Research Article | Open Access

Serap Özcan, "Some Integral Inequalities for Harmonically -Convex Functions", *Journal of Function Spaces*, vol. 2019, Article ID 2394021, 8 pages, 2019. https://doi.org/10.1155/2019/2394021

# Some Integral Inequalities for Harmonically -Convex Functions

**Academic Editor:**Seppo Hassi

#### Abstract

In the paper, the author introduces a new class of harmonically convex functions, which is called harmonically -convex functions and establishes some new integral inequalities of the Hermite-Hadamard type for harmonically -convex functions. The properties of the newly introduced class of harmonically convex functions are also investigated. Finally, some applications to special means are given.

#### 1. Introduction

It is well known in the literature that a function is said to be convex on interval iffor all and . If the inequality in (1) holds in the reverse direction, is said to be concave function.

Inequalities play a fundamental part in many branches of pure and applied sciences. A number of studies have shown that the theory of convex functions has a closely relationship with the theory of inequalities. One of the most famous inequalities for convex functions is named Hermite-Hadamard integral inequality as follows.

Suppose to be a convex function defined on the interval of real numbers and with ; then the double inequalityholds. If is concave function, both inequalities in (2) are reversed. This inequality provides necessary and sufficient condition for a function to be convex.

In recent years, the concept of convexity has been improved, generalized, and extended in many directions; see [1–9]. Several new classes of convex functions have been introduced and new versions of Hermite-Hadamard’s inequality have been obtained. In [10], șcan introduced the class of harmonically convex functions and investigated the Hermite-Hadamard type inequalities for this new class of functions. For several recent results, generalizations, improvements, and refinements concerning harmonically convex functions; see [10–16]. There have been many studies dedicated to generalizing the harmonic convex functions and to establishing their Hermite-Hadamard type inequalities. For some recent studies on Hermite-Hadamard type inequalities, please refer to the monographs [10–12, 16–24].

The aim of this paper is to introduce the concept of harmonically -convex functions and to establish several new Hermite-Hadamard type inequalities based on these new class of functions.

#### 2. Preliminaries

In this section, we recall some basic concepts and results of harmonically convex functions.

*Definition 1 ([10]). *Let be a real interval. A function is said to be harmonically convex function if for all and .

*Definition 2 ([13]). *A function is said to be harmonically -convex function in second sense, where , if for all and .

*Definition 3 ([15]). *A function is said to be harmonically -convex function in first sense, where , if for all and .

*Definition 4 ([12]). *The function , , is said to be harmonically -convex function, where and , if for all and .

Note that if in Definition 4, we have definition of harmonically -convex function in first sense for .

*Definition 5 ([25]). *A function is said to be harmonically -function if for all and .

Theorem 6 ([10]). *Let be harmonically convex and with . If , then the following inequalities hold:*

Theorem 7 ([10]). *Let be a differentiable function on , with and . If is harmonically convex on for , thenwhereand *

Theorem 8 ([10]). *Let be a differentiable function on , with and . If is harmonically convex on for , , thenwhere*

*Definition 9 ([9]). *Let and be two functions. If holds for all , then and are said to be similarly ordered functions.

*Definition 10 ([23]). *A function is said to be -convex if for all and with and .

In order to prove some of our main results, we need the following lemma.

Lemma 11 ([10]). *Let be a differentiable function on and with . If , then*

#### 3. Main Results

In this section we define the concept of harmonically -convex functions and derive our main results. Throughout this section is the interval and is the interior of .

*Definition 12. *A function is said to be harmonically -convex, where , iffor all and . If the inequality in (18) is reversed, then is said to be harmonically -concave.

Now we give some special cases of our proposed definition of harmonically -convex functions.

(I) If , then we have the definition of harmonically -convex functions in second sense.

(II) If , then we have the definition of harmonically -convex functions or in other words harmonically -convex functions in first sense by writing instead of .

(III) If , then we have the definition of harmonically convex functions. Thus, every harmonically convex function is also harmonically -convex function.

The following proposition is obvious.

Proposition 13. *Let be a function.*

(1)If is -convex and nondecreasing function, then is harmonically -convex.(2)If is harmonically -convex and nonincreasing function, then is -convex.

*Proof. *For all and , we have So, the following inequality holds:By inequality (20), the proof is completed.

*Remark 14. *According to Proposition 13, every nondecreasing convex function (or -convex function) is also harmonically convex function (or harmonically -convex function).

*Example 15. *The function , is a nondecreasing -convex function. According to Remark 14, is harmonically -convex function (or harmonically -convex function for ).

Proposition 16. *Let and be two harmonically -convex functions. If and are similarly ordered functions and , then the product is also harmonically -convex function.*

*Proof. *Let and be harmonically -convex functions. Then

Proposition 17. *Let and defined by ; then is harmonically -convex on , where , , if and only if is -convex on .*

*Proof. *Since for all , , where , , .

This demonstrates necessary condition is provided. Now let us show the sufficient condition is also provided.

For all and , we have This completes the proof.

Theorem 18. *Let be harmonically -convex function with , . If and , then*

*Proof. *Since is a harmonically -convex function, we have for all which gives for all .

Adding the above inequalities, we haveIntegrating the above inequality on , we obtain which implies which is the required result.

*Remark 19. *If we take in Theorem 18, then inequality (24) becomes the right-hand side of inequality (2).

If we take in Theorem 18, we have the following result for harmonically -convex functions in second sense.

Corollary 20. *Let be harmonically -convex function in second sense where with and . If , then*

If we take , then Theorem 18 collapses to the following result.

Corollary 21. *Let be harmonically -function where with . If , then*

Our coming result is the Hermite-Hadamard inequality for product of two harmonically -convex functions.

Theorem 22. *Let be two harmonically -convex functions where with , , . If , then whereand is Euler Beta function defined by and is hypergeometric function defined by*

*Proof. *Let be harmonically -convex functions; then we have This completes the proof.

Theorem 23. *Under the conditions of Theorem 22, if and are similarly ordered functions, then we have where and are given by (33) and (34), *

*Proof. *Integrating inequality (21) completes the proof.

Theorem 24. *Let be a differentiable function on , with and . If is harmonically -convex on for , with and , thenwhere is given by (10),and*

*Proof. *Using Lemma 11 and power mean inequality, we have This completes the proof.

Corollary 25. *Under the conditions of Theorem 24 if , then we have where and are given by (43) and (44), respectively.*

*Remark 26. *If we take in Theorem 24, then inequality (42) becomes inequality (9) of Theorem 7.

If we take in Theorem 24, then we have result for harmonically -convex functions in second sense.

Corollary 27. *Let be a differentiable function on , with and . If is harmonically -convex in second sense on for , thenwhere is given by (10),andrespectively.*

If we take in Theorem 24, then we have following result for harmonically -convex functions in first sense for .

Corollary 28. *Let be a differentiable function on , with and . If is harmonically -convex function in first sense, thenwhere is given by (10), and and*

If we take in Theorem 24, we have the following result for harmonically -functions.

Corollary 29. *Let be a differentiable function on , with and . If is harmonically -function for , then where is given by (10).*

Theorem 30. *Let be a differentiable function on , with and . If is harmonically -convex on for , , with , , thenwhere and respectively.*

*Proof. *From Lemma 11, Hölder’s inequality, and the fact that is harmonically -convex function on , we have where This completes the proof.

*Remark 31. *If we take in Theorem 30, then inequality (54) becomes inequality (13) of Theorem 8.

If we take in Theorem 30, then we have result for harmonically -convex functions in second sense.

Corollary 32. *Let be a differentiable function on , with and . If is harmonically -convex function in second sense on for , , with , then whereandrespectively.*

If we take in Theorem 30, then we have the following result for harmonically -convex functions in first sense for .

Corollary 33. *Let be a differentiable function on , with and . If is harmonically -convex in first sense on for , , with , thenwhereandrespectively.*

If in Theorem 30, then we have result for harmonically -functions.

Corollary 34. *Let be a differentiable function on , with and . If is harmonically -function on for , , thenwhere*

#### 4. Some Applications to Special Means

Let us recall the following special means for arbitrary real numbers and .

The arithmetic mean:

The geometric mean:

The -logarithmic mean:

Proposition 35. *Let , and . Then, we have where , , and are given by (10), (48), and (49), respectively.*

*Proof. *The assertion follows from inequality (47) in Corollary 27, for , .

Proposition 36. *Let , , , and . Then, we have *