Hermite–Jensen–Mercer Type Inequalities for Caputo Fractional Derivatives

Zhao, Jinchao; Butt, Saad Ihsan; Nasir, Jamshed; Wang, Zhaobo; Tlili, Iskander

doi:https://doi.org/10.1155/2020/7061549

Journal of Function Spaces

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Abstract Introduction Conclusion Data Availability Conflicts of Interest Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2020 | Article ID 7061549 | https://doi.org/10.1155/2020/7061549

Hermite–Jensen–Mercer Type Inequalities for Caputo Fractional Derivatives

Jinchao Zhao,¹Saad Ihsan Butt,²Jamshed Nasir,²Zhaobo Wang,¹and Iskander Tlili^3,4

Academic Editor: Alberto Fiorenza

Received27 Dec 2019

Accepted11 Feb 2020

Published24 Mar 2020

Abstract

In this article, certain Hermite–Jensen–Mercer type inequalities are proved via Caputo fractional derivatives. We established some new inequalities involving Caputo fractional derivatives, such as Hermite–Jensen–Mercer type inequalities, for differentiable mapping whose derivatives in the absolute values are convex.

1. Introduction

In recent years, inequality theory attracts many researchers due to its applications in our daily life and within the mathematics [1–9]. Let and let be nonnegative weights such that . The Jensen inequality [10] states that h is a convex function on the interval ; then,where and all .

The Hermite–Hadamard inequality asserts that if a mapping is a convex function on J with , , then

The reverse direction in the above inequality holds when h is concave.

Theorem 1 (see [11]). If h is a convex function on , then

Inequality (3) is known as the Jensen–Mercer inequality. Recently, inequality (3) has been generalized, see ([12–15]). For more recent and related results connected with Jensen–Mercer inequality, see ([11, 16–18]).

The previous era of fractional calculus is as old as the history of differential calculus. Several fractional operators are introduced that generalize ordinary integrals. However, the fractional derivatives have some basic properties than the corresponding classical ones. On the contrary, besides the smooth requirement, the Caputo derivative does not coincide with the classical derivative [19]. Caputo fractional derivatives are introduced by the Italian mathematician Caputo in 1967. Since then, a lot of research involves Caputo fractional derivatives [20–22].

The Caputo fractional derivatives are defined as in [23–26].

Definition 1. Suppose and The Caputo fractional derivatives of order α are defined as follows:If and usual derivatives of h of order n exist, then the Caputo fractional derivatives coincide with
Specifically, we getwhere
In this article, by using the Jensen–Mercer inequality, we proved Hermite–Hadamard’s inequalities for fractional integrals and established some Hermite–Hadamard type inequalities for differentiable mappings whose derivatives in absolute value are convex.

2. Hermite–Hadamard–Mercer Type Inequalities for Caputo Fractional Derivatives

By using Jensen–Mercer inequalities, Hermite–Hadamard type inequalities can be expressed in Caputo fractional derivatives as follows.

Theorem 2. Suppose that a positive function with and . If is a convex function on , then the following inequalities for Caputo fractional derivatives hold:, and is the gamma function.

Proof. Using the Jensen–Mercer inequality, we getNow, by changing of variables and , and in (8), we getMultiplying both sides by and then integrating with respect to τ over we getAfter simplification, we getand so the first inequality of (2.1) is proved.
Now, for the proof of the second inequality of (2.1), we first note that if is a convex function, then for it givesMultiplying both sides by and then integrating the resulting inequality with respect to τ over we getMultiplying by and adding both sides in (13), we get the second inequality of (2.1), which completes the proof.

Theorem 3. Suppose that a positive function with and . If is a convex function on , then the following inequalities for Caputo fractional derivatives hold:where , and is the gamma function.

Proof. To prove the first part of inequality, we use convexity of to getfor all . Now by writing variables and for and , we getMultiplying both sides of (16) by and then integrating with respect to τ over we getFurther simplifying givesand so the first inequality of (14) is proved.
Now, for the proof of the second inequality of (14), we first note that if is a convex function, then by employing Jensen–Mercer inequality (3) for givesBy adding the inequalities of (19) and (20), we getMultiplying both sides by and then integrating over we getAfter simplification, we getNow, concatenating (18) and (23), we get (14).
Now, we introduce some new lemmas involving Caputo fractional derivatives.

Lemma 1. Suppose that is a differentiable mapping on with and , then the following equality for Caputo fractional derivatives holds:where , and is the gamma function.

Proof. It suffices to note thatwhereand replacing the values of and in (25), we get (24).

Remark 1. If we take and in Lemma 1, then it reduces to Remark 2.5 in [25].

Lemma 2. Suppose that is a differentiable mapping on with and , then the following equality for Caputo fractional derivatives holds:

Proof. It suffices to note that and combining (29) and (30) with (28), we get (27).

Remark 2. If we take and in Lemma 2, then it reduces to Lemma 2 in [24].

Theorem 4. Suppose that is a differentiable mapping on with and . If is convex function on then the following inequality for Caputo fractional derivatives holds:where , and is the gamma function.

Proof. By using Lemma 1 and the Jensen–Mercer inequality, we getwhereand putting the values of and in (32), we get (31).

Remark 3. If we take and in Theorem 4, then it reduces to Corollary 2.7 in [25].

Theorem 5. Suppose that is a differentiable mapping on with and . If is a convex function on , then the following inequality for Caputo fractional derivatives holds:where , and is the gamma function.

Proof. By using Lemma 2 and the Jensen–Mercer inequality, we getwhich completes the proof.

Theorem 6. Suppose that is a differentiable mapping on with and . If is a convex function on , then the following inequality for Caputo fractional derivatives holds:where , and is the gamma function.

Proof. By using Lemma 2 and applying the famous Hölder integral inequality, we getBy applying Minkowski’s inequality, we getwhich completes the proof.

3. New Hölder’s and Improved Iscan Inequalities

Theorem 7. Suppose that is a differentiable mapping on with and . If is a convex function on , then the following inequality for Caputo fractional derivatives holds:where , and is the gamma function.

Proof. By using Lemma 2 with Jensen–Mercer inequality and applying the Hölder–Iscan integral inequality (Theorem 1.4 [27]), we getBy the convexity of By using calculus tools, one can have required result.

Theorem 8. Suppose that is a differentiable mapping on with and . If is a convex function on , and then the following inequality for Caputo fractional derivatives holds:where is the gamma function.

Proof. By using Lemma 2 with Jensen–Mercer inequality and applying the improved power-mean integral inequality (Theorem 1.5 [27]), we getAfter some simplifications, we get required result.

4. Conclusion

In this paper, we presented Hermite–Hadamard–Mercer inequalities for convex functions via Caputo fractional derivatives. We also developed some new bounds using Hölder–Iscan and improved power-mean integral inequalities. Our results will attract attentions of many researchers working in the field of inequalities and enable them to think further for other generalized convex functions. One may think to extend these results for higher order convex functions. The works above can also build up for a convex function of two variables. For further directions, we refer to [28–33].

Data Availability

All data are included within this paper.

Conflicts of Interest

The authors of this paper declare that they have no conflicts of interest.

Acknowledgments

The research of Saad Ihsan Butt has been fully supported by the H.E.C. of Pakistan under NRPU project 7906.

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Copyright © 2020 Jinchao Zhao et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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