Journal of Function Spaces

Journal of Function Spaces / 2020 / Article

Research Article | Open Access

Volume 2020 |Article ID 5285147 | https://doi.org/10.1155/2020/5285147

Shuang-Shuang Zhou, Saima Rashid, Silvestru Sever Dragomir, Muhammad Amer Latif, Ahmet Ocak Akdemir, Jia-Bao Liu, "Some New Inequalities Involving κ-Fractional Integral for Certain Classes of Functions and Their Applications", Journal of Function Spaces, vol. 2020, Article ID 5285147, 14 pages, 2020. https://doi.org/10.1155/2020/5285147

Some New Inequalities Involving κ-Fractional Integral for Certain Classes of Functions and Their Applications

Academic Editor: Alberto Fiorenza
Received18 Jul 2019
Accepted21 Nov 2019
Published29 Apr 2020

Abstract

In this article, we present several new inequalities involving the κ-fractional integral for the integrable function which satisfies one of the following conditions: is preinvex for some ; is bounded; is a Lipschitz function. As applications, we establish new inequalities for the weighted arithmetic and generalized logarithmic means.

1. Introduction

Let be a nonempty interval. Then, a real-valued function is said to be convex (concave) on E if the inequalitytakes place for any and .

We all know that the convexity theory has penetrated into every branch of pure and applied mathematics [120], and it has more and more practical applications in physics, mechanics, statistics, operations research, and even in economics and meteorology [2140]. Many remarkable inequalities in mathematics, control theory, and game theory can be found in the literature [4160] by use of the convexity theory. In the past half century, to research the generalizations and variants for the convexity has always been a hot topic for mathematicians and physicists as well as engineers. Recently, a great deal of generalizations and variants has been made for the convexity, for example, the GA-convexity and GG-convexity [61], s-convexity [62, 63], preinvex convexity [64], strong convexity [6568], and Schur convexity [69].

When we talk about convex functions, we have to mention a classical and most important inequality, which is the well-known Hermite–Hadamard inequality [70] which states that the double inequalityholds for all with if is a convex (concave) function on J and is a nonempty interval. For a long time, numerous researchers have been devoted to the generalizations, improvements, refinements, and variations for inequality (2) [7173].

The aim of this article is to provide new Hermite–Hadamard-type inequalities for certain classes of functions via the κ-fractional integral and give their applications to the bivariate means.

In order to clearly describe and prove our main results in the next sections, we have to recall some definitions which we present in this section.

Definition 1. Let be a nonempty set and be a mapping. Then, is said to be an invex set with respect to the mapping η iffor all and .

Definition 2. Let be an invex set with respect to the mapping . Then, the mapping is said to be preinvex with respect to the mapping η if the inequalityholds for all and .

Definition 3. (see [74]). Let , with , and . Then, the β order κ-fractional integral operators and of are defined byrespectively, whereis the κ-gamma function.

2. Main Results

Throughout this section, we always assume that is the set of positive integers, , , is an open invex set with respect to the mapping , with , and is a differentiable mapping such that is integrable on for , and

Lemma 1. Let be defined by (7). Then, we have the identity

Proof. Making use of integration by parts and variable transformation, one hasAnalogously, we also haveTherefore, identity (8) follows from multiplying (9) and (10) by , multiplying (11) and (12) by , and then adding them.

Remark 1. Lemma 1 leads to the conclusions as follows:(i)Let . Then, Lemma 1 leads to Lemma 2.1 of [75].(ii)Let . Then, one has(iii)If and , then (13) reduces to(iv)If and , then (13) becomes(v)If and , then Corollary 2.1 of [75] can be derived from Lemma 1.

Theorem 1. Let such that and be preinvex on . Then, the inequalityholds for all , where

Proof. It follows from Lemma 1 and the preinvexity of together with the Hölder’s inequality thatNote thatAnalogously, we haveWe clearly see thatfor ,for , andfor .
Therefore, inequality (16) can be derived from the above inequalities and identities.

Remark 2. Theorem 1 leads to the conclusion as follows:(i)Theorem 2.1 of [75] can be obtained from Theorem 1 if we take .(ii)If and , Theorem 1 reduces to(iii)Let . Then, (24) leads to Corollary 2.2 of [75].

Theorem 2. Let and be preinvex on Ω. Then, the inequalityholds for , where

Proof. It follows from Lemma 1 and the preinvexity of together with the power-mean inequality thatwhereSubstituting the above inequalities in (27), we get inequality (25).

Remark 3. From Theorem 2, we have two conclusions as follows:(i)If , then we get Theorem 2.2 of [76].(ii)If and , then

Theorem 3. If with , and for all , then we have

Proof. It follows from Lemma 1 thatMaking use of the fact thatone hasSimilarly, we haveTherefore,which completes the proof of Theorem 3.

Remark 4. From Theorem 3, we get two conclusions as follows:(i)Let . Then, Theorem 3.1 of [75] can be derived from Theorem 3.(ii)If and , then Theorem 3 leads to(iii)If , then (36) becomes Corollary 3.1 of [75].(iv)If , then (36) leads to(v)If , then (36) leads to the conclusion that(vi)If , then (36) gives

Theorem 4. If is a Lipschitz function on Ω with the Lipschitz constant , then the fractional integral inequalityholds for .

Proof. It follows from Lemma 1 thatSince is a Lipschitz function on with Lipschitz constant , we getSimilarly, we haveTherefore, one has