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Journal of Function Spaces
Volume 2018, Article ID 6595921, 8 pages
https://doi.org/10.1155/2018/6595921
Research Article

Integral Inequalities Involving Strongly Convex Functions

1College of Science, Hunan City University, Yiyang 413000, China
2Department of Mathematics, University of Peshawar, Peshawar 25000, Pakistan
3Department of Mathematics, Huzhou University, Huzhou 313000, China

Correspondence should be addressed to Yu-Ming Chu; nc.ude.uhjz@gnimuyuhc

Received 25 February 2018; Accepted 6 May 2018; Published 11 June 2018

Academic Editor: Alberto Fiorenza

Copyright © 2018 Ying-Qing Song 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.

Abstract

We study the notions of strongly convex function as well as -strongly convex function. We present here some new integral inequalities of Jensen’s type for these classes of functions. A refinement of companion inequality to Jensen’s inequality established by Matić and Pečarić is shown to be recaptured as a particular instance. Counterpart of the integral Jensen inequality for strongly convex functions is also presented. Furthermore, we present integral Jensen-Steffensen and Slater’s inequality for strongly convex functions.

1. Introduction and Preliminaries

The word “convexity” is the most important, natural, and fundamental notations in mathematics. Convex functions were presented by Johan Jensen over years ago. Over the past few years, multiple generalizations and extensions have been made for convexity. These extensions and generalizations in the theory of inequalities have made valuable contributions in many areas of mathematics. Some new generalized concepts in this point of view are quasiconvex [1], strongly convex [2], approximately convex [3], logarithmically convex [4], midconvex functions [5], pseudoconvex [6], -convex [7], -convex [8], -convex [9], delta-convex [10], Schur convex [1115], and others [1619].

The main ingredient of our investigation is the strongly convex function [2]. Let be the real function defined on interval and be positive number, then we say that the function is strongly convex with modulus on iffor all and .

Every strongly convex function is convex, but the converse is not true in general. Strongly convex functions have been utilized for proving the convergence of a gradient type algorithm for minimizing a function. They play a significant role in mathematical economics, approximation theory, and optimization theory. Many applications and properties of them can be found in [2, 9, 20]. In , Adamek [21] further generalized the notion of strongly convex function. They replaced the nonnegative term by a nonnegative real valued function and defined it as follows: function is said to be -strongly convex function iffor all and . From [22], we also havewhere is -strongly convex function.

In literature the following inequality is well-known as Jensen inequality.

Theorem 1 (see [4]). Let be a measure space with and be convex function. Suppose is such that , , then one has

In 1981, Slater proved a companion inequality to the Jensen inequality [23].

Theorem 2. Let be a measure space with and be increasing and convex function. Suppose is such that , , and . If , then one hasIn the case when is strictly convex, then equality holds in (5) if and only if is constant almost everywhere on .

Remark 3. Some improvements and reversions of Slater’s inequality are given in [24, 25].

The following inequality is the integral analogue of another companion inequality to the Jensen inequality.

Theorem 4 (see [26]). Let be a measure space with and be convex function. Suppose is such that , , , and and Then the following inequalities hold:If the function is strictly convex, then equality holds in (7) if and only if is constant almost everywhere on .

Matić and Pečarić established a general inequality from which one can directly obtain inequalities (5) and (7).

Theorem 5 (see [27]). Let all the assumptions of Theorem 4 be fulfilled. If , then one hasAlso, when is strictly convex, then equality in the left side in (8) holds if and only if almost everywhere on , while equality in the right side in (8) holds if and only if almost everywhere on .

Remark 6. Under the assumptions of Theorem 4, let and , then by setting in (8), we get Slater’s inequality (5), and similarly by setting in (8), we get (7).

Merentes and Nikodem improved the Jensen inequality for strongly convex functions as follows.

Theorem 7 (see [28]). Let be a probability measure space and be strongly convex function with modulus . Suppose is a Lebesgue integrable function and . Then the following inequality holds:

For more recent results related to strongly convex function and Jensen type inequalities we recommend [22, 2934].

This paper is organized as follows. In Section 2, we establish general inequalities for -strongly convex function as well as strongly convex functions. As a consequence, we obtain integral Jensen inequality and Slater’s inequality for strongly convex functions. Also by the virtue of these general inequalities we deduce converse of Jensen inequality. In Section 3, we give some properties of strongly convex functions. By using these properties of strongly convex functions we prove Jensen-Steffensen and Slater’s type inequalities.

2. Jensen’s Type Inequalities

We start this section to give the following general theorem.

Theorem 8. Let be a measure space with and be F-strongly convex function. Suppose is such that , , , and and also , . If , then one has

Proof. Since is strongly convex function, thereforeLetting and , in (11), we getTaking integral of (12) and then dividing by , we obtainSimilarly, rearranging (11), we getLetting and , in (14), we haveTaking integral of (15) and then dividing by , we getCombining (13) and (16), we obtain (10).

By virtue of Theorem 8, we can deduce some new and interesting consequences.

Proposition 9. Suppose that all the assumptions of Theorem 8 are satisfied. Then

Proof. If we set in (11) and taking integral over and then dividing by , we haveor equivalentlyTaking the infimum over , we obtain (17).

Proposition 10. Suppose that all the assumptions of Theorem 8 are satisfied and , then

Proof. By setting and in (10), we haveIndeed, the following equivalent form of (21) isTaking the infimum over , we can easily derive the first and the second inequality in (20). The remaining third inequality in (20) follows because

Corollary 11. Let be a measure space with and be strongly convex function with modulus . Suppose is such that , , , and and also , . If , then one has

Remark 12. If we put in (24) and take probability measure space, then we obtain integral Jensen inequality (9) for strongly convex function.

In the following corollary, we obtain integral Slater’s inequality for strongly convex function.

Corollary 13. Suppose , , , , and are stated as in Corollary 11 and assume that ; also if , then

Proof. By setting in (24), we deducedSince , therefore (26) is equivalent to (25).

In the following corollary, we obtain a converse of the Jensen inequality for strongly convex function.

Corollary 14. Suppose , , , , and are stated as in Corollary 11, then one has

Proof. By setting in (24), we obtain (27).

3. Jensen-Steffensen Inequality for Riemann-Stieltjes Integrals

To prove the main results of this section, first we prove the following lemma which will play a key role in the proof of main results.

Lemma 15. Let be strongly convex function with modulus . Suppose is fixed element from interval , then (a)the function defined by(b)and the function defined byare nonnegative on , decreasing on and increasing on .

Proof. By definition of strongly convexity we have It means that is nonnegative on .
Let . Since is strongly convex with modulus , therefore is increasing and henceit followsSetting and in (28) and then taking the difference, we getHence, is decreasing on .
Also, if , then similarly as above by strongly convexity we havefrom which it follows thatSetting and in (28), then taking the difference we getHence, is increasing on .
In the same manner as above, we also obtain is nonnegative on .
Let , then by strongly convexity we haveSetting and in (29) and then taking the difference we getHence, is decreasing on .
Also, if , then by strongly convexity we haveSetting and in (29) and then taking the difference we getHence, is increasing on . This completes the proof.

The following lemma is given in [35].

Lemma 16. Let be a nonnegative function and suppose is either a bounded variation or continuous. Also assume that the functions and have no common discontinuity points.(a)If is increasing on , then(b)If is decreasing on , then

In the next result, we prove some general integral inequalities for strongly convex functions.

Theorem 17. Suppose is monotonic and continuous function and let be a strongly convex function with modulus . If is either a bounded variation or continuous and satisfying for all , , then and given by are well defined and . Also, if and have no common discontinuity points, then for , one has

Proof. Under the given conditions in [35], it has been shown that We define the function by , where is defined as in Lemma 15(a), i.e., From Lemma 15(a) it follows that is nonnegative and since and are continuous therefore the integral exists. Thus we discuss the following three cases:(i)If , since is increasing on and by Lemma 15(a) is decreasing on , therefore is decreasing on . So using Lemma 16 (b), we obtain (ii)If , since is increasing on by Lemma 15(a), therefore is increasing on . So using Lemma 16 (a), we have (iii)If , since is continuous on , there exists at least one point such that . Also by Lemma 15(a), is decreasing on and is increasing on . Using Lemma 16, we have From the above three subcases we conclude that Dividing (53) by , we obtain the left side of the inequality (46). Similarly, if is decreasing we consider the cases ( is increasing on ), ( is decreasing on ), and ( is decreasing on and increasing on ). In all three cases we obtain from the first inequality in (46) which directly follows. Similarly, we can prove the right side of the inequality (46). We define the function by , where is defined as in Lemma 15(b), i.e., Since is monotonic and continuous, is continuous and is monotonic which have no common discontinuity point with . Therefore the integral exists. Using the same process as above we have , which means that If we divide by , we obtain the second inequality of (46). So, both inequalities of (46) are proved.

Now, we are in a situation to obtain the following result.

Corollary 18. Suppose all the assumptions of Theorem 17 are satisfied, then one has

Proof. By setting in (46), we obtain (56).

Remark 19. If we set in (56), we obtain Theorem 3.2 in [35].

In the following corollary, we obtain integral Slater’s inequality for strongly convex functions.

Corollary 20. Suppose all the assumptions of Theorem 17 are satisfied and assume ; also if , then

Proof. Similar to the proof of Corollary 13, setting in the right hand side of (46), we get (57).

Remark 21. If we set in (57), we obtain Slater’s inequality for convex functions given in [35].

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

The research was supported by the Natural Science Foundation of China (Grants nos. 61673169 and 11601485).

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