New Weighted Hermite–Hadamard Type Inequalities for Differentiable <span class="nowrap"><svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.2723999pt" id="M1" height="12.533pt" version="1.1" viewBox="-0.0657574 -12.2606 9.01926 12.533" width="9.01926pt"><g transform="matrix(.017,0,0,-0.017,0,0)"><path id="g113-105" d="M499 88L487 113C459 86 422 65 413 65C403 65 403 74 409 98L461 318C487 426 460 448 431 448C408 448 383 441 352 424C305 398 223 344 157 257H155L243 674C249 702 249 712 240 712C227 712 170 680 87 673L82 648H117C155 648 162 644 150 590L23 -4L30 -12C55 -4 81 3 105 8C113 53 131 150 143 187C199 281 323 387 372 387C390 387 393 369 380 311L328 86C313 19 319 -12 347 -12C379 -12 442 24 499 88Z"/></g></svg>-</span>Convex and Quasi <span class="nowrap"><svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.2723999pt" id="M2" height="12.533pt" version="1.1" viewBox="-0.0657574 -12.2606 9.01926 12.533" width="9.01926pt"><g transform="matrix(.017,0,0,-0.017,0,0)"><use xlink:href="#g113-105"/></g></svg>-</span>Convex Mappings

Latif, Muhammad Amer

doi:https://doi.org/10.1155/2021/4495588

Journal of Mathematics

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Abstract Introduction Applications Conclusion Data Availability Conflicts of Interest Authors’ Contributions Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2021 | Article ID 4495588 | https://doi.org/10.1155/2021/4495588

New Weighted Hermite–Hadamard Type Inequalities for Differentiable -Convex and Quasi -Convex Mappings

Muhammad Amer Latif¹

Academic Editor: Xiaolong Qin

Received23 Apr 2021

Accepted16 Jun 2021

Published08 Jul 2021

Abstract

In this paper, new weighted Hermite–Hadamard type inequalities for differentiable -convex and quasi -convex functions are proved. These results generalize many results proved in earlier works for these classes of functions. Applications of some of our results to -divergence and to statistics are given.

1. Introduction

The theory of convex functions is based on convex functions stated as follows.

A function is said to be convex on a convex set if the inequality given as follows:holds for all , and . If (1) holds in reverse direction, then is said to be concave.

The inequality which can be considered as the necessary and sufficient condition of a function to be convex on is given by [1]where , with .

Inequality (2) is known as Hermite–Hadamard inequality, and it holds in reversed direction if the function is concave on .

Over the past three decades, the definition of convex functions and inequality (2) has been subjected to immense research. The definition of convex functions has been modified in various forms, and hence a number of different weighted and nonweighted forms of inequality (2) have been obtained by many researchers.

Kirmachi [2] obtained the following estimate for .

Theorem 1. (see [2]). Let be a differentiable mapping on ; let with and . If is convex on , thenPearce and Pecaric [3] improved this estimate by proving the following result.

Theorem 2. (see [3]). Let be a differentiable mapping on , let with , and let . If and is convex on , thenThe weighted version of the results in Theorems 1 and 2 was obtained in [4].

Theorem 3. (see [4]). Let be a differentiable mapping on , let with , and let be a continuous positive mapping symmetric with respect to . If and is convex on , thenwhere and .

Theorem 4. (see [4]). Let be a differentiable mapping on , let with , and let be a continuous positive mapping symmetric with respect to . If and is convex on for , thenwhere and are as defined in Theorem 3.
Under the assumptions of Theorem 2, a bound of was proposed by Pearce and Pecaric in [3].

Theorem 5. (see [3]). Let be a differentiable mapping on , let with , and let . If and is convex on , thenThe bound of the result of Theorem 5 in weighed form was given by Hwang in [5].

Theorem 6. (see [5]). Let be a differentiable mapping on , let with , and let be a continuous positive mapping symmetric with respect to . If and is convex on for , thenwhere and .
The concept of quasiconvex functions generalizes the concept of convex functions.

Definition 1. (see [6]). A function is said quasiconvex on ifholds for all and .
There are quasiconvex functions which are not convex functions (see, for example, [6]).
Alomari et al. [7] obtained the bound of the result of Theorem 5 by using the quasiconvexity of the differentiable mappings.

Theorem 7. (see [7]). Let be a differentiable mapping on , and let with . If and is quasiconvex on , thenA general form of the result of Theorem 7 has been proved by Hwang in [5].

Theorem 8. (see [5]). Under the assumptions of Theorem 6, if is quasiconvex on , thenwhere and are as defined in Theorem 6.
Gavrea [8] extended inequality (10) to weighted form and generalized inequalities (5) and (6) in such a way that the weight function is not necessarily symmetric with respect to the midpoint .
Varošanec [9] generalized the concept of convex functions by giving the concept of -convex functions.

Definition 2. Let and be intervals in with and be a nonnegative function, where . A is an -convex function or that belongs to the class if is nonnegative, and for all , , the inequalityholds. If inequality (12) is reversed, then is said to be -concave or is said to belong to the class .
The class of -convex functions contains all nonnegative convex functions, -convex functions in the second sense [10], Godunova–Levin functions [11], -Godunova–Levin type, -convex, and -functions [12] as special cases.
Inspired by the research towards this direction, the main objectives of this paper are to introduce the notion of quasi -convex functions and to acquire new weighted Hermite–Hadamard type inequalities for -convex and quasi -convex mappings. The results of this paper generalize the results of Gavrea [8] and in particular contain the results for all nonnegative convex functions, -convex functions, Godunova–Levin functions, -Godunova–Levin functions, -convex, quasi convex functions, and -functions.
In Section 2, we recall some integral identities for a differentiable mapping and a symmetric function with respect to defined over an interval . In Section 2, an important inequality for positive linear functional on and an -convex function is proved to obtain some very stimulating results of this manuscript. Section 3 contains some new weighted Hermite–Hadamard type integral inequalities related with the left and right parts of Hermite–Hadamard inequalities (2). The results of Section 3 provide weighted generalization of a number of results proved so far in the field of mathematical inequalities for differentiable -convex and quasi -convex functions [13–22].

2. Some Auxiliary Results

The following notations and results have been used in [8].

Let be a continuous function withand the integral is denoted by , that is,

In case, when is symmetric with respect to , that is, ifthen the following result holds.

Lemma 1. (see [8]). If is symmetric with respect to , thenNow, we introduce the notion of the quasi -convex functions as follows.

Definition 3. Let and be intervals in with and be a nonnegative function, where . A is an quasi -convex function, or that belongs to the class if is nonnegative, and for all , , the inequalityholds, where and . If inequality (17) is reversed, then is said to be quasi -concave or is said to belong to the class .

Example 1. Consider the function defined asThen, is quasi -convex but not -convex on .
Now onwards, we suppose that and .

Lemma 2. Let be a differentiable mapping on and , where . Let be a continuous mapping and be a real nonnegative function, such that . Then,where.

Proof. The following identities hold:where is the Heavyside function defined byMultiplying both sides of (21) with and integrating over , we haveSimilarly, multiplying both sides of (22) with and integrating over , we also haveFrom (24) and (25), we getIn the last identity, we set for the first integral and for the second integral, and we obtain (19).

Remark 1. If we take in Lemma 2, then we get the result for nonnegative convex functions similar to that of (see page 94 in Lemma 2.2. in [8]).

Corollary 1. If we take , for all , then (19) reduces towhere

Proof. We know thatHence,

Corollary 2. If the function is symmetric with respect to on , thenwhere

Proof. Since the function is symmetric with respect to on , we haveMoreover,Hence, from (19), we get the required identity (31).
Now, we will discuss some cases for Lemma 2.(1)If , then we have the result for -functions.

Corollary 3. Under the assumptions of Corollary 1, if is -function on , then(2)If , then we obtain the following result for -convex functions.

Corollary 4. Suppose that the conditions of Corollary 1 are fulfilled and if is -convex function on , then where(3)If , , and , then we obtain the result for function of -Godunova–Levin type.

Corollary 5. Under the assumptions of Corollary 1, if is function of -Godunova–Levin type on , then where(4)If , , then we obtain the result for -convex functions.

Corollary 6. Under the assumptions of Corollary 1, if is result for -convex functions on , then

Lemma 3. Let be a positive linear functional on , and let be monomials , , . Let be a -convex function on , then

Proof. By using the -convexity of on and the given equalitywe getSince is a positive linear functional, we get inequality (41) by applying on both sides of (43).

3. Main Results

The following theorem generalizes the result given by Gavrea in [8].

Theorem 9. Let be a differentiable mapping on and , where . If is a continuous mapping and is -convex on , then the following inequality holds:where

Proof. We can writeFrom (46), we obtainTaking absolute value on both sides of (47) and applying Lemma 3, we haveWe notice thatIn a similar way,We get the result from (49) and (50).

Corollary 7. Suppose that the assumptions of Theorem 9 are satisfied and that is symmetric with respect to on , thenwhere

Proof. Since the function is symmetric with respect to on so and the function is symmetric with respect on , this fact givesThus,

Corollary 8. If we take in (44) and is symmetric with respect to . Then, the following inequality holds:

Theorem 10. Let be a differentiable mapping on and , where . If is a continuous mapping and is -convex on for , thenwhere and are given in Theorem 9.

Proof. Application of Hölder inequality in (47) yields thatApplying Lemma 3, we haveOn the other hand, we haveA combination of (57)–(59) gives (56).

Corollary 9. If is symmetric with respect to on , then from (56), we obtainwhere is as defined in Corollary 7.

Corollary 10. If and is symmetric with respect to on , then the following inequality holds:where is as defined in Corollary 8.
For our next results, we use the following notations.It is clear from (62) thatThe next result gives upper bound of when the function is quasi -convex.

Theorem 11. Let be a differentiable mapping on and , where . If be a continuous mapping and is quasi -convex on , then the following inequality holds:where is defined as in Lemma 2, , and .

Proof. Since is quasi -convex on , we havefor all . Hence, inequality (64) follows from (19).

Theorem 12. Let be a differentiable mapping on and , where . If be a continuous mapping and is quasi -convex on , thenwhere and are defined as in Theorem 11.

Proof. The symmetry of with respect to on givesWe also observe thatConsider the function defined byThen,This shows that is an increasing function on andNow, it is easy to see thatHence, inequality (66) follows from the inequality.

Remark 2. If we choose in Theorem 9, Corollary 7 and 8, Theorem 10, Corollaries 9 and 10, and Theorems 11 and 12, we get the results for nonnegative convex functions and quasi-convex functions (see Theorem 3.1, Corollary 3.1, Theorem 3.2, Remark 3.2, Theorem 3.3 and Theorem 3.4 in [8]). Moreover, one can obtain inequalities for -convex functions, Godunova–Levin functions, -Godunova–Levin functions, -convex, -functions, and quasiconvex functions from the result of Theorem 9, Corollaries 7 and 8, Theorem 10, Corollaries 9 and 10, and Theorems 11 and 12 by choosing , , , and 1, respectively.

4. Applications

4.1. -Divergence Measures

Here, we provide some applications on -divergence measure and probability density function by using the results proved in Section 3. Let the set and the finite measure be given, and let the set of all probability densities on to be defined on . Let be given mapping and consider defined by

If is convex, then (74) is called as the Csisźar -divergence. Consider the following Hermite–Hadamard divergence:where is convex on with . Note that with the equality holds if and only if .

Proposition 1. Let be a differentiable mapping on , where . If , , is -convex on for with |No Image for this Article and , then

Proof. Let , , and . Obviously, if , then equality holds in (76). Now, if , then for and in Theorem 10, multiplying both sides to the obtained result by and integrating over , we haveSimilarly, if , then for and in Theorem 9, multiplying both sides to the obtained result by and integrating over , we haveAdding inequalities (77) and (78) and utilizing triangle inequality, we get the desired result (76).

4.2. Applications to Statistics

Let be the probability density function of a continuous random variable symmetric to with . The th moment of is defined aswhich is assumed to be finite.

Theorem 13. Suppose that and , then the following inequality holds:

Proof. Let on for , and we have is convex on [n₁, n₂]. Since the functions are symmetric with respect to ,Therefore, from inequality (44), we obtain inequality (80).

5. Conclusion

In this study, we propose new definition, namely, the definition of quasi -convex functions and provide an example of such type of functions. We prove some new weighted Hermite–-Hadamard type inequalities for differentiable for -convex and quasi -convex functions when the weight function is not necessarily symmetric about the midpoint of the interval. These results generalize many results proved in earlier works for these classes of functions. Applications of some of our results to -divergence and to statistics are given. We believe that the results of the current study may be a motivation to explore more new results relevant to this field of research for people working in the rich field of mathematical inequalities.

Data Availability

Data sharing is not applicable to this paper as no datasets were generated or analyzed during the current study.

Conflicts of Interest

The author declares that there are no conflicts of interest.

Authors’ Contributions

This study was carried out in collaboration of all authors. The author read and approved the final manuscript.

Acknowledgments

This work was supported by the Deanship of Scientific Research, King Faisal University under the Nasher Track (Research Project Number 206078).

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Copyright

Copyright © 2021 Muhammad Amer Latif. 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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