpha },\beta ]\to \mathbb{R}$ is twice differentiable on $(\mathfrak{\alpha },\beta )$ and $|\mathcal{K}''|$ or $|\mathcal{K}''|^{q}, q>1$ is η-quasiconvex on $[\mathfrak{\alpha },\beta ]$. By taking the bifunction $\eta (x,y)=x-y$, our results boil down to inequalities of the Hermite–Hadamard type for functions whose second derivatives in absolute value are quasiconvex (see Remarks 12, 15, and 18 of Sect. 2).

This article is organized in the following manner: Sect. 2 contains the main results and the proofs. In Sect. 3, we apply our results to some special means.

2 Main results

The following lemma will be useful in the proofs of our main results:

Lemma 9

([11])

Let $\mathfrak{\alpha },\beta ,\omega \in \mathbb{R}$ with $\mathfrak{\alpha }<\beta$, and $\rho , \lambda >0$. If $\mathcal{K}:[ \mathfrak{\alpha },\beta ]\to \mathbb{R}$ is a twice differentiable function on $(\mathfrak{\alpha },\beta )$ and $\mathcal{K}''\in L([ \mathfrak{\alpha },\beta ])$, then the following equality for generalized fractional integrals holds:

$$\begin{aligned} &\mathfrak{F}^{\sigma }_{\rho ,\lambda +1} \bigl[\omega (\beta - \mathfrak{\alpha })^{\rho } \bigr]\frac{\mathcal{K}( \mathfrak{\alpha })+\mathcal{K}(\beta )}{2}-\frac{1}{2(\beta - \mathfrak{\alpha })^{\lambda }} \bigl[ \bigl(\mathfrak{J}^{\sigma }_{ \rho ,\lambda ,\beta ^{-};\omega }\mathcal{K} \bigr) ( \mathfrak{ \alpha })+ \bigl(\mathfrak{J}^{\sigma }_{\rho ,\lambda , \mathfrak{\alpha }^{+};\omega }\mathcal{K} \bigr) ( \beta ) \bigr] \\ &\quad=\frac{( \beta -\mathfrak{\alpha })^{2}}{2} \int _{0}^{1} \bigl\{ \bigl(\tau \mathfrak{F}^{\sigma }_{\rho ,\lambda +2}\bigl[\omega (\beta - \mathfrak{\alpha })^{\rho }\bigr]-\tau ^{\lambda +1}\mathfrak{F}^{\sigma } _{\rho ,\lambda +2}\bigl[\omega (\beta -\mathfrak{\alpha })^{\rho }\tau ^{\rho }\bigr] \bigr) \\ & \qquad{}\times \bigl(\mathcal{K}''\bigl(\tau \mathfrak{ \alpha }+(1-\tau )\beta \bigr)+ \mathcal{K}''\bigl( \tau \beta +(1-\tau )\mathfrak{\alpha }\bigr) \bigr) \bigr\} \,d \tau. \end{aligned}$$

(7)

For the sake of convenience, we will make use of the following notations: for $q\geq 1$, denote

$$ \mathbf{M}^{\beta }_{\mathfrak{\alpha }}\bigl( \bigl\vert \mathcal{K}'' \bigr\vert ^{q},\eta \bigr):= \max \bigl\{ \bigl\vert \mathcal{K}''(\beta ) \bigr\vert ^{q}, \bigl\vert \mathcal{K}''(\beta ) \bigr\vert ^{q}+ \eta \bigl( \bigl\vert \mathcal{K}''( \mathfrak{\alpha }) \bigr\vert ^{q}, \bigl\vert \mathcal{K}''( \beta ) \bigr\vert ^{q} \bigr) \bigr\} $$

and

$$ \mathbf{N}^{\beta }_{\mathfrak{\alpha }}\bigl( \bigl\vert \mathcal{K}'' \bigr\vert ^{q},\eta \bigr):= \max \bigl\{ \bigl\vert \mathcal{K}''(\mathfrak{\alpha }) \bigr\vert ^{q}, \bigl\vert \mathcal{K}''( \mathfrak{\alpha }) \bigr\vert ^{q}+\eta \bigl( \bigl\vert \mathcal{K}''(\beta ) \bigr\vert ^{q}, \bigl\vert \mathcal{K}''(\mathfrak{\alpha }) \bigr\vert ^{q} \bigr) \bigr\} . $$

We now state and justify our first result.

Theorem 10

Let $\mathfrak{\alpha },\beta ,\omega \in \mathbb{R}$ with $\mathfrak{\alpha }<\beta$, and $\rho , \lambda >0$. Suppose that $\mathcal{K}:[\mathfrak{\alpha },\beta ]\to \mathbb{R}$ is a twice differentiable function on $(\mathfrak{\alpha },\beta )$ and $\mathcal{K}''\in L([\mathfrak{\alpha },\beta ])$. If, in addition, $|\mathcal{K}''|$ is η-quasiconvex on $[\mathfrak{\alpha }, \beta ]$, then the following inequality for generalized fractional integrals holds:

$$\begin{aligned} & \biggl\vert \mathfrak{F}^{\sigma }_{\rho ,\lambda +1} \bigl[ \omega ( \beta -\mathfrak{\alpha })^{\rho } \bigr]\frac{\mathcal{K}( \mathfrak{\alpha })+\mathcal{K}(\beta )}{2}- \frac{1}{2(\beta - \mathfrak{\alpha })^{\lambda }} \bigl[ \bigl(\mathfrak{J}^{\sigma }_{ \rho ,\lambda ,\beta ^{-};\omega } \mathcal{K} \bigr) ( \mathfrak{\alpha })+ \bigl(\mathfrak{J}^{\sigma }_{\rho ,\lambda , \mathfrak{\alpha }^{+};\omega } \mathcal{K} \bigr) (\beta ) \bigr] \biggr\vert \\ &\quad\leq \frac{(\beta -\mathfrak{\alpha })^{2}}{4}\sum_{j=0}^{\infty } \frac{\sigma (j)(\lambda +j\rho )}{\varGamma (j\rho +\lambda +3)} \vert \omega \vert ^{j}(\beta -\mathfrak{\alpha })^{j\rho } \bigl[{\mathbf{M}}^{ \beta }_{\mathfrak{\alpha }}\bigl( \bigl\vert \mathcal{K}'' \bigr\vert ,\eta \bigr)+ \mathbf{N}^{ \beta }_{\mathfrak{\alpha }}\bigl( \bigl\vert \mathcal{K}'' \bigr\vert ,\eta \bigr) \bigr]. \end{aligned}$$

Proof

Using the η-quasiconvexity of $|\mathcal{K}''|$ on $[ \mathfrak{\alpha },\beta ]$, we obtain

$$ \bigl\vert \mathcal{K}'' \bigl(\tau \mathfrak{\alpha }+(1-\tau )\beta \bigr) \bigr\vert \leq { \mathbf{M}}^{\beta }_{\mathfrak{\alpha }}\bigl( \bigl\vert \mathcal{K}'' \bigr\vert ,\eta \bigr) $$

(8)

and

$$ \bigl\vert \mathcal{K}'' \bigl(\tau \beta +(1-\tau )\mathfrak{\alpha } \bigr) \bigr\vert \leq { \mathbf{N}}^{\beta }_{\mathfrak{\alpha }}\bigl( \bigl\vert \mathcal{K}'' \bigr\vert ,\eta \bigr) $$

(9)

for $\tau \in [0, 1]$.

Now applying Lemma 9, inequalities (8)–(9), and relation (3), one obtains

$$\begin{aligned} & \biggl\vert \mathfrak{F}^{\sigma }_{\rho ,\lambda +1} \bigl[ \omega ( \beta -\mathfrak{\alpha })^{\rho } \bigr]\frac{\mathcal{K}( \mathfrak{\alpha })+\mathcal{K}(\beta )}{2}- \frac{1}{2(\beta - \mathfrak{\alpha })^{\lambda }} \bigl[ \bigl(\mathfrak{J}^{\sigma }_{ \rho ,\lambda ,\beta ^{-};\omega } \mathcal{K} \bigr) ( \mathfrak{\alpha })+ \bigl(\mathfrak{J}^{\sigma }_{\rho ,\lambda , \mathfrak{\alpha }^{+};\omega } \mathcal{K} \bigr) (\beta ) \bigr] \biggr\vert \\ &\quad\leq \frac{(\beta -\mathfrak{\alpha })^{2}}{2} \int _{0}^{1} \bigl\{ \bigl\vert \tau \mathfrak{F}^{\sigma }_{\rho ,\lambda +2}\bigl[\omega (\beta - \mathfrak{\alpha })^{\rho }\bigr]-\tau ^{\lambda +1}\mathfrak{F}^{\sigma } _{\rho ,\lambda +2}\bigl[\omega (\beta -\mathfrak{\alpha })^{\rho }\tau ^{\rho }\bigr] \bigr\vert \\ &\qquad{} \times \bigl\vert \mathcal{K}''\bigl(\tau \mathfrak{\alpha }+(1-\tau )\beta \bigr)+ \mathcal{K}'' \bigl(\tau \beta +(1-\tau )\mathfrak{\alpha }\bigr) \bigr\vert \bigr\} \,d \tau \\ &\quad\leq \frac{(\beta -\mathfrak{\alpha })^{2}}{2} \\ &\qquad{} \times \int _{0}^{1} \Biggl\{ \Biggl\vert \sum _{j=0}^{\infty }\frac{\sigma (j)}{ \varGamma (j\rho +\lambda +2)}\omega ^{j}( \beta -\mathfrak{\alpha })^{j \rho }\tau -\sum_{j=0}^{\infty } \frac{\sigma (j)}{\varGamma (j\rho + \lambda +2)}\omega ^{j}(\beta -\mathfrak{\alpha })^{j\rho } \tau ^{j\rho +\lambda +1} \Biggr\vert \\ & \qquad{}\times \bigl\vert \mathcal{K}''\bigl(\tau \mathfrak{\alpha }+(1-\tau )\beta \bigr)+ \mathcal{K}'' \bigl(\tau \beta +(1-\tau )\mathfrak{\alpha }\bigr) \bigr\vert \Biggr\} \,d \tau \\ &\quad\leq \frac{(\beta -\mathfrak{\alpha })^{2}}{2} \int _{0}^{1} \Biggl\{ \Biggl\vert \sum _{j=0}^{\infty }\frac{\sigma (j)}{\varGamma (j\rho + \lambda +2)}\omega ^{j}( \beta -\mathfrak{\alpha })^{j\rho } \bigl[\tau - \tau ^{j\rho +\lambda +1} \bigr] \Biggr\vert \\ &\qquad{} \times \bigl\vert \mathcal{K}''\bigl(\tau \mathfrak{\alpha }+(1-\tau )\beta \bigr)+ \mathcal{K}'' \bigl(\tau \beta +(1-\tau )\mathfrak{\alpha }\bigr) \bigr\vert \Biggr\} \,d \tau \\ &\quad\leq \frac{(\beta -\mathfrak{\alpha })^{2}}{2}\sum_{j=0}^{ \infty } \frac{\sigma (j)}{\varGamma (j\rho +\lambda +2)} \vert \omega \vert ^{j}( \beta -\mathfrak{\alpha })^{j\rho } \\ &\qquad{} \times \int _{0}^{1} \bigl\vert \tau -\tau ^{j\rho +\lambda +1} \bigr\vert \bigl[ \bigl\vert \mathcal{K}'' \bigl( \tau \mathfrak{\alpha }+(1-\tau )\beta \bigr) \bigr\vert + \bigl\vert \mathcal{K}''\bigl( \tau \beta +(1-\tau )\mathfrak{ \alpha }\bigr) \bigr\vert \bigr] \,d\tau \\ & \quad\leq \frac{(\beta -\mathfrak{\alpha })^{2}}{2}\sum_{j=0}^{\infty } \frac{ \sigma (j)}{\varGamma (j\rho +\lambda +2)} \vert \omega \vert ^{j}(\beta - \mathfrak{\alpha })^{j\rho } \\ &\qquad{} \times \int _{0}^{1} \bigl\vert \tau - \tau ^{j\rho +\lambda +1} \bigr\vert \bigl[{\mathbf{M}}^{\beta }_{ \mathfrak{\alpha }} \bigl( \bigl\vert \mathcal{K}'' \bigr\vert ,\eta \bigr)+ \mathbf{N}^{\beta }_{ \mathfrak{\alpha }}\bigl( \bigl\vert \mathcal{K}'' \bigr\vert ,\eta \bigr) \bigr] \,d\tau \\ &\quad= \frac{( \beta -\mathfrak{\alpha })^{2}}{2}\sum_{j=0}^{\infty } \frac{\sigma (j)}{ \varGamma (j\rho +\lambda +2)} \vert \omega \vert ^{j}(\beta -\mathfrak{\alpha })^{j \rho } \\ & \qquad{}\times \bigl[{\mathbf{M}}^{\beta }_{\mathfrak{\alpha }}\bigl( \bigl\vert \mathcal{K}'' \bigr\vert ,\eta \bigr)+ \mathbf{N}^{\beta }_{\mathfrak{\alpha }}\bigl( \bigl\vert \mathcal{K}'' \bigr\vert ,\eta \bigr) \bigr] \int _{0}^{1} \bigl(\tau - \tau ^{j\rho +\lambda +1} \bigr) \,d\tau \\ &\quad= \frac{(\beta - \mathfrak{\alpha })^{2}}{4}\sum_{j=0}^{\infty } \frac{\sigma (j)( \lambda +j\rho )}{\varGamma (j\rho +\lambda +3)} \vert \omega \vert ^{j}(\beta - \mathfrak{\alpha })^{j\rho } \bigl[{\mathbf{M}}^{\beta }_{ \mathfrak{\alpha }}\bigl( \bigl\vert \mathcal{K}'' \bigr\vert ,\eta \bigr)+ \mathbf{N}^{\beta }_{ \mathfrak{\alpha }}\bigl( \bigl\vert \mathcal{K}'' \bigr\vert ,\eta \bigr) \bigr], \end{aligned}$$

since

$$ \int _{0}^{1} \bigl(\tau -\tau ^{j\rho +\lambda +1} \bigr) \,d\tau =\frac{ \lambda +j\rho }{2(\lambda +j\rho +2)}. $$

Hence, the intended inequality is obtained. □

Corollary 11

Let $\mathfrak{\alpha },\beta \in \mathbb{R}$ with $\mathfrak{\alpha }<\beta$, and $\lambda >0$. Suppose that $\mathcal{K}:[\mathfrak{\alpha },\beta ]\to \mathbb{R}$ is a twice differentiable function on $(\mathfrak{\alpha },\beta )$ and $\mathcal{K}''\in L([\mathfrak{\alpha },\beta ])$. If, in addition, $|\mathcal{K}''|$ is η-quasiconvex on $[\mathfrak{\alpha }, \beta ]$, then the following inequality for fractional integrals holds:

$$\begin{aligned} & \biggl\vert \frac{\mathcal{K}(\mathfrak{\alpha })+\mathcal{K}(\beta )}{2}-\frac{ \varGamma (\lambda +1)}{2(\beta -\mathfrak{\alpha })^{\lambda }} \bigl[ \mathcal{J}^{\lambda }_{\beta ^{-}}\mathcal{K}(\mathfrak{\alpha })+ \mathcal{J}^{\lambda }_{\beta ^{-}}\mathcal{K}(\mathfrak{\alpha }) \bigr] \biggr\vert \\ &\quad\leq \frac{(\beta -\mathfrak{\alpha })^{2}\lambda }{4( \lambda +1)(\lambda +2)} \bigl[{\mathbf{M}}^{\beta }_{ \mathfrak{\alpha }}\bigl( \bigl\vert \mathcal{K}'' \bigr\vert ,\eta \bigr)+ \mathbf{N}^{\beta }_{ \mathfrak{\alpha }}\bigl( \bigl\vert \mathcal{K}'' \bigr\vert ,\eta \bigr) \bigr]. \end{aligned}$$

(10)

Proof

By taking $\sigma (0)=1$ and $w=0$, one observes that

$$\begin{aligned} &\mathfrak{F}^{\sigma }_{\rho ,\lambda +1} \bigl[\omega (\beta - \mathfrak{\alpha })^{\rho } \bigr]=\frac{1}{\varGamma (\lambda +1)}, \\ &\bigl(\mathfrak{J}^{\sigma }_{\rho ,\lambda ,\beta ^{-};\omega } \mathcal{K} \bigr) (\mathfrak{ \alpha })=\mathcal{J}^{\lambda }_{\beta ^{-}}\mathcal{K}(\mathfrak{\alpha }), \end{aligned}$$

and

$$ \bigl(\mathfrak{J}^{\sigma }_{\rho ,\lambda ,\mathfrak{\alpha }^{+}; \omega }\mathcal{K} \bigr) (\beta )= \mathcal{J}^{\lambda }_{ \mathfrak{\alpha }^{+}}\mathcal{K}(\beta ). $$

Now, using Theorem 10, we arrive at the intended inequality. □

Remark 12

By setting the bifunction $\eta (x,y)=x-y$, the inequality in (10) boils down to

$$\begin{aligned} & \biggl\vert \frac{\mathcal{K}(\mathfrak{\alpha })+\mathcal{K}(\beta )}{2}-\frac{ \varGamma (\lambda +1)}{2(\beta -\mathfrak{\alpha })^{\lambda }} \bigl[ \mathcal{J}^{\lambda }_{\beta ^{-}}\mathcal{K}(\mathfrak{\alpha })+ \mathcal{J}^{\lambda }_{\mathfrak{\alpha }^{+}}\mathcal{K}(\beta ) \bigr] \biggr\vert \\ &\quad\leq \frac{(\beta -\mathfrak{\alpha })^{2}\lambda }{2( \lambda +1)(\lambda +2)}\max \bigl\{ \bigl\vert \mathcal{K}''( \mathfrak{\alpha }) \bigr\vert , \bigl\vert \mathcal{K}''( \beta ) \bigr\vert \bigr\} . \end{aligned}$$

(11)

Theorem 13

Let $\mathfrak{\alpha },\beta ,\omega \in \mathbb{R}$ with $\mathfrak{\alpha }<\beta$, and $\rho , \lambda >0$. Suppose that $\mathcal{K}:[\mathfrak{\alpha },\beta ]\to \mathbb{R}$ is a twice differentiable function on $(\mathfrak{\alpha },\beta )$, $\mathcal{K}''\in L([\mathfrak{\alpha },\beta ])$, and $p>1$ with $1/p+1/q=1$. If, in addition, $|\mathcal{K}''|^{q}$ is η-quasiconvex on $[\mathfrak{\alpha },\beta ]$, then the following inequality for generalized fractional integrals holds:

$$\begin{aligned} & \biggl\vert \mathfrak{F}^{\sigma }_{\rho ,\lambda +1} \bigl[ \omega ( \beta -\mathfrak{\alpha })^{\rho } \bigr]\frac{\mathcal{K}( \mathfrak{\alpha })+\mathcal{K}(\beta )}{2}- \frac{1}{2(\beta - \mathfrak{\alpha })^{\lambda }} \bigl[ \bigl(\mathfrak{J}^{\sigma }_{ \rho ,\lambda ,\beta ^{-};\omega } \mathcal{K} \bigr) ( \mathfrak{\alpha })+ \bigl(\mathfrak{J}^{\sigma }_{\rho ,\lambda , \mathfrak{\alpha }^{+};\omega } \mathcal{K} \bigr) (\beta ) \bigr] \biggr\vert \\ &\quad\leq \frac{(\beta -\mathfrak{\alpha })^{2}}{2}\sum_{j=0}^{\infty } \frac{\sigma (j)}{\varGamma (j\rho +\lambda +2)} \vert \omega \vert ^{j}(\beta - \mathfrak{\alpha })^{j\rho } \biggl[\frac{1}{\lambda +j\rho } \mathcal{B} \biggl( \frac{p+1}{\lambda +j\rho },p+1 \biggr) \biggr]^{ \frac{1}{p}} \\ &\qquad{} \times \bigl[ \bigl(\mathbf{M}^{\beta }_{ \mathfrak{\alpha }}\bigl( \bigl\vert \mathcal{K}'' \bigr\vert ^{q},\eta \bigr) \bigr)^{\frac{1}{q}}+ \bigl(\mathbf{N}^{\beta }_{\mathfrak{\alpha }}\bigl( \bigl\vert \mathcal{K}'' \bigr\vert ^{q}, \eta \bigr) \bigr)^{\frac{1}{q}} \bigr], \end{aligned}$$

where $\mathcal{B}(\cdot ,\cdot )$ is the Euler beta function.

Proof

We start by first observing that if one lets $u=\tau ^{\lambda +j \rho }$, one gets

$$\begin{aligned} \int _{0}^{1}\tau ^{p}\bigl(1-\tau ^{\lambda +j\rho }\bigr)^{p} \,d\tau &=\frac{1}{ \lambda +j\rho } \int _{0}^{1}u^{\frac{p+1}{\lambda +j\rho }-1}(1-u)^{(p+1)-1} \,du \\ &=\frac{1}{\lambda +j\rho }\mathcal{B} \biggl(\frac{p+1}{\lambda +j\rho },p+1 \biggr). \end{aligned}$$

(12)

Employing Lemma 9 and mimicking the idea from the proof of Theorem 10, we get

$$\begin{aligned} & \biggl\vert \mathfrak{F}^{\sigma }_{\rho ,\lambda +1} \bigl[\omega ( \beta -\mathfrak{\alpha })^{\rho } \bigr]\frac{\mathcal{K}( \mathfrak{\alpha })+\mathcal{K}(\beta )}{2}- \frac{1}{2(\beta - \mathfrak{\alpha })^{\lambda }} \bigl[ \bigl(\mathfrak{J}^{\sigma }_{ \rho ,\lambda ,\beta ^{-};\omega } \mathcal{K} \bigr) ( \mathfrak{\alpha })+ \bigl(\mathfrak{J}^{\sigma }_{\rho ,\lambda , \mathfrak{\alpha }^{+};\omega } \mathcal{K} \bigr) (\beta ) \bigr] \biggr\vert \\ &\quad\leq \frac{(\beta -\mathfrak{\alpha })^{2}}{2}\sum_{j=0}^{\infty } \frac{\sigma (j)}{\varGamma (j\rho +\lambda +2)} \vert \omega \vert ^{j}(\beta - \mathfrak{\alpha })^{j\rho } \\ &\qquad{} \times \int _{0}^{1} \bigl\vert \tau - \tau ^{j\rho +\lambda +1} \bigr\vert \bigl[ \bigl\vert \mathcal{K}'' \bigl(\tau \mathfrak{\alpha }+(1-\tau )\beta \bigr) \bigr\vert + \bigl\vert \mathcal{K}''\bigl( \tau \beta +(1-\tau )\mathfrak{ \alpha }\bigr) \bigr\vert \bigr] \,d\tau. \end{aligned}$$

(13)

Using Hölder’s inequality, the η-quasiconvexity of $|\mathcal{K}''|$ on $[\mathfrak{\alpha },\beta ]$, and (12), one has

$$\begin{aligned} & \int _{0}^{1} \bigl\vert \tau -\tau ^{j\rho +\lambda +1} \bigr\vert \bigl[ \bigl\vert \mathcal{K}'' \bigl( \tau \mathfrak{\alpha }+(1-\tau )\beta \bigr) \bigr\vert + \bigl\vert \mathcal{K}''\bigl( \tau \beta +(1-\tau )\mathfrak{ \alpha }\bigr) \bigr\vert \bigr] \,d\tau \\ &\quad \leq \biggl[ \int _{0}^{1} \bigl(\tau -\tau ^{j\rho +\lambda +1} \bigr) ^{p} \,d\tau \biggr]^{\frac{1}{p}} \biggl[ \biggl( \int _{0}^{1} \bigl\vert \mathcal{K}'' \bigl( \tau \mathfrak{\alpha }+(1-\tau )\beta \bigr) \bigr\vert ^{q} \,d\tau \biggr) ^{\frac{1}{q}} \\ &\qquad{} + \biggl( \int _{0}^{1} \bigl\vert \mathcal{K}'' \bigl(\tau \beta +(1-\tau )\mathfrak{\alpha }\bigr) \bigr\vert ^{q} \,d\tau \biggr)^{\frac{1}{q}} \biggr] \\ &\quad\leq \biggl[ \int _{0} ^{1} \tau ^{p} \bigl(1-\tau ^{j\rho +\lambda } \bigr)^{p} \,d\tau \biggr] ^{\frac{1}{p}} \bigl[ \bigl(\mathbf{M}^{\beta }_{\mathfrak{\alpha }}\bigl( \bigl\vert \mathcal{K}'' \bigr\vert ^{q},\eta \bigr) \bigr)^{\frac{1}{q}}+ \bigl(\mathbf{N}^{ \beta }_{\mathfrak{\alpha }}\bigl( \bigl\vert \mathcal{K}'' \bigr\vert ^{q}, \eta \bigr) \bigr)^{ \frac{1}{q}} \bigr] \\ &\quad= \biggl[\frac{1}{\lambda +j\rho }\mathcal{B} \biggl(\frac{p+1}{\lambda +j\rho },p+1 \biggr) \biggr]^{\frac{1}{p}} \bigl[ \bigl(\mathbf{M}^{\beta }_{\mathfrak{\alpha }}\bigl( \bigl\vert \mathcal{K}'' \bigr\vert ^{q}, \eta \bigr) \bigr)^{\frac{1}{q}}+ \bigl(\mathbf{N}^{\beta }_{ \mathfrak{\alpha }} \bigl( \bigl\vert \mathcal{K}'' \bigr\vert ^{q},\eta \bigr) \bigr)^{\frac{1}{q}} \bigr]. \end{aligned}$$

(14)

Combining (13) and (14) gives the desired result. □

Substituting $\sigma (0)=1$ and $w=0$ in Theorem 13, we derive the following corollary.

Corollary 14

Let $\mathfrak{\alpha },\beta \in \mathbb{R}$ with $\mathfrak{\alpha }<\beta$, and $\lambda >0$. Suppose that $\mathcal{K}:[\mathfrak{\alpha },\beta ]\to \mathbb{R}$ is a twice differentiable function on $(\mathfrak{\alpha },\beta )$, $\mathcal{K}''\in L([\mathfrak{\alpha },\beta ])$, and $p>1$ with $1/p+1/q=1$. If, in addition, $|\mathcal{K}''|^{q}$ is η-quasiconvex on $[\mathfrak{\alpha },\beta ]$, then the following inequality for fractional integrals holds:

$$\begin{aligned} & \biggl\vert \frac{\mathcal{K}(\mathfrak{\alpha })+\mathcal{K}(\beta )}{2}-\frac{ \varGamma (\lambda +1)}{2(\beta -\mathfrak{\alpha })^{\lambda }} \bigl[ \mathcal{J}^{\lambda }_{\beta ^{-}}\mathcal{K}(\mathfrak{\alpha })+ \mathcal{J}^{\lambda }_{\mathfrak{\alpha }^{+}}\mathcal{K}(\beta ) \bigr] \biggr\vert \\ &\quad\leq \frac{(\beta -\mathfrak{\alpha })^{2}}{2(\lambda +1)} \biggl[\frac{1}{\lambda }\mathcal{B} \biggl( \frac{p+1}{\lambda },p+1 \biggr) \biggr] ^{\frac{1}{p}} \bigl[ \bigl( \mathbf{M}^{\beta }_{\mathfrak{\alpha }}\bigl( \bigl\vert \mathcal{K}'' \bigr\vert ^{q},\eta \bigr) \bigr)^{\frac{1}{q}}+ \bigl( \mathbf{N}^{ \beta }_{\mathfrak{\alpha }}\bigl( \bigl\vert \mathcal{K}'' \bigr\vert ^{q},\eta \bigr) \bigr)^{ \frac{1}{q}} \bigr], \end{aligned}$$

(15)

where $\mathcal{B}(\cdot ,\cdot )$ is the Euler beta function.

Remark 15

If the bifunction $\eta (x,y)=x-y$, then (15) becomes

$$\begin{aligned} & \biggl\vert \frac{\mathcal{K}(\mathfrak{\alpha })+\mathcal{K}(\beta )}{2}-\frac{ \varGamma (\lambda +1)}{2(\beta -\mathfrak{\alpha })^{\lambda }} \bigl[ \mathcal{J}^{\lambda }_{\beta ^{-}}\mathcal{K}(\mathfrak{\alpha })+ \mathcal{J}^{\lambda }_{\mathfrak{\alpha }^{+}}\mathcal{K}(\beta ) \bigr] \biggr\vert \\ &\quad\leq \frac{(\beta -\mathfrak{\alpha })^{2}}{\lambda +1} \biggl[\frac{1}{\lambda }\mathcal{B} \biggl( \frac{p+1}{\lambda },p+1 \biggr) \biggr] ^{\frac{1}{p}} \bigl(\max \bigl\{ \bigl\vert \mathcal{K}''(\mathfrak{\alpha }) \bigr\vert ^{q}, \bigl\vert \mathcal{K}''(\beta ) \bigr\vert ^{q} \bigr\} \bigr)^{\frac{1}{q}}. \end{aligned}$$

(16)

Theorem 16

Let $\mathfrak{\alpha },\beta ,\omega \in \mathbb{R}$ with $\mathfrak{\alpha }<\beta$, and $\rho , \lambda >0$. Suppose that $\mathcal{K}:[\mathfrak{\alpha },\beta ]\to \mathbb{R}$ is a twice differentiable function on $(\mathfrak{\alpha },\beta )$ and $\mathcal{K}''\in L([\mathfrak{\alpha },\beta ])$. If, in addition, $|\mathcal{K}''|^{q}$ is an η-quasiconvex function on $[\mathfrak{\alpha },\beta ]$ for $q\geq 1$, then the following inequality for generalized fractional integrals holds:

$$\begin{aligned} & \biggl\vert \mathfrak{F}^{\sigma }_{\rho ,\lambda +1} \bigl[ \omega ( \beta -\mathfrak{\alpha })^{\rho } \bigr]\frac{\mathcal{K}( \mathfrak{\alpha })+\mathcal{K}(\beta )}{2}- \frac{1}{2(\beta - \mathfrak{\alpha })^{\lambda }} \bigl[ \bigl(\mathfrak{J}^{\sigma }_{ \rho ,\lambda ,\beta ^{-};\omega } \mathcal{K} \bigr) ( \mathfrak{\alpha })+ \bigl(\mathfrak{J}^{\sigma }_{\rho ,\lambda , \mathfrak{\alpha }^{+};\omega } \mathcal{K} \bigr) (\beta ) \bigr] \biggr\vert \\ &\quad\leq \frac{(\beta -\mathfrak{\alpha })^{2}}{4}\sum_{j=0}^{\infty } \frac{\sigma (j)(\lambda +j\rho )}{\varGamma (j\rho +\lambda +3)} \vert \omega \vert ^{j}(\beta -\mathfrak{\alpha })^{j\rho } \bigl[ \bigl(\mathbf{M} ^{\beta }_{\mathfrak{\alpha }}\bigl( \bigl\vert \mathcal{K}'' \bigr\vert ^{q},\eta \bigr) \bigr)^{ \frac{1}{q}}+ \bigl(\mathbf{N}^{\beta }_{\mathfrak{\alpha }} \bigl( \bigl\vert \mathcal{K}'' \bigr\vert ^{q},\eta \bigr) \bigr)^{\frac{1}{q}} \bigr]. \end{aligned}$$

Proof

Utilizing Lemma 9, the η-quasiconvexity of $|\mathcal{K}''|^{q}$ on $[\mathfrak{\alpha },\beta ]$, and the power mean inequality, the following inequalities are established:

$$\begin{aligned} & \biggl\vert \mathfrak{F}^{\sigma }_{\rho ,\lambda +1} \bigl[\omega ( \beta -\mathfrak{\alpha })^{\rho } \bigr]\frac{\mathcal{K}( \mathfrak{\alpha })+\mathcal{K}(\beta )}{2}- \frac{1}{2(\beta - \mathfrak{\alpha })^{\lambda }} \bigl[ \bigl(\mathfrak{J}^{\sigma }_{ \rho ,\lambda ,\beta ^{-};\omega } \mathcal{K} \bigr) ( \mathfrak{\alpha })+ \bigl(\mathfrak{J}^{\sigma }_{\rho ,\lambda , \mathfrak{\alpha }^{+};\omega } \mathcal{K} \bigr) (\beta ) \bigr] \biggr\vert \\ &\quad\leq \frac{(\beta -\mathfrak{\alpha })^{2}}{2}\sum_{j=0}^{\infty } \frac{\sigma (j)}{\varGamma (j\rho +\lambda +2)} \vert \omega \vert ^{j}(\beta - \mathfrak{\alpha })^{j\rho } \\ &\qquad{} \times \int _{0}^{1} \bigl\vert \tau - \tau ^{j\rho +\lambda +1} \bigr\vert \bigl[ \bigl\vert \mathcal{K}'' \bigl(\tau \mathfrak{\alpha }+(1-\tau )\beta \bigr) \bigr\vert + \bigl\vert \mathcal{K}''\bigl( \tau \beta +(1-\tau )\mathfrak{ \alpha }\bigr) \bigr\vert \bigr] \,d\tau \\ &\quad= \frac{( \beta -\mathfrak{\alpha })^{2}}{2}\sum_{j=0}^{\infty } \frac{\sigma (j)}{ \varGamma (j\rho +\lambda +2)} \vert \omega \vert ^{j}(\beta -\mathfrak{\alpha })^{j \rho } \\ &\qquad{} \times \int _{0}^{1} \bigl(\tau -\tau ^{j\rho +\lambda +1} \bigr) \bigl[ \bigl\vert \mathcal{K}''\bigl(\tau \mathfrak{\alpha }+(1- \tau )\beta \bigr) \bigr\vert + \bigl\vert \mathcal{K}''\bigl(\tau \beta +(1-\tau ) \mathfrak{ \alpha }\bigr) \bigr\vert \bigr] \,d\tau \\ &\quad\leq \frac{(\beta - \mathfrak{\alpha })^{2}}{2}\sum_{j=0}^{\infty } \frac{\sigma (j)}{ \varGamma (j\rho +\lambda +2)} \vert \omega \vert ^{j}(\beta -\mathfrak{\alpha })^{j \rho } \\ &\qquad{} \times \biggl[ \int _{0}^{1} \bigl(\tau -\tau ^{j\rho +\lambda +1} \bigr) \,d\tau \biggr]^{1-\frac{1}{q}} \biggl[ \biggl( \int _{0}^{1} \bigl(\tau - \tau ^{j\rho +\lambda +1} \bigr) \bigl\vert \mathcal{K}''\bigl(\tau \mathfrak{ \alpha }+(1-\tau )\beta \bigr) \bigr\vert ^{q} \,d\tau \biggr)^{ \frac{1}{q}} \\ &\qquad{} + \biggl( \int _{0}^{1} \bigl(\tau -\tau ^{j\rho +\lambda +1} \bigr) \bigl\vert \mathcal{K}''\bigl( \tau \beta +(1-\tau )\mathfrak{\alpha }\bigr) \bigr\vert ^{q} \,d\tau \biggr) ^{\frac{1}{q}} \biggr] \\ &\quad\leq \frac{(\beta -\mathfrak{\alpha })^{2}}{2}\sum_{j=0}^{\infty } \frac{ \sigma (j)}{\varGamma (j\rho +\lambda +2)} \vert \omega \vert ^{j}(\beta - \mathfrak{\alpha })^{j\rho } \biggl[\frac{\lambda +j\rho }{2(\lambda +j \rho +2)} \biggr]^{1-\frac{1}{q}} \\ &\qquad{} \times \biggl[ \biggl( \int _{0}^{1} \bigl(\tau -\tau ^{j\rho +\lambda +1} \bigr) \mathbf{M}^{\beta }_{\mathfrak{\alpha }}\bigl( \bigl\vert \mathcal{K}'' \bigr\vert ^{q},\eta \bigr) \,d \tau \biggr)^{\frac{1}{q}} \\ & \qquad{}+ \biggl( \int _{0}^{1} \bigl(\tau -\tau ^{j\rho +\lambda +1} \bigr) \mathbf{N}^{\beta }_{\mathfrak{\alpha }}\bigl( \bigl\vert \mathcal{K}'' \bigr\vert ^{q},\eta \bigr) \,d \tau \biggr)^{\frac{1}{q}} \biggr] \\ &\quad=\frac{(\beta - \mathfrak{\alpha })^{2}}{4}\sum_{j=0}^{\infty } \frac{\sigma (j)( \lambda +j\rho )}{\varGamma (j\rho +\lambda +3)} \vert \omega \vert ^{j}(\beta - \mathfrak{\alpha })^{j\rho } \\ &\qquad{} \times \bigl[ \bigl(\mathbf{M}^{\beta }_{\mathfrak{\alpha }}\bigl( \bigl\vert \mathcal{K}'' \bigr\vert ^{q},\eta \bigr) \bigr)^{\frac{1}{q}}+ \bigl(\mathbf{N}^{ \beta }_{\mathfrak{\alpha }}\bigl( \bigl\vert \mathcal{K}'' \bigr\vert ^{q},\eta \bigr) \bigr)^{ \frac{1}{q}} \bigr]. \end{aligned}$$

 □

Proceeding in a similar fashion, we get from Theorem 16 the succeeding drop out.

Corollary 17

Let $\mathfrak{\alpha },\beta \in \mathbb{R}$ with $\mathfrak{\alpha }<\beta$, and $\lambda >0$. Suppose that $\mathcal{K}:[\mathfrak{\alpha },\beta ]\to \mathbb{R}$ is a twice differentiable function on $(\mathfrak{\alpha },\beta )$ and $\mathcal{K}''\in L([\mathfrak{\alpha },\beta ])$. If, in addition, $|\mathcal{K}''|^{q}$ is an η-quasiconvex function on $[\mathfrak{\alpha },\beta ]$ for $q\geq 1$, then the following inequality for fractional integrals holds:

$$\begin{aligned} & \biggl\vert \frac{\mathcal{K}(\mathfrak{\alpha })+\mathcal{K}(\beta )}{2}-\frac{ \varGamma (\lambda +1)}{2(\beta -\mathfrak{\alpha })^{\lambda }} \bigl[ \mathcal{J}^{\lambda }_{\beta ^{-}}\mathcal{K}(\mathfrak{\alpha })+ \mathcal{J}^{\lambda }_{\mathfrak{\alpha }^{+}}\mathcal{K}(\beta ) \bigr] \biggr\vert \\ &\quad\leq \frac{(\beta -\mathfrak{\alpha })^{2}\lambda }{4( \lambda +1)(\lambda +2)} \bigl[ \bigl(\mathbf{M}^{\beta }_{ \mathfrak{\alpha }} \bigl( \bigl\vert \mathcal{K}'' \bigr\vert ^{q},\eta \bigr) \bigr)^{\frac{1}{q}}+ \bigl(\mathbf{N}^{\beta }_{\mathfrak{\alpha }} \bigl( \bigl\vert \mathcal{K}'' \bigr\vert ^{q}, \eta \bigr) \bigr)^{\frac{1}{q}} \bigr]. \end{aligned}$$

(17)

Remark 18

From Corollary 17, one gets

$$\begin{aligned} & \biggl\vert \frac{\mathcal{K}(\mathfrak{\alpha })+\mathcal{K}(\beta )}{2}-\frac{ \varGamma (\lambda +1)}{2(\beta -\mathfrak{\alpha })^{\lambda }} \bigl[ \mathcal{J}^{\lambda }_{\beta ^{-}}\mathcal{K}(\mathfrak{\alpha })+ \mathcal{J}^{\lambda }_{\mathfrak{\alpha }^{+}}\mathcal{K}(\beta ) \bigr] \biggr\vert \\ &\quad\leq \frac{(\beta -\mathfrak{\alpha })^{2}\lambda }{2( \lambda +1)(\lambda +2)} \bigl(\max \bigl\{ \bigl\vert \mathcal{K}''( \mathfrak{\alpha }) \bigr\vert ^{q}, \bigl\vert \mathcal{K}''(\beta ) \bigr\vert ^{q} \bigr\} \bigr)^{ \frac{1}{q}}. \end{aligned}$$

(18)

3 Applications to special means

We now apply inequalities (11), (16), and (18) to the following special means of distinct real numbers:

• 1. Arithmetic mean:

  $$ \mathcal{A}(u,v)=\frac{u+v}{2}. $$

• 2. Geometric mean:

  $$ \mathcal{G}(u,v)=\sqrt{uv},\quad u,v>0. $$

• 3. Harmonic mean:

  $$ \mathcal{H}(u,v)=\frac{2uv}{u+v}. $$

• 4. Logarithmic mean:

  $$ \mathcal{L}(u,v)=\frac{u-v}{\ln \vert u \vert -\ln \vert v \vert },\quad \vert u \vert \neq \vert v \vert , \text{ and } u, v\neq 0. $$

• 5. Generalized logarithmic mean:

  $$ \mathcal{L}_{m}(u,v)= \biggl[\frac{v^{m+1}-u^{m+1}}{(m+1)(v-u)} \biggr] ^{\frac{1}{m}},\quad m\in \mathbb{N}. $$

• 6. Identric mean:

  $$ \mathcal{I}(u,v)=\frac{1}{e} \biggl(\frac{v^{v}}{u^{u}} \biggr)^{ \frac{1}{v-u}}. $$

Proposition 19

Suppose $u,v\in \mathbb{R}$ with $u< v$ and $m\geq 2$. Then the following inequality holds:

$$ \bigl\vert \mathcal{A}\bigl(u^{m},v^{m}\bigr)- \mathcal{L}_{m}^{m}(u,v) \bigr\vert \leq \frac{(v-u)^{2}}{12})\max \bigl\{ \vert u \vert ^{m-2}, \vert v \vert ^{m-2} \bigr\} . $$

Proof

Let $\mathcal{K}(x)=x^{m}$. In this case, $|\mathcal{K}''(x)|=m(m-1)|x|^{m-2}$ which is quasiconvex on $[u,v]$. Now applying inequality (11), with $\lambda =1$, to the function $\mathcal{K}$, the desired result is established. □

Proposition 20

Suppose $u,v\in \mathbb{R}$ with $0< u< v$. Then the following inequality holds:

$$ \bigl\vert \mathcal{A}(u,v)-\mathcal{L}(u,v) \bigr\vert \leq \frac{(\ln v -\ln u)^{2}}{12}\max \{u, v \}. $$

Proof

The result follows by applying (11) to the function $\mathcal{K}(x)=e^{x}$ with $\lambda =1$. Since the function $|\mathcal{K}''(x)|=e^{x}$ is convex for all $x\in \mathbb{R}$, it is also quasiconvex. □

Proposition 21

Suppose $u,v\in \mathbb{R}$, $0< u< v$, and $q>1$ with $\frac{1}{p}+ \frac{1}{q}=1$. Then the following inequality holds:

$$ \bigl\vert \ln \mathcal{G}(u,v)-\ln \mathcal{I}(u,v) \bigr\vert \leq \frac{(v -u)^{2}}{2} \bigl[\mathcal{B}(p+1,p+1) \bigr]^{\frac{1}{p}} \biggl(\max \biggl\{ \frac{1}{u^{2q}}, \frac{1}{v^{2q}} \biggr\} \biggr) ^{\frac{1}{q}}. $$

Proof

Set $\lambda =1$ in inequality (16) and take $\mathcal{K}(x)=\ln x$. □

Proposition 22

Suppose $u,v\in \mathbb{R}$ with $0< u< v$ and $q\geq 1$. Then the following inequality holds:

$$ \bigl\vert \mathcal{H}^{-1}(u,v)-\mathcal{L}^{-1}(u,v) \bigr\vert \leq \frac{(v -u)^{2}}{12} \biggl[\max \biggl\{ \biggl( \frac{2}{u^{3}} \biggr) ^{q}, \biggl(\frac{2}{v^{3}} \biggr)^{q} \biggr\} \biggr]^{ \frac{1}{q}}. $$

Proof

The intended inequality is obtained by taking $\lambda =1$ in (18) and using the function $\mathcal{K}(x)=\frac{1}{x}$. □

4 Conclusion

The main contribution of this paper is to establish new inequalities of the Hermite–Hadamard kind for functions with second derivatives, involving generalized Riemann–Liouville fractional integrals introduced by Raina [10] and Agarwal et al. [1], via η-quasiconvexity. Applications to some special means are also provided. By taking $\omega =0$ and $\sigma (0)=1$, we extend some already known theorems to a larger class. To the best of our knowledge, the results obtained herein are novel, and we hope that they would trigger further interest in this direction. For more recent results around η-(quasi)convexity, we refer the interested reader to [2, 6,7,8].