Journal of Function Spaces
Volume 2019 (2019), Article ID 2394021, 8 pages
https://doi.org/10.1155/2019/2394021

Research Article

Some Integral Inequalities for Harmonically -Convex Functions

Serap Özcan

Department of Mathematics, Faculty of Art and Science, Kırklareli University, 39100, Kırklareli, Turkey

Correspondence should be addressed to Serap Özcan; serapozcann@yahoo.com

Received 26 February 2019; Accepted 11 July 2019; Published 30 July 2019

Academic Editor: Seppo Hassi

Copyright © 2019 Serap Özcan. 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.

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 if

for 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 inequality

holds. 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 , then

where

and

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

where

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 , if

for 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 have

Integrating 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

where

and

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 , then

where 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 , then

where is given by (10),

and

respectively.

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, then

where 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 , , then

where

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

where

and

respectively.

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 , then

where

and

respectively.

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 , , then

where

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

where and are given by (63) and (64).

Proof. The assertion follows from inequality (62) in Corollary 33, for , .

5. Conclusion

In this paper, we introduce and investigate a new class of harmonically convex functions, which is called harmonically -convex functions. Some new results of Hermite-Hadamard type for the newly introduced class of harmonically convex functions are established. Some applications of these results to special means have also been presented. The results of this paper may stimulate further research for the researchers working in this field.

Data Availability

No data were used to support this study.

Conflicts of Interest

The author declares no conflicts of interest.

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