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The main aim of this present note is to establish three new Hermite-Hadamard type integral inequalities for r-convex functions. The three new Hermite-Hadamard type integral inequalities for r-convex functions improve the result of original one by H?lder’s integral inequality, Stolarsky mean and convexity of function.

The inequalities

which discovered by C. Hermite and Hadamard for all convex functions are known in the literature as Hermite-Hadamard inequalities.

We note that Hermite-Hadamard inequalities may be regarded as a refinement of the concept of convexity and they follows easily from Jenson’s inequality. HermiteHadamard inequalities for convex functions has received renewed attention in recent years and a remarkable variety of refinements and generalizations have been found in [1-6].

Let be integrable functions on

then the well known Hölder’s integral inequality is given as

The following definition is well known in the literature.

Definition 1.1. Suppose

.

If following inequality holds

for any, then we sayis convex function on.

In [

Definition 1.2. ([

is said to be -convex function on, if

holds for any and.

We have that 0-convex functions are simply log-convex functions and 1-convex functions are ordinary convex functions.

The integral power mean (see [

The Stolarsky mean (see [

In [

Theorem 1.1. ([

Then

(i) is monotonically increasing on;

(ii).

In [

Theorem 1.2. ([

Theorem 1.3. ([

Theorem 1.4. ([

for and

In this paper we obtain some new Hermite-Hadamard type integral inequalities for -convex functions and improve the results of Theorems 1.2-1.4.

The following are extensions of Hermite-Hadamard type inequality:

Theorem 2.1. Let be - convex function on with,. Then

Proof. Let, then

If, by the -convexity of, we have

for any So the conclusion is valid.

If, we have to discuss three cases as following:

Case 1. If, we have

for any Hence, we obtain

Case 2. If, we have

for any Hence, we obtain

Case 3. If we have

for any Hence, we get

The proof of Theorem 2.1 is complete.

Corollary 2.1.1. If in Theorem 2.1, we have

Theorem 2.2. Let

be -convex and -convex functions respectively on with . Then the following inequality holds

for any and.

Proof. Let, then we have

.

If then 1) when by the -convexity and -convexity of functions respectively, we have

and

for any So we obtain

By the Hölder’s integral inequality and Theorem 2.1, we have

2) when we just prove for which is similar to and. By the Hölder’s integral inequality, Theorem 2.1 and -convexity and -convexity of functions respectively, we have

If or, by Theorem 2.1 we obtain the conclusion, which the proof of Theorem 2.2 is completed.

Corollary 2.2.1. Under the conditions of Theorem 2.2if for any, then we have

In particular, if, then we have

If, , we have

Corollary 2.2.2. Under conditions of Theorem 2.2, if and then we have

In particular, if, then we have

Theorem 2.3. Let and be -convex and -convex functions respectively on with Then the following inequality holds

for any and.

Proof. Let, then we have

.

By the Hölder’s integral inequality, Theorem 2.1 and -convexity and -convexity of function respectively, we have

This completed the proof of Theorem 2.3.

Corollary 2.3.1. Under the conditions of Theorem 2.3if and, then we have

In particular, if, we have

In this paper, we obtained three new Hermite-Hadamard type integral inequalities for -convex functions, which improved the results of Theorems 1.2-1.4 by Hölder’s integral inequality, Stolarsky mean and convexity of function. The special case of new Hermite-Hadamard type integral inequalities is classical Hermite-Hadamard type integral inequality. So it improved the classical one.

The first author was supported in part by the National Natural Science Foundation of China under Grant No. 11161033 and Inner Mongolia Natural Science of China under Grant No. 2010MS0119.