Inequalities Involving Conformable Approach for Exponentially Convex Functions and Their Applications

Abstract

In the article, we present several Hermite–Hadamard-type inequalities for the exponentially convex functions via conformable integrals. As applications, we give new inequalities for certain bivariate means.

1. Introduction

Recently, the fractional calculus has attracted the attention of several researchers [1–14]. The affect and motivation of the fractional calculus in both theoretical and applied science and engineering arose out substantially. Numerous studies are related with the discrete versions of the fractional calculus which benefit from countless applications in the theory of time scales, physics, different fields of engineering, chemistry, and so forth.

In the past decades, the subject of fractional integrals has attracted the attention for mathematicians working on inequality theory and convexity. Fractional integral operators are sometimes the gateway to physical problems that cannot be expressed by classical integral, sometimes for the solution of problems expressed in fractional order. In recent years, a lot of new operator definitions have been given, and the properties and structures of these operators have been examined. Some of these operators are very close to classical operators in terms of their characteristics and definitions. In general, they have nonlocality property and defined with singular kernels.

The derivatives in this calculus seemed complicated and lost some of the basic properties that usual derivatives have such as the product rule and the chain rule. However, the semigroup properties of these operators behave well in some cases. Recently, the authors in [15] defined a new well-behaved simple derivative called “conformable fractional derivative” which depends just on the basic limit definition of the derivative. It will define the derivative of higher order (i.e., order ) and also define the integral of order only. It will also prove the product rule and the mean value theorem and solve some (conformable) differential equations where the fractional exponential function plays an important rule.

Almost every mathematician knows the importance of convex sets and convex functions in many fields of mathematics, for example, in nonlinear programming and optimization theory. By using the concept of convexity, several integral inequalities have been introduced such as Jensen, Hermite–Hadamard, and Slater inequalities. But, the well-known one of them is the celebrated Hermite–Hadamard inequality.

Let be an interval and be a convex function. Then, the double inequalityholds for all with .

It is easy to see that if χ is concave on K, then one has the reverse of inequality (1). Many upper and lower bounds for the mean value of a convex function can be derived by use of inequality (1). Recently, the generalizations, improvements, refinements, extensions, and variants for Hermite–Hadamard inequality (1) can be found in the literature [16–19].

It is well known that convex functions have wide applications in pure and applied mathematics [20–35] and many other natural sciences [21–55]. The development in the history of convex function is the minimum of the differentiable convex functions that can be characterized by variational inequalities.

Motivated and inspired by the previous study, a comprehensive investigation of exponentially convex functions for conformable integral in the present paper is new. The class of exponentially convex functions was introduced by Antczak [56] and Dragomir and Gomm [57]. Inspired by these facts, Awan et al. [58] defined a new class of convexity, namely, the exponentially convex function. The growth of research on big data analysis, time space management, and deep learning has recently increased interest in information theory involving exponentially convex functions. The smoothness of exponentially convex functions is exploited for statistical learning, sequential prediction, and stochastic optimization.

Now, we recall and introduce some definitions for various convex functions.

Definition 1. A nonempty set is said to be convex if for all and .

Definition 2. Let be a real-valued function. Then, χ is said to be convex (concave) if the inequality holdswhich holds for all and . χ is said to be concave if is convex.

Definition 3. A positive real-valued function is said to be exponentially convex if the inequalityholds for all and .
It is well known that is the minimum of the differentiable exponentially convex functions χ if and only if it satisfiesfor all . Inequality (4) is known as the exponentially variational inequality.
The Riemann–Liouville fractional integral, conformable derivative, and conformable integral are very important in the theory of fractional calculus which are defined as follows.

Definition 4. Let and . Then, the Riemann–Liouville integral and of order α is defined byrespectively, where is the Euler gamma function [59–61].

Definition 5. Let and be a real-valued function. Then, the α-order conformable derivative of f at is defined byIf exists, then we defineAdditionally, if χ is differentiable, thenwhereWe write for to denote the α-order conformable derivative of χ at ξ. Additionally, we say that χ is α-differentiable if the α-order conformable derivative of χ exists.

Definition 6. Let and . Then, the real-valued function is said to be α integrable on if the integralexists and is finite.

Remark 1. Let . Then, it is well known thatwhere the integral is the classical Riemann improper integral.
Anderson [62] gave a variant of the Hermite–Hadamard inequality which is as follows.

Theorem 1. Let and be an α-differentiable function such that is monotone. Then, the double inequalityholds for all with .

Note that, if , then inequality (12) leads to the classical Hermite–Hadamard inequality (1).

The main purpose of the article is to present new Hermite–Hadamard-type inequalities for exponentially convex functions via the conformable integrals and find new inequalities for certain bivariate means.

2. Main Results

First of all, we provide an integral identity which can be proved easily by the readers.

Lemma 1. Let and be an α-fractional exponentially differentiable function on . The identityholds for any if .

Remark 2. Let . Then, identity (13) reduces to

Theorem 2. Let with , , and be an differentiable exponentially function on Then, the inequalityholds, where

Proof. It follows from Lemma 1 and the convexity of the functions , , and that

Remark 3. Let . Then, inequality (15) leads to

Theorem 3. Let with , , such that , and be an α-differentiable function on . Then, the inequalityholds if and is convex on , where

Proof. From (13) and (20)–(22), we clearly see thatMaking use of Hölder’s inequality, one hasSimilarly, we haveTherefore, inequality (19) follows from (24) and (25).

Remark 4. Let . Then, inequality (19) becomes

Theorem 4. Let with , , such that , and be an α-differentiable function on . Then, the inequalityholds if and is convex on , whereand and are given in Theorem 3.

Proof. It follows from Lemma 1 thatMaking use of power mean inequality, we getSimilarly, one hasFrom the convexity of , we getwhere we have used the facts thatTherefore, Theorem 4 follows from (30)–(33).

Remark 5. Let . Then, inequality (27) leads to

Theorem 5. Let with , , such that , and be an α-differentiable function on . Then, the inequalityholds if and is convex on , where

Proof. From Definition 3 and Lemma 1, we getMaking use of power mean inequality, we getSimilarly, we haveIt follows from the convexity of thatwhere we have used the facts thatTherefore, inequality (36) follows from (38)–(42).

Theorem 6. Let with , , such that , and be an differentiable function on . Then, the inequalityholds if and is concave on , where

Proof. We clearly see that is concave due to is concave for . From Definition 3, Lemma 1, the concavity of , and Jensen’s integral inequality, we getwhere we have used the identitiesTherefore, inequality (44) follows from (46)–(48).

3. Applications to Special Means

It is very important to give applications in terms of efficiency and usefulness of the obtained results. At the same time, the accuracy of the findings will be confirmed by the applications.

A bivariate function is said to be a mean if for all . Recently, the bivariate means have been the subject of intensive research [63–75]; in particular, many remarkable inequalities and properties for the bivariate means and their related special functions can be found in the literature [76–85].

In this section, we use the results obtained in Section 2 to derive some inequalities for the weighted arithmetic meanand -th generalized logarithmic meanfor with , and .

Let . Then, Theorems 2–4 and (50) and (51) lead to Theorems 7–9 immediately.

Theorem 7. Let with . Then, the inequalityholds for all , , and .

Theorem 8. Let with . Then, the inequalityholds for all , and .

Theorem 9. Let with . Then, the inequalityholds for all , , and , where and for are defined in Theorem 4.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

Acknowledgments

This work was supported by the Natural Science Foundation of China (Grant nos. 11971142, 61673169, 11871202, 11701176, 11626101, and 11601485) and the Natural Science Foundation of Zhejiang Province (Grant no. LY19A010012).