Research Article | Open Access

# New Hermite-Hadamard Type Inequalities for -Times Differentiable and -Logarithmically Preinvex Functions

**Academic Editor:**Cristina Pignotti

#### Abstract

The concept of *s*-logarithmically preinvex function is introduced, and by creating an integral identity involving an *n*-times differentiable function, some new Hermite-Hadamard type inequalities for *s*-logarithmically preinvex functions are established.

#### 1. Introduction

Throughout this paper, let , , denote the set of all positive integers, denote the interval in , and denote the set in .

Let us recall some definitions of various convex functions.

*Definition 1. *A function is said to be convex if
holds for all and . If inequality (1) reverses, then is said to be concave on .

*Definition 2 (see [1]). *A set is said to be invex with respect to the map , if for every and
The invex set is also called a -connected set.

It is obvious that every convex set is invex with respect to the map , but there exist invex sets which are not convex (see [1], e.g.).

*Definition 3 (see [1]). *Let be an invex set with respect to . For every , the -path joining the points and is defined by

*Definition 4 (see [2]). *Let be an invex set with respect to . A function is said to be preinvex with respect to , if for every and
The function is said to be preincave if and only if is preinvex.

Every convex function is preinvex with respect to the map , but not conversely (see [2], e.g.).

*Definition 5 (see [3]). *Let be an invex set with respect to . The function on the set is said to be logarithmically preinvex with respect to , if for every and

For properties and applications of preinvex and logarithmically preinvex functions, please refer to [1–8] and closely related references therein.

The most important inequality in the theory of convex functions, the well known Hermite-Hadamard’s integral inequality, may be stated as follows. Let and with . If is a convex function on , then If is concave on , then inequality (6) is reversed.

Inequality (6) has been generalized by many mathematicians. Some of them may be recited as follows.

Theorem 6 (see [9, Theorem 2.2]). *Let be a differentiable mapping on and with . If is convex on , then
*

Theorem 7 (see [10, Theorem 1]). *Let and with . If is differentiable on such that is a convex function on for , then
*

Theorem 8 (see [11, Theorem 2.3]). *Let be differentiable on , with and . If is convex on , then
*

Theorem 9 (see [6]). *Let be an open invex set with respect to and with for all . If is a preinvex function on , then the following inequality holds:
*

Theorem 10 (see [4, Theorem 4.3]). *Let be an open invex set with respect to and with for all . Suppose that is a twice differentiable function on and is preinvex on . If and is integrable on the -path for , then
*

Theorem 11 (see [12, Theorem 3.1]). *For and , let be an open invex set with respect to and with for all . Suppose that is an -times differentiable function on and is integrable on the -path for . If is preinvex on for , then
*

Theorem 12 (see [13, Theorem 5]). *For and , let be an open invex set with respect to and with for all . Suppose that is a function such that exits on and is integrable on . If is logarithmically preinvex on for , then we have the inequality
**
where
*

Recently, some related inequalities for preinvex functions were also obtained in [14, 15].

In the paper, the concept of -logarithmically preinvex function is introduced, and by creating an integral identity involving an -times differentiable function, some new Hermite-Hadamard type inequalities for -logarithmically preinvex functions are established which generalize some known results.

#### 2. New Definition and Lemmas

Now we introduce concepts of -logarithmically preinvex functions.

*Definition 13. *Let be an invex set with respect to . The function on the set is said to be -logarithmically preinvex with respect to , if for every , and some

Clearly, when taking in (15), then becomes the standard logarithmically convex function on .

In order to obtain our main results, we need the following lemmas.

Lemma 14. *For , let be an open invex set with respect to and with for all . If is an -times differentiable function on and is integrable on the -path for , then
**
where and the above summation is zero for .*

*Proof. *Since and is an invex set with respect to , for every , we have . When , by integrating by part in the right-hand side of (16), one gives
Hence, the identity (16) holds for .

When and , suppose that the identity (16) is valid.

When , by the hypothesis, we have

Therefore, when , the identity (16) holds. By induction, the proof of Lemma 14 is complete.

*Remark 15. *Under the conditions of Lemma 14, we have

*Proof. *These are special cases of Lemma 14 for , , , respectively.

*Remark 16. *Adding the identities (19) and (21) and then dividing by 2 result in Lemma 14 from [12].

Lemma 17. *Let and . Then
**
for , .*

*Proof. *When , the proof is straightforward.

When , for , we have
which coincides with the right-hand side of (22) for .

For , we get
which coincides with the right-hand side of (22) for .

Suppose that (22) is true for , , then, for , it follows that
Therefore, when , the identity (22) holds. By induction, the proof of Lemma 17 is complete.

Lemma 18 (see [16]). *Let , , and . Then
**
where .*

By Lemmas 17 and 18, a straightforward computation gives the following lemmas.

Lemma 19. *Let and . Then
**
for , .*

Lemma 20. *Let , , and . Then
**
where .*

Lemma 21 (see [17]). *Let and , . Then
*

#### 3. Hermite-Hadamard Type Inequalities

Now we start out to establish some new Hermite-Hadamard type inequalities for -times differentiable and -logarithmically preinvex functions.

Theorem 22. *For and , let be an open invex set with respect to and with for all . Suppose that is an -times differentiable function on and is integrable on the -path for . If is -logarithmically preinvex on for , then for and some **
where , is defined in Lemma 19, and
*

*Proof. *Since and is an invex set with respect to , for every , we have . Using Lemma 14, Hölder’s inequality, and -logarithmically preinvexity of , it yields that
By Lemmas 17 and 21, for , we give
For , we get
For , we obtain
For , we have
Using Lemma 19 and substituting (33) to (39) into (32), we get inequality (30).

Theorem 22 is thus proved.

Corollary 23. *Under the assumptions of Theorem 22,**(1) if , then
**(2) if , then
*

Theorem 24. *For and , let be an open invex set with respect to and with for all . Suppose that is an -times differentiable function on and is integrable on the -path for . If is -logarithmically preinvex on for , then for and some **
where , and are given in Theorem 22 and is defined by (26).*

*Proof. *Since and is an invex set with respect to , for every , we have . Using Lemma 14, Hölder’s inequality, and -logarithmically preinvexity of , it follows that