ABSTRACT

In the paper, the authors find some new inequalities of Hermite-Hadamard type for functions whose third derivatives are s-convex and apply these inequalities to discover inequalities for special means.

1. Introduction

The following definition is well known in the literature.

Definition 1.1. A function is said to be convex if

holds for all and.

In [1,2], among others, the concepts of so-called quasiconvex and s-convex functions in the second sense was introduced as follows.

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

holds for all and.

Definition 1.3 ([2]). Let A function is said to be s-convex in the second sense if

for all and.

If is a convex function on with and, Then we have Hermite-Hardamard’s inequality

. (1.1)

Hermite-Hadamard inequality (1.1) has been refined or generalized for convex, s-convex, and quasi-convex functions by a number of mathematicians. Some of them can be reformulated as follows.

Theorem 1.1 ([3, Theorems 2.2 and 2.3]). Let be a differentiable mapping on, with.

(1) If is convex on, then

. (1.2)

(2) If the new mapping is convex on

for, then

Theorem 1.2 ([4, Theorems 1 and 2]). Let be a differentiable function on and with, and let. If is convex on, then

(1.3)

and

(1.4)

Theorem 1.3 ([5, Theorems 2.3 and 2.4]). Let be differentiable on, with, and let. If is convex on, then

and

(1.5)

Theorem 1.4 ([6, Theorems 1 and 3]). Let be differentiable on and with.

(1) If is s-convex on for some fixed and, then

(1.6)

(2) If is s-convex on for some fixed and, then

(1.7)

Theorem 1.5 ([7, Theorem 2]). Let be an absolutely continuous function on such that for with. If is quasi-convex on, then

In recent years, some other kinds of Hermite-Hadamard type inequalities were created in, for example, [8-17], especially the monographs [18,19], and related references therein.

In this paper, we will find some new inequalities of Hermite-Hadamard type for functions whose third derivatives are s-convex and apply these inequalities to discover inequalities for special means.

2. A Lemma

For finding some new inequalities of Hermite-Hadamard type for functions whose third derivatives are -convex, we need a simple lemma below.

Lemma 2.1. Let be a three times differentiable function on with and. If, then

(2.1)

Proof. By integrating by part, we have

The proof of Lemma 2.1 is complete.

3. Some New Hermite-Hadamard Type Inequalities

We now utilize Lemma 2.1, Hölder’s inequality, and others to find some new inequalities of Hermite-Hadamard type for functions whose third derivatives are s-convex.

Theorem 3.1. Let be a three times differentiable function on such that for with. If is s-convex on for some fixed and, then

(3.1)

Proof. Since is s-convex on, by Lemma 2.1 and Hölder’s inequality, we have

where

and

Thus, we have

The proof of Theorem 3.1 is complete.

Corollary 3.1.1. Under conditions of Theorem 3.11) if, then

(3.2)

2) if, then

Theorem 3.2. Let be a three times differentiable function on such that for with. If is s-convex on for some fixed and, then

(3.3)

where

Proof. Using Lemma 2.1, the s-convexity of on, and Hölder’s integral inequality yields

where an easy calculation gives

(3.4)

and

(3.5)

Substituting Equations (3.4) and (3.5) into the above inequality results in the inequality (3.3). The proof of Theorem 3.2 is complete.

Corollary 3.2.1. Under conditions of Theorem 3.2, if, then

Theorem 3.3. Under conditions of Theorem 3.2, we have

(3.6)

Proof. Making use of Lemma 2.1, the s-convexity of on, and Hölder’s integral inequality leads to

where

(3.7)

and

(3.8)

Substituting Equations (3.7) and (3.8) into the above inequality derives the inequality (3.6). The proof of Theorem 3.3 is complete.

Corollary 3.3.1. Under conditions of Theorem 3.3, if s = 1, then

Theorem 3.4. Under conditions of Theorem 3.2, we have

Proof. Since is s-convex on, by Lemma 2.1 and Hölder’s inequality, we have

and

where a straightforward computation gives

Substituting these equalities into the above inequality brings out the inequality (3.10). The proof of Theorem 3.4 is complete.

Corollary 3.4.1. Under conditions of Theorem 3.4, if, then

4. Applications to Special Means

For positive numbers and, define

(4.1)

and

(4.2)

It is well known that A and are respectively called the arithmetic and generalized logarithmic means of two positive number and.

Now we are in a position to construct some inequalities for special means A and by applying the above established inequalities of Hermite-Hadamard type.

Let

(4.3)

for and. Since and

for and then is s-convex function on and

Applying the function (4.3) to Theorems 3.1 to 3.3 immediately leads to the following inequalities involving special means and.

Theorem 4.1. Let , and. Then

Theorem 4.2. For, , and, we have

(4.4)

Theorem 4.3. For, , and, we have

5. Acknowledgements

The first author was supported by Science Research Funding of Inner Mongolia University for Nationalities under Grant No. NMD1103.

REFERENCES