Abstract

In the present paper, some fractional integral inequalities of Hermite–Hadamard type for functions whose derivatives are generalized -convex are established. Moreover, several particular cases are also discussed which can be deduced from our results. As special cases, one can obtain several new versions of the results of generalized -convexity for other various generalizations of convex functions.

1. Introduction

The theory of inequalities is in process of continuous development state, and inequalities have become very effectual and potent tools for analyzing a large number of problems in different branches of mathematics. In last decades, the theory of inequalities has drawn the attention of most of the researchers, motivated new research directions, and affected several features of mathematical analysis and applications [1–4]. Among a variety of inequalities, some inequalities such as Jensen, Hilbert, Hadamard, Hardy, Poincare, Sobolev, Opial, and Fejér have made a significant influence on numerous branches of mathematics [5–8]. One good strategy for investigating any convexity is first to discuss main inequalities such as Hermite–Hadamard and Fejér and then to derive fractional integral inequalities for this. The fractional inequalities for convex function are also important to calculate different means. So, it is always interesting to develop fractional integral inequalities for some generalized convexity. For recent results concerning fractional Hermite–Hadamard inequalities and for different generalizations of convexity, see [9–11] and references therein.

In 2019, some new inequalities based on harmonic log-convex functions have been studied by Baloch et al. in [12]. In the same year, Sarikaya and Alp [13] studied the Hermite–Hadamard–Fejér type integral inequalities for generalized convex functions via local fractional integrals. Kwun et al. [14] studied generalized Riemann–Liouville -fractional integrals associated with Ostrowski-type inequalities and error bounds of Hadamard inequalities. In [15], Kang et al. studied Hadamard and Fejér–Hadamard inequalities for extended generalized fractional integrals involving special functions. In [15], -convex functions and associated fractional Hadamard and Fejér–Hadamard inequalities via an extended generalized Mittag–Leffler function were studied. Simpson’s type inequalities for strongly convex functions in the second sense has been studied in [16] by Kermausuor.

In the present paper, we aim to study some Hermite–Hadamard-type fractional integral inequalities for functions whose derivatives are generalized -convex. Our results can be considered as generalization of many existing results as many existing results can be obtained directly from our results.

The paper is organized as follows. In Section 2, we give some basic definitions and known results that will help to prove our main results. In Section 3, we will develop Fejér and Hermite–Hadamard-type inequalities for generalized -convex function. The Section 4 is devoted for fractional integral inequalities for generalized -convex function. The results are concluded in Section 5.

2. Preliminaries

Throughout this research, assume be an interval. Also, be a bifunction, where . Before turning to main results, we first review the basic definitions.

Definition 1. (see [17]). A function is called -convex function iffor all and .
This definition was introduced in [17] and motivated by works carried out in [18, 19]; Delavar and Dragomir proved few basic results of -convex functions.

Definition 2. (see [20]). Let be a nonnegative function. A function is known as modified -convex iffor all and .
We now define certain convexity which is unification of generalized and modified -convexity.

Definition 3. (see [21]). Let be a function defined as above. A function is called generalized -convex function iffor all and .

Remark 1. It is worth pointing out that the Definition 3 can ofcourse be reduced to(1)-convexity for (2)-convexity for (3)Modified -convex function when we take (4)Convex function when we take and A convex function , with satisfies the following inequality:known as Hermite–Hadamard’s inequality for convex function.
In [21], Fejér generalized (4) as below.
Assume a convex function and a function which is nonnegative, integrable, and symmetric with respect to ; then,Furthermore, Pachpatte developed the following inequalities for product of convex functions, see [22].

Theorem 1. Let be convex functions on and . Then,whereFor establishing new fractional integral inequalities, the following Lemmas are required.

Lemma 1. (see [23]). Assume a function , differentiable on , , with . If , , , and , then

Lemma 2. (see [23]). For and , one has

3. Hermite–Hadamard and Fejér-Type Inequalities

Here, we present our main results.

Theorem 2. Assume a generalized -convex function , , with and . Then,

Proof. Since is a generalized -convex function, soSubstituting , , and in (11), we obtainBy integrating the above inequality, we haveNow, finally upon recalling (11) with and and integrating over , we haveOf course (11) and (14) yields (10).

Remark 2. By putting and in (10), we get classical Hermite–Hadamard inequality (4) for convex function.

Theorem 3. Assume two nonnegative generalized -convex functions and , , and such that , thenwhere

Proof. First, we notefor all . By using nonnegativity of and , we obtainOn integrating the above inequality over , we obtainThe above yields (15) for and as in (16) and (17).

Remark 3. If we take and in Theorem 3, then (15) reduces to (14).

Theorem 4. Assume a generalized -convex function , , with and and a function which is nonnegative, integrable, and symmetric, with respect to ; then,

Proof. Since is a generalized -convex function and is nonnegative, integrable, and symmetric, with respect to , we find that

Remark 4. In Theorem 4, if(1), h(t) = t, and , then (21) becomes second inequality in (3)(2) and h(t) = t; then, (21) becomes second inequality in (4).

4. Fractional Integral Inequalities

Theorem 5. Assume a function , differentiable on , , with , , , and . If for , is generalized -convex on , then

Proof. For , by using Lemma 3, generalized - convexity of and the noted Holder’s integral inequality, we obtainBy using Lemma 2, a direct calculation yieldsOn putting the above two inequalities in (24) and using Lemma 2, we obtain inequality (23) for .
For , by using Lemmas 3 and 2, we obtain