New Inequalities for Strongly -Convex Functions

Abstract

In this study, firstly we introduce a new concept called “strongly -convex function.” After that we establish Hermite-Hadamard-like inequalities for this class of functions. Moreover, by using an integral identity together with some well known integral inequalities, we establish several new inequalities for -times differentiable strongly -convex functions. In special cases, the results obtained coincide with the well-known results in the literature.

1. Introduction

A function is said to be convex if the inequality is valid for all and . If this inequality reverses, then the function is said to be concave on interval . This definition is well known in the literature. Convexity theory has appeared as a powerful technique to study a wide class of unrelated problems in pure and applied sciences. Many articles have been written by a number of mathematicians on convex functions and inequalities for their different classes, using, for example, the last articles [1–6] and the references in these papers.

Let be a convex function; then the inequality is known as the Hermite-Hadamard inequality (see [7] for more information). Since then, some refinements of the Hermite-Hadamard inequality on convex functions have been extensively investigated by a number of authors (e.g., [1, 4, 8]). In [9], the first author obtained a new refinement of the Hermite-Hadamard inequality for convex functions. The Hermite-Hadamard inequality was generalized in [10] to an -convex positive function which is defined on an interval .

Definition 1. A positive function is called -convex function on , if, for each and , If the equality is reversed, then the function is said to be -concave.

It is obvious that -convex functions are simply log-convex functions, -convex functions are ordinary convex functions, and -convex functions are arithmetically harmonically convex. One should note that if the function is -convex on , then the function is a convex function for and is a concave function for . We note that if the functions and are convex and is increasing, then is convex; moreover, since , it follows that a -convex function is convex.

The definition of -convexity naturally complements the concept of -concavity, in which the inequality is reversed [11] and plays an important role in statistics.

It is easily seen that if is -convex on ,

Some refinements of the Hadamard inequality for -convex functions could be found in [12–16]. In [14], the authors showed that if the function is -convex in and , then

Theorem 2 (see [17]). Suppose that is a positive -convex function on . Then

If the function is a positive -concave function, then the inequality is reversed, where

Definition 3. Let be an interval and be a positive number. A function is called strongly convex with modulus if for all and .

In this definition, if we take , we get the definition of convexity in the classical sense. Strongly convex functions have been introduced by Polyak [18], and they play an important role in optimization theory and mathematical economics. Since strong convexity is a strengthening of the notion of convexity, some properties of strongly convex functions are just “stronger versions” of known properties of convex functions. For more information on strongly convex functions, see [19–21] and references therein.

Lemma 4. Let , . Then .

Lemma 5 (Minkowski’s integral inequality). Let . If , then Minkowski’s integral inequality states that

Let , throughout this paper we will usefor the arithmetic, geometric, and generalized logarithmic mean, respectively. Also for shortness we will use the following notation: where an empty sum is understood to be nil.

2. Main Results

In this section we introduce a new concept, which is called strongly -convex function, as follows.

Definition 6. A positive function is called strongly -convex function with modulus on , if, for each and ,

In this definition, if we take , we get the definition of -convexity in the classical sense.

Theorem 7. Let be strongly -convex function with modulus on with . Then the following inequality holds for :

Proof. Since the function is strongly -convex function and , we havefor all . It is easy to observe thatUsing Minkowski’s integral inequality, we obtainThusThis proof is complete.

Theorem 8. Suppose that is a positive strongly -convex function with modulus on . Then where

Proof. Firstly, assume that By (13) we haveSecondly, for , we getThis proof is complete.

We will use the following lemma for obtaining our main results.

Lemma 9 (see [6]). Let be -times differentiable mapping on for and , where with ; we have the identity where an empty sum is understood to be nil.

We note that the authors obtained several new integeral inequalities for -times differentiable log-convex, -convex functions in the first sense, strongly convex, -Convex and -Concave, and convex and concave functions using the above lemma (see [5, 6, 22–24]). In this paper, we consider -times differentiable strongly -convex function and establish several new inequalities for this class of functions. Obtained results in this paper coincide with the results of papers ([6, 23, 24]).

Theorem 10. For in , let be -times differentiable function on , and with . If and for is strongly -convex function with modulus on , then the following inequality holds:

Proof. If for is strongly -convex function on and , using Lemma 9, the Hölder integral inequality and the inequalitywe have This completes the proof of theorem.

Remark 11. The following results are remarkable for Theorem 10.
(i) The results obtained in this paper reduce to the results of [24] in case of .
(ii) The results obtained in this paper reduce to the results of [23] in case of .
(iii) The results obtained in this paper reduce to the results of [6] in case of and .

Corollary 12. Under the conditions Theorem 10 for we have the following inequality:

Theorem 13. For in , let be -times differentiable function on , and with . If and for is strongly -convex function with modulus on , then the following inequality holds: where

Proof. From Lemma 9 and Power-Mean integral inequality, we get Here, using Lemma 4 we obtain, respectively, the following.
For For , using Minkowski’s integral inequality, we getThis completes the proof of theorem.

Remark 14. The following results are remarkable for Theorem 13.
(i) The results obtained in this paper reduce to the results of [24] in case of .
(ii) The results obtained in this paper reduce to the results of [23] in case of .
(iii) The results obtained in this paper reduce to the results of [6] in case of and .

Corollary 15. Under the conditions Theorem 13 for we have the following inequalities: where .

Corollary 16. Under the conditions Theorem 13 for we have the following inequalities:

Theorem 17. For in , let be -times differentiable function on , and with . If and for is strongly -convex function with modulus on , then the following inequality holds:where

Proof. Since for is strongly -convex function on , using Lemma 9 and the Hölder integral inequality, we have the following inequality: Here, using Lemma 4 we obtain
For , For , using Minkowski’s integral inequality, we get This completes the proof of theorem.

Remark 18. The following results are remarkable for Theorem 17.
(i) The results obtained in this paper reduce to the results of [24] in case of .
(ii) The results obtained in this paper reduce to the results of [23] in case of .
(iii) The results obtained in this paper reduce to the results of [6] in case of and .

Corollary 19. Under the conditions Theorem 17 for we have the following inequalities:

Theorem 20. For in , let be -times differentiable function on (interior of ), and with . If and for is strongly -convex function with modulus on , then the following inequalities hold:

Proof. For , since for is strongly -convex function on , with respect to inequality (6), we have Using Lemma 9 and the Hölder integral inequality we have For , using Lemma 9 and Theorem 2, we have This completes the proof of theorem.

Remark 21. The following results are remarkable for Theorem 20.
(i) The results obtained in this paper reduce to the results of [24] in case of .
(ii) The results obtained in this paper reduce to the results of [23] in case of .
(iii) The results obtained in this paper reduce to the results of [6] in case of and .

Corollary 22. Under the conditions Theorem 20 for we have the following inequalities:

Data Availability

No data were used to support this study.

Conflicts of Interest

The author declares that they have no conflicts of interest.

Acknowledgments

A part of this article is presented in the “International Conference on Advanced Manufacturing and Materials” symposium. Then, it was expanded by me and presented as an article to this journal.