Abstract and Applied Analysis

Volume 2014 (2014), Article ID 725987, 11 pages

http://dx.doi.org/10.1155/2014/725987

Research Article

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

Shuhong Wang1,2 and Ximin Liu¹

¹School of Mathematical Sciences, Dalian University of Technology, Dalian 11602, China

²College of Mathematics, Inner Mongolia University for Nationalities, Tongliao 028043, China

Received 6 March 2014; Revised 27 June 2014; Accepted 29 June 2014; Published 20 July 2014

Academic Editor: Cristina Pignotti

Copyright © 2014 Shuhong Wang and Ximin Liu. 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.

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

The rest is the same as the proof of Theorem 22.

Corollary 25. Under the assumptions of Theorem 24, if , then

Theorem 26. 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 (28).

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 turns out that

The rest is also similar to the proof of Theorem 22.

Corollary 27. Under the assumptions of Theorem 26,

(1) if , then

(2) if , then

(3) if , then

Conflict of Interests

The authors declare that they have no conflict of interests.

Authors’ Contribution

All authors contributed equally to the paper and read and approved the final paper.

Acknowledgments

This work was supported by the Fundamental Research Funds for the Central Universities (no. DUT14ZD208) and by the NNSF of China (no. 11361038).

References

T. Antczak, “Mean value in invexity analysis,” Nonlinear Analysis. Theory: Methods & Applications, vol. 60, no. 8, pp. 1473–1484, 2005.

T. Weir and B. Mond, “Pre-invex functions in multiple objective optimization,” Journal of Mathematical Analysis and Applications, vol. 136, no. 1, pp. 29–38, 1988.

M. A. Noor, “Variational-like inequalities,” Optimization, vol. 30, no. 4, pp. 323–330, 1994.

A. Barani, A. G. Ghazanfari, and S. S. Dragomir, “Hermite-Hadamard inequality for functions whose derivatives absolute values are preinvex,” Journal of Inequalities and Applications, vol. 2012, article 247, 9 pages, 2012.

S. R. Mohan and S. K. Neogy, “On invex sets and preinvex functions,” Journal of Mathematical Analysis and Applications, vol. 189, no. 3, pp. 901–908, 1995.

M. A. Noor, “Hermite-Hadamard integral inequalities for log-preinvex functions,” Journal of Mathematical Analysis and Approximation Theory, vol. 2, no. 2, pp. 126–131, 2007.

M. A. Noor, “On Hadamard integral inequalities involving two log-preinvex functions,” Journal of Inequalities in Pure and Applied Mathematics, vol. 8, no. 3, article 75, pp. 1–6, 2007.

M. Z. Sarikaya, H. Bozkurt, and N. Alp, “On Haermite-Hadamard type integral inequalities for preinvex and log-preinvex functions,” Contemporary Analysis and Applied Mathematics, vol. 1, no. 2, pp. 237–252, 2013.

S. S. Dragomir and R. P. Agarwal, “Two inequalities for differentiable mappings and applications to special means of real numbers and to trapezoidal formula,” Applied Mathematics Letters, vol. 11, no. 5, pp. 91–95, 1998.

C. E. M. Pearce and J. Pečarić, “Inequalities for differentiable mappings with application to special means and quadrature formulae,” Applied Mathematics Letters, vol. 13, no. 2, pp. 51–55, 2000.

U. S. Kirmaci, “Inequalities for differentiable mappings and applications to special means of real numbers and to midpoint formula,” Applied Mathematics and Computation, vol. 147, no. 1, pp. 137–146, 2004.

S. H. Wang and F. Qi, “Hermite-Hadamard type integral inequalities for n-times differentiable and preinvex functions,” Journal of Inequalities and Applications, vol. 2014, article 49, 2014.

M. A. Latif and S. S. Dragomir, “On Hermite-Hadamard type integral inequalities for n-times differentiable log-preinvex functions,” Filomat. Submitted

Y. Wang, S.-H. Wang, and F. Qi, “Simpson type integral inequalities in which the power of the absolute value of the first derivative of the integrand is -preinvex,” Facta Universitatis: Mathematics and Informatics, vol. 28, no. 2, pp. 151–159, 2013.

Y. Wang, B.-Y. Xi, and F. Qi, “Hermite-Hadamard type integral inequalities when the power of the absolute value of the first derivative of the integrand is preinvex,” Le Matematiche, vol. 69, no. 1, pp. 89–96, 2014.

J. Wang, J. Deng, and M. Fečkan, “Hermite-Hadamard-type inequalities for -convex functions based on the use of Riemann-Liouville fractional integrals,” Ukrainian Mathematical Journal, vol. 65, no. 2, pp. 193–211, 2013.

R. F. Bai, F. Qi, and B. Y. Xi, “Hermite-Hadamard type inequalities for the m- and (α, m)-logarithmically convex functions,” Filomat, vol. 27, no. 1, pp. 1–7, 2013.