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American Journal of Public Health Research
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The Implementation of a Tele-Homecare System with Service Oriented Architecture and HL7 Message Transmission Standard
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Clinical and Histological Features in Wegener Granulomatosis
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Anne Lanham, Lillian Mwanri

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Some Inequalities of the Hermite-Hadamard Type for Harmonically Quasi-Convex Functions

Chun-Long Li*, Gui-Hua Gu, Bai-Ni Guo

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

Abstract: In the paper, by Holder’s integral inequality, the authors establish some Hermite-Hadamard type integral inequalities for harmonically quasi-convex functions.

Keywords: Hermite-Hadamard inequality, harmonically quasi-convex function, Holder’s integral inequality

Received October 31, 2017; Revised December 01, 2017; Accepted December 06, 2017

Citation: Chun-Long Li, Gui-Hua Gu, Bai-Ni Guo, Some Inequalities of the Hermite-Hadamard Type for Harmonically Quasi-Convex Functions. Turkish Journal of Analysis and Number Theory, 5 (6). 226-229. January 2017. doi:10.12691/tjant-5-6-4

1. Introduction

The following definitions for various convex functions are well known in the literature.

Definition 1.1 A function is said to be convex if

holds for all and .

Definition 1.2 ([1,2,3]) A function is said to be quasi-convex if

holds for all and .

Definition 1.3 ([4]) For with and if

is valid for all and , then we say that is an -convex function on .

Definition 1.4 ([5]) Let with and . If

is valid for all and , then we say that is an -convex function on .

Definition 1.5 ([9]) A function is said to be a harmonically quasi-convex function on if

holds for all and .

In [7,8], the following inequalities of Hermite-Hadamard type were established.

Theorem 1.1 ([7], Theorems 2.2 and 2.3]). Let be a differentiable mapping and with . Then

(i) if is convex on , then

(ii) if is convex on for , then

Theorem 1.2 ([8], Theorem 2.3]) Let be differentiable on and with . If is s-convex on for , then

In this paper, we will create some integral inequalities of Hermite-Hadamard type for harmonically quasi-convex functions.

2. A Lemma

For creating some integral inequalities of Hermite-Hadamard type for harmonically quasi-convex functions, we need the following lemma.

Lemma 2.1 Let be differentiable on with , and . If , then

where

Proof Integrating by part and changing variables for yield

Lemma 2.1 is thus proved.

3. Some New Integral Inequalities of Hermite-Hadamard Type

Now we set off to create some integral inequalities of Hermite-Hadamard type for harmonically quasi-convex functions.

Theorem 3.1 Let be a differentiable function, with and . If is harmonically quasi-convex on , then

Proof Using Lemma 2.1 and the harmonic quasi-convexity of , we have

The proof of Theorem 3.1 is complete.

Theorem 3.2 Let be a differentiable function, with , and . If is harmonically quasi-convex on and , then

Proof Since is harmonically quasi-convex on , by Lemma 2.1 and Hölder's inequality, we have

Theorem 3.2 is thus proved.

Theorem 3.3. Let be a differentiable function, with , and . If is harmonically quasi-convex on for , then

Proof From the GA-inequality, we have

for all . By Lemma 2.1 and the harmonic quasi-convexity of and Hölder's inequality, we have

The proof of Theorem 3.3 is complete.

References

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[13]   | Bo-Yan Xi and Feng Qi. Integral inequalities of Hermite-Hadamard type for ((α,m), log)-convex functions on co-ordinates. Problemy Analiza-Issues of Analysis, 2015, 22 (2): 73-92.
[14]   | Bo-Yan Xi and Feng Qi. Hermite-Hadamard type inequalities for geometrically r-convex functions. Studia Scientiarum Mathematicarum Hungarica, 2014, 51(4): 530-546.
[15]   | Bo-Yan Xi and Feng Qi. Some Hermite-Hadamard type inequalities for differentiable convex functions and applications. Hacettepe Journal of Mathematics and Statistics, 2013, 42(3): 243-257.
[16]   | Bo-Yan Xi and Feng Qi. Integral inequalities of Simpson type for logarithmically convex functions. Advanced Studies in Contemporary Mathematics, 2013, 23(4): 559-566.
[17]   | Bo-Yan Xi and Feng Qi. Hermite-Hadamard type inequalities for functions whose derivatives are of convexities. Nonlinear Functional Analysis and Applications, 2013, 18(2),: 163-176.