Some Ostrowski Type Inequalities for Harmonically -Convex Functions in Second Sense

Abstract

The authors introduce the concept of harmonically -convex functions in second sense and establish some Ostrowski type inequalities of these classes of functions.

1. Introduction

Let be a mapping differentiable in and with . If , for all , then the following inequality holds:for . This inequality is known in the literature as the Ostrowski inequality [1], which gives an upper bound for the approximation of the integral average by the value at the point . For some results which generalize, improve, and extend inequality (1), we refer the reader to recent papers (see [2, 3]) and monograph of [4].

In [5], İşcan introduced class of harmonically convex functions. This class of functions is defined as follows.

Definition 1. Let be a real interval. A function is said to be harmonically convex, iffor all and . If inequality in (2) is reversed, then is said to be harmonically concave.

In [6], İşcan introduced the concept of harmonically -convex function in second sense and established a variant of Ostrowski type inequalities which hold for these classes of functions as follows.

Definition 2. A function is said to be harmonically -convex in second sense, iffor all and for some fixed . If inequality in (3) is reversed, then is said to be harmonically -concave.

Remark 3. Note that, for , harmonically -convexity reduces to ordinary harmonic convexity.

Theorem 4. Let be a differentiable function on , with and . If is harmonically -convex on for , then, for all , we havewhere is Euler Beta function defined byand is hypergeometric function (see [4]) defined by

Corollary 5. In Theorem 4, additionally, if , , then inequalityholds.

Theorem 6. Let be a differentiable function on , with and . If is harmonically -convex on for , then, for all , one haswhere , , , and are defined as in Theorem 4.

Corollary 7. In Theorem 6, additionally, if , , then the inequalityholds.

Theorem 8. Let be a differentiable function on , with and . If is harmonically -convex on for , then, for all , one haswhereand , , , and are defined as in Theorem 4.

Corollary 9. In Theorem 8, additionally, if , , then the inequalityholds.

Theorem 10. Let be a differentiable function on , with and . If is harmonically -convex on for , , then, for all , one haswhere , , , and are defined as in Theorem 4.

Corollary 11. In Theorem 10, additionally, if, , then the inequalityholds.

Theorem 12. Let be a differentiable function on , with and . If is harmonically -convex on for , , then, for all , one haswhere and are defined as in Theorem 4.

Corollary 13. In Theorem 12, additionally, if, , then the inequalityholds.

In [7], İşcan introduced the concept of harmonically -convex functions and established some Hermite-Hadamard type inequalities for this class of function. This class of functions is defined as follows.

Definition 14. The function is said to be harmonically -convex, where and , iffor all and . If the inequality in (18) is reversed, then is said to be harmonically -concave.

In [8], Park considered the class of -convex functions in second sense. This class of functions is defined as follows.

Definition 15. For some fixed and , a mapping is said to be -convex in the second sense on ifholds, for all and .

Now, we introduce the concept of harmonically -convex functions in second sense, which generalize the notion of harmonically convex and harmonically -convex functions in second sense introduced by İşcan in [5, 6], as follows.

Definition 16. The function is said to be harmonically -convex in second sense, where and if and .

Remark 17. Note that, for , harmonic -convexity reduces to harmonic -convexity and for harmonic -convexity reduces to harmonic -convexity in second sense (see [6]) and for harmonic -convexity reduces to ordinary harmonic convexity (see [5]).

Proposition 18. Let be a function:(a)if is -convex function in second sense and nondecreasing, then is a harmonically -convex function in second sense.(b)If is a harmonically -convex function in second sense and nonincreasing, then is -convex function in second sense.

Remark 19. According to Proposition 18, every nondecreasing -convex function in second sense is also a harmonically -convex function in second sense.

Example 20 (see [9]). Let and ; then, function defined byis a nondecreasing -convex function in second sense for and . Hence, by Proposition 18, is a harmonically -convex function.

Proposition 21. Let , , and be an increasing function and and , . Then, is harmonically -convex in second sense on if and only if is -convex in second sense on .

The following result of the Hermite-Hadamard type holds.

Theorem 22. Let be a harmonically -convex function in second sense with and . If and , then one has the following inequality:

Corollary 23. If one takes in Theorem 22, then one gets

Corollary 24. If one takes in Theorem 22, then one gets

In this paper, we obtain similar inequalities (1) for harmonically -convex functions and establish some new results of Ostrowski type inequalities for harmonically -convex functions such that results given in [6] by İşcan are obtained for particular value of .

2. Main Results

For finding some new inequalities of Ostrowski type for the functions whose derivatives are harmonically -convex in second sense, we need the following lemma.

Lemma 25. Let be a differentiable function on and with . If , then

Theorem 26. Let be a differentiable function on , with , , and . If is harmonically -convex in second sense on for with , thenwhere , , , are defined as in Theorem 4.

Proof. From Lemma 25 and using power mean inequality, we have Since is harmonically -convex function in second sense, we have It is easy to check thatThis completes the proof.

Remark 27. If we take in Theorem 26, we get Theorem 4.

Corollary 28. In Theorem 26, additionally, if , , then inequality holds.

Remark 29. If we take in Corollary 28, we get Corollary 5.

Theorem 30. Let be a differentiable function on , with , , and . If is harmonically -convex in second sense on for with , then, for all , one haswhere, , , , and are defined as in Theorem 4.

Proof. From Lemma 25, power mean inequality, and harmonic -convexity in second sense of on , we have This completes the proof.

Remark 31. If we take in Theorem 30, we get Theorem 6.

Corollary 32. In Theorem 30, additionally, if , , then inequality holds.

Remark 33. If we take in Corollary 32, then we get Corollary 7.

Theorem 34. Let be a differentiable function on , with , , and . If is harmonically -convex in second sense on for with , then, for all , one haswhereand , , , and are defined as in Theorem 4.

Proof. From Lemma 25, power mean inequality, and harmonic -convexity in second sense of on , we have It is easy to check thatHence, by use of (29), (30), (31), and (32) for and (40) in (39), we get the desired result.

Remark 35. If we take in Theorem 34, we get Theorem 8.

Corollary 36. In Theorem 34, additionally, if , , then inequality holds.

Remark 37. If we take in Corollary 36, we get Corollary 9.

Theorem 38. Let be a differentiable function on , with , , and . If is harmonically -convex in second sense on for , , with , then where, , , , and are defined as in Theorem 4.

Proof. From Lemma 25, Hölder’s inequality, and harmonic -convexity of on , one has This completes the proof.

Remark 39. If we take in Theorem 38, we get Theorem 10.

Corollary 40. In Theorem 38, additionally, if , , then inequality holds.

Remark 41. If we take in Corollary 40, we get Corollary 11.

Theorem 42. Let be a differentiable function on , with , , and . If is harmonically -convex in second sense on for , , with , then one has the following inequality:where, and are defined as in Theorem 4.

Proof. From Lemma 25, Hölder’s inequality, and harmonic -convexity of on , we have

Remark 43. If we take in Theorem 42, we get Theorem 12.

Corollary 44. In Theorem 42, additionally, if , , then inequality holds.

Remark 45. Corollary 44 is exactly as Corollary 13.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.