Ostrowski Inequalities for Functions Whose First Derivatives Are Logarithmically Preinvex

Abstract

Some Ostrowski type inequalities for functions whose first derivatives are logarithmically preinvex are established.

1. Introduction

In 1938, A. M. Ostrowski proved the following important inequality.

Theorem 1 (see [1]). Let be a differentiable mapping on (interior of ), and let with . If for all , then

This is well known in the literature as Ostrowski’s inequality. Due to its wide range of applications in numerical analysis and in probability, many researchers have established generalizations, extensions, and variants of inequality (1); we refer readers to [2–10] and the references cited therein.

In recent years, a lot of efforts have been made by many mathematicians to generalize classical convexity. Hanson [11] introduced a new class of generalized convexity, called invexity. In [12], the authors gave the concept of preinvex function which is a special case of invexity. Pini [13], Noor [14, 15], Yang and Li [16], and Weir and Mond [17] have studied the basic properties of the preinvex functions and their role in optimization, variational inequalities, and equilibrium problems.

In [5], Işcan established some Ostrowski type inequalities for functions whose derivatives in absolute value are preinvex, by using the following identity.

Lemma 2 (see [5]). Let be an open invex subset with respect to and with . Suppose that is a differentiable function. If is integrable on , then the following equality holds: for all .

Motivated by the results given in [5], in the present paper, we establish some new Ostrowski type inequalities for functions whose first derivatives in absolute value are logarithmically preinvex.

2. Preliminaries

In this section, we recall some concepts of convexity that are well known in the literature. Throughout this section, is an interval of .

Definition 3 (see [18]). A positive function is said to be logarithmically convex, if, for all and all , we have

Definition 4 (see [17]). A set is said to be invex at with respect to , if, for all and , we have is said to be an invex set with respect to if is invex at each .

Definition 5 (see [14]). A positive function on the invex set is said to be -preinvex with respect to , if, for all and , we have

Lemma 6 (see [19]). For , and , we have where .

3. Main Results

Theorem 7. Let be an invex subset with respect to and ( interior of ) with and . Let be a differentiable function such that and . If is logarithmically preinvex function, then the following inequality holds: for all , where .

Proof. From Lemma 2 and properties of modulus, we haveSince is a logarithmically preinvex function, we deduceIf , then (9) givesIn the case where , (9) giveswhere we use the fact that The desired result follows from (10) and (11).

Corollary 8. In Theorem 7, if we choose , we obtain the following midpoint inequality:

Corollary 9. Let be a differentiable function such that and . If is a logarithmically convex function, then the following inequality holds: for all , where .

Example 10. In Theorem 7, if we choose , the geometric mean, we obtain the following inequality:

Theorem 11. Let be an invex subset with respect to and ( interior of ) with and . Let be a differentiable function such that and ; let with . If is a logarithmically preinvex function, then the following inequality holds: for all , where .

Proof. From Lemma 2, properties of modulus, and Hölder’s inequality, we haveSince is a logarithmically preinvex function, we deduceIf , then (18) givesIn the case where , (18) becomeswhere we use the fact thatFrom (19) and (20), we get the desired result.

Corollary 12. In Theorem 11, if we choose , we obtain the following midpoint inequality:

Corollary 13. Let be a differentiable function such that and ; let with . If is a logarithmically convex function, then the following inequality holds: for all , where .

Example 14. In Theorem 11, if we choose , the arithmetic mean, we obtain the following inequality:

Theorem 15. Let be an invex subset with respect to and   ( interior of ) with and . Let be a differentiable function such that and ; let If is a logarithmically preinvex function, then the following inequality holds:for all , where .

Proof. From Lemma 2, properties of modulus, and power mean inequality, we haveSince is a logarithmically preinvex function, we deduceIn the case where , (27) givesFor , (27) giveswhere we use the fact that From (28) and (29), we obtain the desired result.

Corollary 16. In Theorem 15, if we choose , we obtain the following midpoint inequality:

Corollary 17. Let be a differentiable function such that and ; let If is a logarithmically convex function, then the following inequality holds: for all , where .

Example 18. In Theorem 15, if we choose with , the logarithmic mean, we obtain the following inequality: where .

Theorem 19. Suppose that all the assumptions of Theorem 15 are satisfied, then the following inequality holds:for all , where and .

Proof. From Lemma 2, properties of modulus, and power mean inequality, we haveSince is a logarithmically preinvex function, we deduceIf , (36) givesIn the case where , we can restate (36) asApplying Lemma 6 with , we getSubstituting (39) into (38), we obtainThe desired result follows from (37) and (40).

Corollary 20. In Theorem 19, if we choose , we obtain the following midpoint inequality:

Corollary 21. In Theorem 19, if we choose , we obtain the following inequality:

Remark 22. In all the above theorems, inequalities for nonconvex functions could be drawn by just replacing by other means than those in the previously mentioned examples.

Competing Interests

The author declares that they have no competing interests.