Abstract

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

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