Abstract
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.
1. Introduction
The 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]. The 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 [3–5].
Fractional integral inequalities are considered one of the important tools to study the behavior and properties of solutions of various fractional problems [6–14]. There are many interesting generalization of fractional derivatives as per need of practical problems or some theoretical approach, for example, Raina’s fractional integral operator, Caputo-Fabrizio fractional integral, and extended Caputo-Fabrizio fractional integral. For recent work on it, we refer [15–20].
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. -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. This 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.
The 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.
2. Preliminaries
Here, we present some basic definitions and known results.
Definition 1 (convex function). A function is said to be convex function if the following inequality holds:for and .
One of the novel generalization of convexity is -convexity introduced by M. R. Delavar and S. S. Dragomir in [24].
Definition 2. A function is said to be generalized convex function with respect to for appropriate iffor and
In [25], Zhang and Wan gave definition of -convex function as follows.
Definition 3. Let be a -convex set. A function is said to be -convex function ifholds, for all and .
In [26], the authors gave the definition of the generalized p-convex function as follows.
Definition 4. A function is said to be generalized p-convex function with respect to for appropriate iffor and .
For some important properties and results about generalized -convexity, see [26]. Moreover, in [26], the following Hermite–Hadamard type inequality for p-convex functions can be found.
Theorem 1. Let be generalized -convex function for with condition ; then, we obtain the inequality
In [27], the author introduced a class of functions defined formally bywhere , ( is the set of real numbers), and is a bounded sequence of positive real numbers.
Using (6), in [28], the authors defined the following left-sided and right-sided fractional integral operators, respectively:where , and is such that the integral on the right side exits.
It is easy to verify that and are bounded integral operators on if
In fact, for , we havewhere
The importance of these operators stems indeed from their generality. Many useful fractional integral operators can be obtained by specializing the coefficient . Let The right-hand side and left-hand side Riemann–Liouville fractional integral of order with are defined byrespectively, where is the Gamma function defined as dk.
Lemma 1 (see [29, 30]). Let , and be a sequence of nonnegative real numbers. Let be a differentiable mapping on with and . If , the following equality for the fractional integral operator holds:
3. Main Results
In this section, we establish new Hermite–Hadamard type inequalities for generalized -convex functions in terms of Raina’s fractional integral operators.
Theorem 2. Let be generalized -convex function and provided (.,.) is bounded from above on and with and . Then, following fractional integral inequality holds:where , for all and are bounds of .
Proof. From inequality (6), we havewhere are bounds of . Substitute and ; then, (6) can be written as
Multiplying both sides by , we obtain
Integrate over , we obtain
With the convenient change of the variable, we can observe that
Similarly, the second integral can be written as
Now, equation (17) becomeswhich is the left-hand side of inequality (13). To prove right-hand side of (13), using the Definition 4 of generalized p-convex function,
Multiplying both inequalities by and then adding, we obtain
Integrate over , we obtainwhere . This completes the proof.
Corollary 1. Let be generalized -convex function and provided (.,.) is bounded from above on and with and . Then, the following inequality holds:where are bounds of .
Proof. By taking , , , and , we obtain
Making the substitution in (13), we obtain (26).
Remark 1. In Corollary 1, if we take , , , , and , then we get Theorem 1.4 of [29, 31].
Theorem 3. Let be a differentiable function on with . If is a generalized p-convex function on , then the following inequality for fractional integral operator holds:where .
Proof. Using Lemma 1 and definition of generalized p-convexity of , we have
It is easy to verify that
Equation (27) becomes
Finally, we can write it aswhere , which is our required result.
Corollary 2. Let be a differentiable function on with . If is a generalized p-convex function on ; then, the following inequality for fractional integral operator holds:
Proof. By taking , , , and , we obtain
Making the substitution in (35), we obtain (31).
Remark 2. In Corollary 2, if we take , , , , and , then we obtain Theorem 1.5 in [29, 31].
Theorem 4. Let be a differentiable function on with . If is a generalized p-convex function on with for some fixed , then the following inequality for fractional integral operator holds:where , , andfor all .
Proof. Using Lemma 1, definition of generalized -convexity of , and Hölder inequality, we havewhich completes the proof.
Data Availability
All data required for this research are included within the paper.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Authors’ Contributions
Changyue Chen wrote the final version of this paper, verified the results, and arranged the funding for this paper, Muhammad Shoaib Saleem proposed the problem, proved the results, and supervised the work, and Muhammad Sajid Zahoor wrote the first version of the paper.
Acknowledgments
The authors are thankful to the University of Okara, Okara, Pakistan, for providing funds for this research. This work was funded by University of Okara, Okara, Pakistan.