Fractional Hermite–Jensen–Mercer Integral Inequalities with respect to Another Function and Application

Butt, Saad Ihsan; Umar, Muhammad; Khan, Khuram Ali; Kashuri, Artion; Emadifar, Homan

doi:https://doi.org/10.1155/2021/9260828

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Volume 2021 | Article ID 9260828 | https://doi.org/10.1155/2021/9260828

Fractional Hermite–Jensen–Mercer Integral Inequalities with respect to Another Function and Application

Saad Ihsan Butt,¹Muhammad Umar,²Khuram Ali Khan,³Artion Kashuri,⁴and Homan Emadifar⁵

Academic Editor: Muhammad Imran Asjad

Received18 Jun 2021

Accepted18 Aug 2021

Published06 Sept 2021

Abstract

In this paper, authors prove new variants of Hermite–Jensen–Mercer type inequalities using –Riemann–Liouville fractional integrals with respect to another function via convexity. We establish generalized identities involving –Riemann–Liouville fractional integral pertaining first and twice differentiable convex function , and these will be used to derive novel estimates for some fractional Hermite–Jensen–Mercer type inequalities. Some known results are recaptured from our results as special cases. Finally, an application from our results using the modified Bessel function of the first kind is established as well.

1. Introduction and Preliminaries

The theory of fractional integrals and derivatives has occurred in many fields and directions such as partial differential equations, difference equations, probability, and stochastic processes (see [1–6]). Behind it, the theory of convex functions with integral inequalities is also useful.

Definition 1. A function is said to be convex on ifholds for every and .
One of the best-known inequalities for convex functions is the following Hermite–Hadamard’s inequality: if is a convex function in , where and , thenIt is also known as classical inequality. A number of mathematicians in the field of applied and pure mathematics have dedicated their efforts to extend, generalize, counterpart, and refine Hermite–Hadamard’s inequality for different classes of convex functions. For more recent results obtained on inequality , we refer the reader to references [7–10].
Let be nonnegative weights such that . The Jensen inequality states that, if is convex function on , thenholds for all and all , see [11].
In the literature, Jensen’s inequality and Hermite–Hadamard’s inequality are highly familiar results pertaining convex functions. One of the well-known and most significant inequalities in mathematical analysis is Jensen’s and related inequalities. Jensen’s inequality for differentiable convex functions plays a significant role in the field of inequalities as several other inequalities can be seen as special cases of it. It is used in order to make claims regarding the function while just a little is known or is needed to be known about the distribution. Furthermore, this inequality has been used in various areas of sciences and technology to solve several problems, such as engineering, mathematical statistics, financial economics, and computer science. Some recent results can be seen in [12–14].
Jensen’s inequality has a following variant gave by Mercer (see [15]).

Theorem 1. Let be a convex function on , thenholds for all and all .

Jensen–Mercer’s type inequality is a topic of supreme interest as it gives more information with explicit boundary conditions. It is quite effective for applications in operator analysis in higher dimensions [16–18]. Moradi et al. established some new improvements and generalization of Jensen–Mercer’s type inequalities [19]. Recently, in [20], Adil et al. gave applications of Jensen–Mercer’s inequality in information theory. They computed new estimates for Csiszár and related divergences. Taking into consideration the wonderful packages of Jensen’s and associated inequalities in diverse fields of mathematics and engineering sciences, their generalizations and upgrades were a subject of an excellent hobby for the researchers in the last few years as obvious from a massive variety of investigation on it (see [21–24]).

In [25], Vanterler da Costa Sousa and Capelas de Oliveira introduced –fractional integrals and –Hilfer fractional derivative with respect to another function. They also studied Gronwall inequalities using –Hilfer operator (see [26]).

Definition 2 (see [23]). Suppose that and . Also let be an increasing and positive monotone function on , having a continuous derivative on . Then, the left-sided and right-sided –Riemann–Liouville fractional integrals of a function with respect to another function on are defined as follows:respectively.
If we choose and , then we get, respectively, Riemann–Liouville and Hadamard fractional integrals.
Motivated by previous results, we will establish several new Hermite–Hadamard–Mercer type inequalities involving –Riemann–Liouville fractional integrals (i.e., Riemann–Liouville fractional integral of any function with respect to another function). Moreover, our results recover several known results. Finally, an application using the modified Bessel function of the first kind will be established as well.

2. Hermite–Jensen–Mercer Type Inequalities

Throughout the paper, the following assumption will be used in the sequel.

: Let , be a positive function and . Also suppose, is an increasing and positive monotone function on , having a continuous derivative on and .

Theorem 2. If is satisfied and is a convex function on , thenfor all , where is the gamma function.

Proof. Using Jensen–Mercer’s inequality, we havefor all .
Now, by change of variables and , for all and in (8), we getMultiplying the above inequality by on both sides and integrating with respect to on , we obtainwhereNow, let , then . Using the above equality, we obtainSo, the final form will be of this type as follows:and so the first inequality of (6) is proved.
Regarding the second inequality of (6), since is convex function, then for , we haveMultiplying the above inequality by on both sides and integrating with respect to on , we getLet and . Then, we haveMultiplying by , we will getAdding both sides in (17), we obtain our second inequality of (6).
To prove the first inequality of (7) by using the convexity of , we havefor all . By change of variables and , we getMultiplying the above inequality by on both sides and integrating with respect to over , we getHence, by change of variables, we obtainand so the first inequality of (7) is proved.
About the second inequality of (7), since is convex function, then for , we obtainBy adding inequalities (22) and (23), we haveMultiplying the above inequality by on both sides and integrating with respect to on , we getMultiplying by , we will getFrom inequalities (21) and (26), we get the desired double inequality (7).

Remark 1. Taking in Theorem 2, we will get Theorem 2 proved in [27].

Remark 2. Taking and in Theorem 2, we will obtain Theorem 2 proved by Kian and Moslehian in [28].

Theorem 3. If is satisfied and is a convex function on , thenfor all .

Proof. About the first inequality (27) by using the convexity of , we havefor all . By change of variables and we getMultiplying the above inequality by on both sides and integrating with respect to over , we haveHence, by change of variables, we obtainwhich proved the first inequality of (27).
Regarding the second inequality of (27), since is convex function, then for , we haveBy adding inequalities (32) and (33), we getMultiplying the above inequality by on both sides and integrating with respect to on , we obtainThen, we have the following inequality:Multiplying by , we will getFrom inequalities (31) and (37), we get the desired double inequality (27).

Remark 3. Taking in Theorem 3, we will get Theorem 3 proved in [27].

Remark 4. Taking and in Theorem 3, we will obtain Theorem 2.1 proved by Kian and Moslehian in [28].

Theorem 4. If is satisfied and is a convex function on , thenfor all .

Proof. Regarding the first part of inequality (38) by using the convexity of , we havefor all . By change of variables and we getMultiplying the above inequality by on both sides and integrating with respect to , we obtainHence, by change of variables, we havewhich concludes the first inequality of (38).
About the second inequality of (38), since is convex function, then for , we getBy adding inequalities (43) and (44), we haveMultiplying the above inequality by on both sides and integrating with respect to over , we obtainThen, we have the following inequality:Multiplying by , we will getSo, the second inequality of (38) holds.

Remark 5. Taking in Theorem 4, we will get Theorem 2 proved in [29].

Remark 6. Taking and in Theorem 4, we will obtain Theorem 2.1 proved by Kian and Moslehian in [28].

3. New Generalized Identities and Their Integral Inequalities

In this section, the following lemmas will play a basic role in our next results.

Lemma 1. If is satisfied and is a differentiable function on , thenfor all .

Proof. It suffices to note thatwhereSubstituting (51) and (52) in (50), we get the desired equality (49).

Remark 7. For and in Lemma 1, we will get Lemma 3.1 proved in [30].

Theorem 5. If is satisfied and is a convex function on , thenfor all .

Proof. Here, we will use Lemma 1, properties of modulus, and Jensen–Mercer’s inequality.
For every , we have .
Let , and then . So, we getwhereSubstituting (55) and (56) in (54), we get (53).

Remark 8. For and in Theorem 5, we will get Theorem 3.4 proved in [30].

Remark 9. Taking in Theorem 5, we will get Theorem 4 proved in [27].

Lemma 2. If is satisfied and is a differentiable function on , then

Proof. It suffices to note thatwhereSubstituting (59) and (60) in (58), we get (57).

Remark 10. Taking in Lemma 2, we will get Lemma 1 proved in [29].

Theorem 6. If is satisfied and is a convex function on , thenfor all .

Proof. From Lemma 2 and using mean value theorem for , we havewhere . This leads us to

Remark 11. For in Theorem 6, we will get Theorem 3 proved in [29].

Theorem 7. If is satisfied and is a convex function on , thenfor all .

Proof. By using Lemma 2, properties of modulus, and Jensen–Mercer inequality, we haveand after integration, we get required result.

Remark 12. For in Theorem 7, we will get Theorem 4 proved in [29].

Lemma 3. If is satisfied and is a differentiable function on , then

Proof. See the proof of Lemma 2.

Remark 13. For , , and in Lemma 3, we will get Lemma 3 proved in [31].

Remark 14. For in Lemma 3, we will get Lemma 2 proved in [27].

Theorem 8. If is satisfied and is a convex function on , thenfor all .

Proof. By using Lemma 3, properties of modulus, and Jensen–Mercer inequality, we haveAfter integration, we get required result.

Remark 15. Taking in Theorem 8, we will get Theorem 5 proved in [27].

Theorem 9. If is satisfied and is convex function, thenwhere and for all .

Proof. Applying Lemma 3, Hölder and Jensen–Mercer inequalities, the fact that is convex function, and properties of modulus, we haveAfter further simplifications, we get required result.

Remark 16. For in Theorem 9, we will get Theorem 6 proved in [27].

Lemma 4. If is satisfied and is a twice differentiable function on , then

Proof. It suffices to note thatwhereSubstituting (73) and (74) in (72), we get (71).

Corollary 1. If we set and , we get

Remark 17. If we set in Lemma 4, we get Lemma 2 of [29].
Moreover, if we set and , we obtain Lemma 1 of [32].

Remark 18. For , , , and in Lemma 4, it reduces to Lemma 2 proved in [32].

Theorem 10. If is satisfied and is a convex function on , then

Proof. By using Lemma 4, properties of modulus, and Jensen–Mercer inequality, we haveand after integration, we get required result.

Corollary 2. If we set and in Theorem 10, we get

Remark 19. If we set in Theorem 10, we obtain Theorem 5 of [29].
Moreover, if we set and , we get Theorem 5 of [32].

Corollary 3. If we set , , , and in Theorem 10, we get Proposition 1 of [33]:

Theorem 11. If is satisfied and is convex function, thenwhere and for all .

Proof. From Lemma 4, Hölder and Jensen–Mercer inequalities, the fact that is convex function, and properties of modulus, we have

Remark 20. If we set in Theorem 11, we get Theorem 6 of [29].

Lemma 5. If satisfied and is a twice differentiable function on , then

Proof. It suffices to note thatwhereSubstituting (84) and (85) in (83), we get (82).

Corollary 4. If we set and , we get

Corollary 5. If we set , we getMoreover, if we set and , we get

Remark 21. By using Lemma 5, we can get the same results of Theorems 10 and 11, so we omit their proof here.

Lemma 6. If is satisfied and is a twice differentiable function on , then

Proof. It suffices to note thatwhereSubstituting (91) and (92) in (90), we get (89).

Corollary 6. If we set and in Lemma 6, we get

Corollary 7. If we set in Lemma 6, we getMoreover, if we set and , we obtain Lemma 2.1 of [34] for .

Theorem 12. If is satisfied and is a convex function on , then

Proof. By using Lemma 6, properties of modulus, and Jensen–Mercer inequality, we haveand after integration, we get required result.

Corollary 8. If we set and in Theorem 12, we get

Corollary 9. If we set in Theorem 12, we get

Remark 22. If we set , , and in Theorem 12, we get Theorem 2.1 of [34].
Moreover, if we set , we obtain Proposition 1 of [33].

Theorem 13. If is satisfied and is convex function, thenwhere and for all .

Proof. Applying Lemma 6, Hölder and Jensen–Mercer inequalities, the fact that is convex function, and properties of modulus, we have

Corollary 10. If we set and in Theorem 12, we get

Corollary 11. If we set in Theorem 12, we get

Theorem 14. If is satisfied and is convex function, thenwhere for all .

Proof. From Lemma 2, power-mean and Jensen–Mercer inequalities, the fact that is convex function, and properties of modulus, we have

Remark 23. If we set , , and in Theorem 14, we get Theorem 2.2 of [34] for .
Moreover, if we set , we obtain Proposition 5 of [33].

4. Application

In this last section, we will give an application of our results using modified Bessel function of the first kind.

Let the function be defined by

For this, we recall the modified Bessel function of the first kind which is defined as follows [35]:

The first and the th order derivative formula of is, respectively, given by the following [36]:where is the hypergeometric function defined by the following [36]:and for some parameter , the Pochhammer symbol is defined as

Proposition 1. Let be real numbers and , then

Proof. Let . Note that the function is convex on the interval for each . Using Corollary 3 and relations (107) and (108), we obtain the desired inequality (111).

Remark 24. Using the same technique like Proposition 1, we can obtain some new interesting inequalities pertaining modified Bessel function of the first kind or the well-known –digamma function for from our generic results. We omit here their proofs, and the details are left to the interested reader.

5. Conclusion

In this article, some new Hermite–Jensen–Mercer type inequalities involving –Riemann–Liouville fractional integrals are found. Several –Riemann–Liouville fractional integral inequalities using identities as auxiliary results are provided, and the known results are recaptured as special cases as well. Finally, the efficiency of our results is showed with an application via modified Bessel function of the first kind. We hope that current work using our idea and technique will attract the attention of researchers working in mathematical analysis and other related fields in pure and applied sciences.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

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Copyright © 2021 Saad Ihsan Butt 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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