1. Introduction
The inequalities
, (1.1)
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
(1.2)
The following definition is well known in the literature.
Definition 1.1. Suppose
.
If following inequality holds
(1.3)
for any
, then we say
is convex function on
.
In [1], C. E. M. Pearce, J. Pecaric and V. Simic introduced the definition of 
-convex function and studied the inequalities of Hermite-Hadamard type for 
-convex functions.
Definition 1.2. ([1]) A function
is said to be 
-convex function on
, if
(1.4)
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 [2]) of a positive function 
on 
is a functional given by
(1.5)
The Stolarsky mean 
(see [7]) of two positive numbers 
is given by
(1.6)
In [2], following theorem is given.
Theorem 1.1. ([2]) Let 
be a positive 
-convex function on 
and 
is defined by
(1.7)
Then
(i) 
is monotonically increasing on
;
(ii)
.
In [4], following theorems are given.
Theorem 1.2. ([3]) Let
be
-convex function on 
with
. Then the following inequality holds for 
Theorem 1.3. ([3]) Let
be 
- convex and 
-convex functions respectively on 
with 
Then the following inequality holds for
,
Theorem 1.4. ([3]) Let 
be 
convex and 
-convex functions respectively on 
with 
Then the following inequality holds
for 
and 
2. Main Results
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
(2.1)
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
(2.2)
Theorem 2.2. Let
be 
-convex and 
-convex functions respectively on 
with 
. Then the following inequality holds
(2.3)
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
(2.4)
In particular, if
, then we have
If
, 
, we have
Corollary 2.2.2. Under conditions of Theorem 2.2, if 
and 
then we have
(2.5)
In particular, if
, then we have
Theorem 2.3. Let 
and 
be 
-convex and 
-convex functions respectively on 
with 
Then the following inequality holds
(2.6)
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
(2.7)
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.
3. Acknowledgements
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.