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Some Generalized Hermite-Hadamard Type Inequalities for Quasi-Geometrically Convex Functions

İmdat İşcan

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

Abstract: In this paper, by Hölder’s integral inequality, some new generalized Hermite-Hadamard type integral inequalities for quasi-geometrically convex functions are obtained.

Keywords: Hermite-Hadamard type inequality, Hölder’s integral inequality, quasi-geometrically convex function

Received November 05, 2013; Revised November 22, 2013; Accepted December 10, 2013

Citation: İmdat İşcan, Some Generalized Hermite-Hadamard Type Inequalities for Quasi-Geometrically Convex Functions. American Journal of Mathematical Analysis, 1 (3). 48-52. January 2013. doi:10.12691/ajma-1-3-5

1. Introduction

Let I be on interval in . It is general knowledge that if is a convex function and a,b∈I with a<b, then

This inequality is well known in the literature as Hermite-Hadamard’s inequality for convex functions.

In [4,5], Niculescu defined the GA-convex and GG-convex functions as follows:

Definition 1.1. The function is said to be geometric arithmetically convex, simply speaking, GA-convex on I if the following inequality

holds for all and , where and are respectively called the weighted geometric mean of two positive numbers andand the weighted arithmetic mean of and .

Definition 1.2. The function is said to be geometrically convex, simply speaking, GG-convex on I if the following inequality

holds for all and .

For some recent results concerning Hermite-Hadamard type inequalities for GA-convex and GG-convex functions we refer interestes reader to [1,2,3,6,7].

In [2], Iscan defined the quasi-geometrically-convex functions as follows:

Definition 1.3. A function is said to be quasi-geometrically convex on I if

For any and .

Clearly, any GA-convex and GG-convex functions are quasi-geometrically convex functions. Furthermore, there exist quasi-geometrically convex functions which are neither GA-convex nor GG-convex [2].

In [6], Park obtained generalized Hermite-Hadamard type integral inequalities for -GA-convex functions by using the following lemma:

Lemma 1.1. Let be a diﬀerentiable function on , the interior of an interval and with . If then the following identity holds:

[picture] | (1.1)

for n≠0.

In [7], Zhang et al. established some Hermite-Hadamard type integral inequalities for GA-convex functions by using inequality (1.1) for . In [3] Ji et al. established some Hermite-Hadamard type integral inequalities for -GA-convex functions by using inequality (1.1) for .

The goal of this article is to establish some new generalized integral inequalities of Hermite-Hadamard type for quasi-geometrically convex functions by using inequality (1.1) and Hölder inequality.

Throughout this paper we will consider

the logarithmic mean for b>a>0.

2. Hermite-Hadamard Type Inequalities for Quasi-geometrically Convex Functions

Theorem 2.1. Let be a diﬀerentiable function on , with and . If is quasi-geometrically convex on for , then

for .

Proof. Since is quasi-geometrically convex on , from Lemma 1.1 and Hölder’s inequality we have

Theorem 2.2. Let be a diﬀerentiable function on , with and . If is quasi-geometrically convex on for , then

for .

Proof. Since is quasi-geometrically convex on , from Lemma 1.1 and Hölder’s inequality we have

Theorem 2.3. Let be a diﬀerentiable function on , with and . If is quasi-geometrically convex on for , then

for .

Proof. Since is quasi-geometrically convex on , from Lemma 1.1 and Hölder’s inequality we have

Theorem 2.4. Let be a diﬀerentiable function on , with and . If is quasi-geometrically convex on for with , then

for .

Proof. Since is quasi-geometrically convex on , from Lemma 1.1 and Hölder’s inequality we have

Theorem 2.5. Let be a diﬀerentiable function on , with and . If is quasi-geometrically convex on for with , then

for .

Proof. Since is quasi-geometrically convex on , from Lemma 1.1 and Hölder’s inequality we have

Theorem 2.6. Let be a diﬀerentiable function on , with and . If is quasi-geometrically convex on for with , then

for .

Proof. Since is quasi-geometrically convex on , from Lemma 1.1 and Hölder’s inequality we have

Theorem 2.7. Let be a diﬀerentiable function on , with and . If is quasi-geometrically convex on for with , then

for and .

Proof. Since is quasi-geometrically convex on , from Lemma 1.1 and Hölder’s inequality we have

Theorem 2.8. Let be a diﬀerentiable function on , with and . If is quasi-geometrically convex on for with , then

for and .

Proof. Since is quasi-geometrically convex on , from Lemma 1.1 and Hölder’s inequality we have

Theorem 2.9. Let be a diﬀerentiable function on , with and . If is quasi-geometrically convex on for with , then

for and .

Proof. Since is quasi-geometrically convex on , from Lemma 1.1 and Hölder’s inequality we have

Theorem 2.10. Let be a diﬀerentiable function on , with and . If is quasi-geometrically convex on for with , then

for and .

Proof. Since is quasi-geometrically convex on , from Lemma 1.1 and Hölder’s inequality we have

Theorem 2.11. Let be a diﬀerentiable function on , with and . If is quasi-geometrically convex on for with , then

for .

Proof. Since is quasi-geometrically convex on , from Lemma 1.1 and Hölder’s inequality we have

3. Applications to Special Means

Finally we apply Hermite-Hadamard type inequalities obtained in the above section to construct some inequalities for special means.

Proposition 3.1. For , and , we have

where and

Proof. Let for , and . Then the function is quasi-geometrically convex function on . Thus by the inequalities in Theorem 2.1, Theorem 2.2 and Theorem 2.3, Proposition 3.1 is proved.

Proposition 3.2. For , and with , we have

where ,and

Proof. Let for , and . Then the function is quasi-geometrically convex function on . Thus by the inequalities in Theorem 2.4, Theorem 2.5, Theorem 2.6, Theorem 2.7, Theorem 2.8, Theorem 2.9, Theorem 2.10 and Theorem 2.11, Proposition 3.2 is proved.

References

[1]   | Iscan, I., “Some new Hermite-Hadamard type inequalities for geometrically convex functions”, Mathematics and Statistics, 1 (2). 86-91. 2013.
[2]   | Iscan, I., “New general integral inequalities for quasi-geometrically convex functions via fractional integrals”, J. Inequal. Appl., 2013 (491). pp 15. 2013.
[3]   | Ji, A.-P., Zhang, T.-Y. and Qi F.,” Integral inequalities of Hermite-Hadamard type for (α,m)-GA-convex functions”. arxiv:1306.0852. Available online at http://arxiv.org/abs/1306.0852.
[4]   | Niculescu, C.P., “Convexity according to the geometric mean”, Math. Inequal. Appl., 3 (2). 571-579. 2000.
[5]   | Niculescu, C.P., “Convexity according to mean”, Math. Inequal. Appl.,6 (4). 155-167. 2003. http://dx.doi.org/10.7153/mia-03-19.
[6]   | Park, J., “Some generalized inequalities of Hermite-Hadamard type for (α,m)-geometric-arithmetically convex functions, Applied Mathematical Sciences, 7 (95). 4743-4759. 2013.
[7]   | Zhang, T.-Y., Ji, A.-P. and Qi, F., “Some inequalities of Hermite-Hadamard type for GA-convex functions with applications to means”, Le Mathematiche, LXVIII (I). 229-239. 2013.