Research Article | Open Access

Huriye Kadakal, "New Inequalities for Strongly -Convex Functions", *Journal of Function Spaces*, vol. 2019, Article ID 1219237, 10 pages, 2019. https://doi.org/10.1155/2019/1219237

# New Inequalities for Strongly -Convex Functions

**Academic Editor:**Henryk Hudzik

#### 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