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Hermite-Hadamard and Simpson Type Inequalities for Differentiable Quasi-Geometrically Convex Functions

İmdat İşcan*, Kerim Bekar, Selim Numan

Department of Mathematics, Faculty of Arts and Sciences, Giresun University, Giresun, Turkey

Abstract: In this paper, the authors define a new identity for differentiable functions. By using of this identity, authors obtain new estimates on generalization of Hadamard and Simpson type inequalities for quasi-geometrically convex functions.

Keywords: quasi-geometrically convex functions, hermite–hadamard type inequalities, simpson type inequality

Received April 08, 2014; revised April 12, 2014; accepted April 13, 2014

Citation: İmdat İşcan, Kerim Bekar, Selim Numan, Hermite-Hadamard and Simpson Type Inequalities for Differentiable Quasi-Geometrically Convex Functions. Turkish Journal of Analysis and Number Theory, 2 (2). 42-46. January 2014. doi:10.12691/tjant-2-2-3

1. Introduction

Let real function be defined on some nonempty interval of real line R. The function is said to be convex on if inequality

holds for all and

Following inequalities are well known in the literature as Hermite-Hadamard inequality and Simpson inequality respectively:

Theorem 1. Let be a convex function defined on the interval of real numbers and with . The following double inequality holds

Theorem 2. Let be a four times continuously differentiable mapping on and Then the following inequality holds:

In recent years, many athors have studied errors estimations for Hermite-Hadamard, Ostrowski and Simpson inequalities; for refinements, counterparts, generalization see [2,9,10].

The following definitions are well known in the literature.

Definition 1 ([7,8]). A function is said to be GA-convex (geometric-arithmatically convex) if

for all and .

Definition 2 ([7,8]). A function is said to be GG-convex (called in [13] geometrically convex function) if

for all and .

In [3], İşcan gave definition of quasi-geometrically convexity as follows:

Definition 3. A function is said to be quasi-geometrically convex on if

for any and

Clearly, any GA-convex and geometrically convex functions are quasi-geometrically convex functions. Furthermore, there exist quasi-geometrically convex functions which are neither GA-convex nor GG-convex [3].

For some recent results concerning Hermite-Hadamard type inequalities for GA-convex, GG-convex, quasi-geometrically convex functions we refer interestes reader to [1,3,4,5,6,11,12,14].

The goal of this article is to establish some new general integral inequalities of Hermite-Hadamard and Simpson type for quasi-geometrically convex functions by using a new integral identity.

2. Main Results

Let be a differentiable function on , the interior of , throughout this section we will take

where with and .

In order to prove our main results we need the following identity.

Lemma 1. Let be a differentiable function on such that , where with Then for all we have:

[picture] | (1)

Proof. By integration by parts and changing the variable, we can state

and similarly we get

Adding the resulting identities we obtain the desired result.

Theorem 3 Let be a differentiable function on such that , where with . If is quasi-geometrically convex on for some fixed and , then the following inequality holds

[picture] | (2)

where

[picture] | (3)

and is logarithmic mean defined by

Proof. Since is quasi-geometrically convex on for all

Hence, using Lemma 1 and power mean inequality we get

where

which completes the proof.

Corollary 1 Under the assumptions of Theorem 3 with the inequality (2) reduced to the following inequality

Corollary 2 Under the assumptions of Theorem 3 with and the inequality (2) reduced to the following inequality

Corollary 3 Under the assumptions of Theorem 3 with and the inequality (2) reduced to the following inequality

Theorem 4 Let be a differentiable function on such that , where with . If is quasi-geometrically convex on for some fixed and , then the following inequality holds.

[picture] | (4)

where

and .

Proof. Since is quasi-geometrically convex on and using Lemma 1 and Hölder inequality, we get

here it is seen by simple computation that

Hence, the proof is completed.

Corollary 4 Under the assumptions of Theorem 4 with the inequality (4) reduced to the following inequality

Corollary 5 Under the assumptions of Theorem 4 with and the inequality (4) reduced to the following inequality.

Corollary 6 Under the assumptions of Theorem 4 with and the inequality (4) reduced to the following inequality

Theorem 5 Let be a differentiable function on such that , where with . If is quasi-geometrically convex on for some fixed and , then the following inequality holds

[picture] | (5)

where are defined as in Theorem 4 and .

Proof. Since is quasi-geometrically convex on and using Lemma 1 and Hölder inequality, we get

Corollary 7 Under the assumptions of Theorem 5 with the inequality (5) reduced to the following inequality

Corollary 8 Under the assumptions of Theorem 5 with and the inequality (5) reduced to the following inequality

Corollary 9 Under the assumptions of Theorem 5 with and the inequality (5) reduced to the following inequality

References

[1]   | Hua, J., Xi, B.-Y. and Qi, F., Hermite-Hadamard type inequalities for Geometric-arithmetically S-convex functions”, Commun. Korean Math. Soc., 29 (1). 51-63. 2014.
[2]   | İşcan, İ., “Generalization of different type integral inequalities for S-convex functions via fractional integrals”, Applicable Analysis, 2013.
[3]   | İşcan, İ., “New general integral inequalities for quasi-geometrically convex functions via fractional integrals”, Journal of Inequalities and Applications, 2013. 491. 15 pages. 2013.
[4]   | İşcan, İ., “Some New Hermite-Hadamard Type Inequalities for Geometrically Convex Functions”, Mathematics and Statistics, 1 (2). 86-91. 2013.
[5]   | İşcan, İ., “Some Generalized Hermite-Hadamard Type Inequalities for Quasi-Geometrically Convex Functions”, American Journal of Mathematical Analysis, 1 (3). 48-52. 2013.
[6]   | İşcan, İ., “Hermite-Hadamard type inequalities for GA-S-convex functions”, Le Matematiche, 2014a. Accepted for publication.
[7]   | Niculescu, C. P., “Convexity according to the geometric mean”, Math. Inequal. Appl., 3 (2). 155-167. 2000.
[8]   | Niculescu, C. P., “Convexity according to means”, Math. Inequal. Appl., 6 (4). 571-579. 2003.
[9]   | Sarıkaya, M. Z. and Aktan, N., “On the generalization of some integral inequalities and their applications”, Math. Comput. Modelling, 54. 2175-2182. 2011.
[10]   | Sarıkaya, M. Z., Set, E. and Ozdemir, M. E., “On new inequalities of Simpson’s type for S-convex functions”, Comput. Math. Appl. 60. 2191-2199. 2010.
[11]   | Shuang, Y., Yin, H.-P. and Qi, F., “Hermite-Hadamard type integral inequalities for geometric-arithmetically S-convex functions”, Analysis (Munich), 33 (2). 197-208. 2013.
[12]   | Zhang, X.-M., Chu, Y.-M. and Zhang, X.-H. “The Hermite-Hadamard Type Inequality of GA-Convex Functions and Its Application”, Journal of Inequalities and Applications. 2010. Article ID 507560. 11 pages. 2010.
[13]   | Zhang, T.-Y., Ji, A.-P. and Qi, F., “On Integral Inequalities of Hermite-Hadamard Type for S-Geometrically Convex Functions”, Abstr. Appl. Anal., 2012. Article ID 560586. 14 pages. 2012.
[14]   | Zhang, T.-Y., Ji, A.-P. and Qi, F., “Some inequalities of Hermite-Hadamard type for GA-convex functions with applications to means”, Le Matematiche, LXVIII- Fasc. I. 229-239. 2013.