Some Inequalities of Generalized p-Convex Functions concerning Raina’s Fractional Integral Operators

Convex functions play an important role in pure and applied mathematics specially in optimization theory. In this paper, we will deal with well-known class of convex functions named as generalized p-convex functions. We develop Hermite–Hadamard-type inequalities for this class of convex function via Raina’s fractional integral operator.


Introduction
e subject of fractional calculus got rapid development in the last few decades. As a matter of fact, fractional calculus give more accuracy to model applied problems in engineering and other sciences then classical calculus. In order to model recent complicated problems, scientists are using fractional inequalities and fractional equations. For more on this, we refer the books [1,2]. e models with fractional calculus have been applied successfully in ecology, aerodynamics, physics, biochemistry, environmental science, and many other branches. For more about fractional calculus and models, we refer [3][4][5].
Convex functions also play an important role in pure and applied mathematics specially in optimization theory. Classical convexity does not fulfil needs of modern mathematics; therefore, several generalizations of convex functions are presented in literature. s-convex function [21], M-convex functions [22], and h-convex function [23] are some examples of generalized convex functions. It is always interesting to study properties of some generalized convex function in the setting of fractional integral operators. is paper is an effort in this direction. In this paper, we study the p-convex functions and present some of its properties in the setting of Raina's fractional integral operators. e paper is organized as follows. In Section 2, we present some basic definition and properties of Raina's fractional integral operator. Section 3 is devoted for Hermite-Hadamard type inequalities for generalized p-convex functions in terms of Raina's fractional integral operators.
One of the novel generalization of convexity is η-convexity introduced by M. R. Delavar and S. S. Dragomir in [24]. Definition 2. A function ϕ: I ⟶ R is said to be generalized convex function with respect to η: for ∀x, y ∈ I and ϑ ∈ [0, 1] In [25], Zhang and Wan gave definition of p-convex function as follows.
In [26], the authors gave the definition of the generalized p-convex function as follows.
Definition 4. A function ϕ: I ⟶ R is said to be generalized p-convex function with respect to η: for ∀x, y ∈ I, p > 0 and ϑ ∈ [0, 1].

Main Results
In this section, we establish new Hermite-Hadamard type inequalities for generalized p-convex functions in terms of Raina's fractional integral operators.
Proof. From inequality (6), we have where N η are bounds of ϕ. Substitute x P � ϑξ

Journal of Mathematics
Integrate over ϑ ∈ [0, 1], we obtain 2ϕ ξ With the convenient change of the variable, we can observe that Similarly, the second integral can be written as Now, equation (17) becomes 4 Journal of Mathematics which is the left-hand side of inequality (13). To prove righthand side of (13), using the Definition 4 of generalized pconvex function, Multiplying both inequalities by and then adding, we obtain Integrate over ϑ ∈ [0, 1], we obtain where σ 1 (k) � σ(k)(kρ + λ). is completes the proof.