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

Research Article

Integral Inequalities Involving Strongly Convex Functions

Ying-Qing Song,¹ Muhammad Adil Khan,² Syed Zaheer Ullah,² and Yu-Ming Chu³

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; chuyuming@zjhu.edu.cn

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 [11–15], and others [16–19].

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 if

for 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 if

for all and . From [22], we also have

where 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 has

In 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 has

Also, 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, 29–34].

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, therefore

Letting and , in (11), we get

Taking integral of (12) and then dividing by , we obtain

Similarly, rearranging (11), we get

Letting and , in (14), we have

Taking integral of (15) and then dividing by , we get

Combining (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 have

or equivalently

Taking 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 have

Indeed, the following equivalent form of (21) is

Taking 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 deduced

Since , 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 by

are 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 hence

it follows

Setting and in (28) and then taking the difference, we get

Hence, is decreasing on .
Also, if , then similarly as above by strongly convexity we have

from which it follows that

Setting and in (28), then taking the difference we get

Hence, is increasing on .
In the same manner as above, we also obtain is nonnegative on .
Let , then by strongly convexity we have

Setting and in (29) and then taking the difference we get

Hence, is decreasing on .
Also, if , then by strongly convexity we have

Setting and in (29) and then taking the difference we get

Hence, 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).

References

B. de Finetti, “Sulle stratificazioni convesse,” Annali di Matematica Pura ed Applicata, vol. 30, no. 4, pp. 173–183, 1949.

B. T. Polyak, “Existence theorems and convergence of minimizing sequences for extremal problems with constraints,” Soviet Doklady Mathematics, vol. 7, pp. 72–75, 1966.

D. H. Hyers and S. M. Ulam, “Approximately convex functions,” Proceedings of the American Mathematical Society, vol. 3, pp. 821–828, 1952.

J. E. Pečarić, F. Proschan, and Y. L. Tong, Convex Functions, Partial Orderings, and Statistical Applications, vol. 187, Academic Press, Boston, MA, USA, 1992.

J. L. W. V. Jensen, “Om konvexe funktioner og uligheder mellem middlvaerdier,” Nyt Tidsskrift for Matematik. B, vol. 16, pp. 49–69, 1905.

O. L. Mangasarian, “Pseudo-convex functions,” Journal of the Society for Industrial and Applied Mathematics Series A Control, vol. 3, pp. 281–290, 1965.

S. S. Dragomir, “Inequalities of Hermite-Hadamard type for φ-convex functions on linear spaces,” Preprint RGMIA Res. Rep. Coll, vol. 87, 2013, http://rgmia.org/papers/v16/v16a87.pdf.

S. S. Dragomir, “Inequalities of Hermite-Hadamard type for λ-convex functions on linear spaces,” Preprint RGMIA Res. Rep. Coll, vol. 17, no. 18, 2014, http://rgmia.org/papers/v17/v17a13.pdf.

S. Varošanec, “On h-convexity,” Journal of Mathematical Analysis and Applications, vol. 326, no. 1, pp. 303–311, 2007.

T. Rajba, “On strong delta-convexity and Hermite-Hadamard type inequalities for delta-convex functions of higher order,” Mathematical Inequalities & Applications, vol. 18, no. 1, pp. 267–293, 2015.

Y. Chu and X. Zhang, “Necessary and sufficient conditions such that extended mean values are Schur-convex or Schur-concave,” Kyoto Journal of Mathematics, vol. 48, no. 1, pp. 229–238, 2008.

N. Elezović and J. Pečarić, “A note on Schur-convex functions,” Rocky Mountain Journal of Mathematics, vol. 30, no. 3, pp. 853–856, 2000.

H.-N. Shi, “Schur-convex functions related to Hadamard-type inequalities,” Journal of Mathematical Inequalities, vol. 1, no. 1, pp. 127–136, 2007.

C. Stępniak, “An effective characterization of Schur-convex functions with applications,” Journal of Convex Analysis, vol. 14, no. 1, pp. 103–108, 2007.

X.-M. Zhang, “Schur-convex functions and isoperimetric inequalities,” Proceedings of the American Mathematical Society, vol. 126, no. 2, pp. 461–470, 1998.

G. A. Anastassiou, “Basic and s-convexity Ostrowski and Grüss type inequalities involving several functions,” Communications in Applied Analysis, vol. 17, no. 2, pp. 189–212, 2013.

E. Set, M. E. Özdemir, and S. S. Dragomir, “On Hadamard-type inequalities involving several kinds of convexity,” Journal of Inequalities and Applications, vol. 2010, Article ID 286845, 12 pages, 2010.

Z.-H. Yang, W.-M. Qian, Y.-M. Chu, and W. Zhang, “On approximating the error function,” Mathematical Inequalities & Applications, vol. 21, no. 2, pp. 469–479, 2018.

Z.-H. Yang, W.-M. Qian, Y.-M. Chu, and W. Zhang, “On approximating the arithmetic-geometric mean and complete elliptic integral of the first kind,” Journal of Mathematical Analysis and Applications, vol. 462, no. 2, pp. 1714–1726, 2018.

A. W. Roberts and D. E. Varberg, Convex Functions, Academic Press, London, UK, 1973.

M. Adamek, “On a problem connected with strongly convex functions,” Mathematical Inequalities & Applications, vol. 19, no. 4, pp. 1287–1293, 2016.

H. R. Moradi, M. E. Omidvar, M. Adil Khan, and K. Nikodem, “Around Jensen's inequality for strongly convex functions,” Aequationes Mathematicae, vol. 92, no. 1, pp. 25–37, 2018.

M. L. Slater, “A companion inequality to Jensen's inequality,” Journal of Approximation Theory, vol. 32, no. 2, pp. 160–166, 1981.

M. Adil Khan and J. Pečarić, “On Slater's integral inequality,” Journal of Mathematical Inequalities, vol. 5, no. 2, pp. 231–241, 2011.

M. Adil Khan and J. E. Pečarić, “Improvement and reversion of Slater's inequality and related results,” Journal of Inequalities and Applications, vol. 2010, Article ID 646034, 14 pages, 2010.

S. S. Dragomir and C. J. Goh, “A counterpart of Jensen's discrete inequality for differentiable convex mappings and applications in information theory,” Mathematical and Computer Modelling, vol. 24, no. 2, pp. 1–11, 1996.

M. Matić and J. Pečarić, “Some companion inequalities to Jensen's inequality,” Mathematical Inequalities & Applications, vol. 3, no. 3, pp. 355–368, 2000.

N. Merentes and K. Nikodem, “Remarks on strongly convex functions,” Aequationes Mathematicae, vol. 80, no. 1-2, pp. 193–199, 2010.

M. Klaričić Bakula and K. Nikodem, “On the converse Jensen inequality for strongly convex functions,” Journal of Mathematical Analysis and Applications, vol. 434, no. 1, pp. 516–522, 2016.

M. Adil Khan, J. Khan, and Pečarić, “Generalization of Jensen's and Jensen-Steffensen's inequalities by generalized majorization theorem,” Journal of Mathematical Inequalities, vol. 11, no. 4, pp. 1049–1074, 2017.

S. S. Dragomir, M. Adil Khan, and A. Abathun, “Refinement of the Jensen integral inequality,” Open Mathematics, vol. 14, pp. 221–228, 2016.

F.-C. Mitroi-Symeonidis and N. Minculete, “On the Jensen functional and strong convexity,” Bulletin of the Malaysian Mathematical Sciences Society, vol. 41, no. 1, pp. 311–319, 2018.

K. Nikodem, On Strongly Convex Functions and Related Classes of Functions, Handbook of Functional Equations, Springer, New York, NY, USA, 2014.

K. Nikodem and Z. Páles, “Characterizations of inner product spaces by strongly convex functions,” Banach Journal of Mathematical Analysis, vol. 5, no. 1, pp. 83–87, 2011.

M. Klaričić Bakula, M. Matić, and J. Pečarić, “Generalizations of the Jensen-Steffensen and related inequalities,” Central European Journal of Mathematics, vol. 7, no. 4, pp. 787–803, 2009.