Some Inequalities of Hermite-Hadamard Type for Functions Whose 3rd Derivatives Are P-Convex ()
1. Introduction
The following definition is well known in the literature.
Definition 1.1. A function 
is said to be convex if
(1.1)
holds for all
.
In [1], the concept of the so-called 
-convex functions was introduced as follows.
Definition 1.2. ([1]) We say that a map
belongs to the class 
if it is nonnegative and satisfies
(1.2)
for all
.
In [2], S. S. Dragomir proved the following theorems.
Theorem 1.1. ([2]) Let 
be a differentiable mapping on 
and
. If 
is convex on
, then
(1.3)
Theorem 1.2. ([2]) Let 
be a differentiable mapping on 
and
. If 
is convex on 
for
, then
(1.4)
Theorem 1.3. ([3], Theorems 2) Let 
be an absolutely continuous function on 
such that
for
. If 
is quasi-convex on
, then
For more information and recent developments on this topic, please refer to [4-14] and closely related references therein.
The concepts of various convex functions have indeed found important places in contemporary mathematics as can be seen in a large number of research articles and books devoted to the field these days.
In this paper, we will establish some new HermiteHadamard type inequalities for functions whose 
rd derivatives are P-convex.
2. A Lemma
In this section, we establish an integral identity.
Lemma 2.1. Let 
be a three times differentiable mapping on 
and
. If
, then
(2.1)
Proof. Integrating by part and changing variable of definite integral yield
and
The proof of Lemma 2.1 is complete.
3. Hermite-Hadamard’s Type Inequalities for P-Convex Functions
Theorem 3.1. Let 
be differentiable on
,
, and 
If 
is 
-convex on
for
, then
(3.1)
Proof. Since 
is a 
-convex function on
, by Lemma 2.1 and Hölder’s inequality, we obtain
The proof of Theorem 3.1 is complete.
Corollary 3.1.1. Under the conditions of Theorem 3.1, if
, we have
Theorem 3.2. Let 
be differentiable on
, 
, and
. If 
is 
-convex on 
for
, then
(2.2)
Proof. From Lemma 2.1, Hölder’s inequality, and the 
-convexity of 
on
, we drive
Theorem 3.2 is proved.
Theorem 3.3. Let 
be differentiable on
, 
, and 
If 
is 
-convex on
for
, then
(2.3)
Proof. From Lemma 2.1, Hölder’s inequality, and the 
-convexity of 
on
, we have
Theorem 3.3 is thus proved.
Theorem 3.4. Let 
be differentiable on
, 
, and 
If 
for 
is 
-convex on
and
, then
(2.5)
Proof. Using Lemma 2.1, Hölder’s inequality, and the 
-convexity of 
on 
yields
The proof of Theorem 3.4 is complete.
Corollary 3.3.1. Under the conditions of Theorem 3.4(1) if
, then
(2) if
, then
(3) if
, then
Finally we would like to note that these Hermite-Hadamard type inequalities obtained in this paper can be applied to the fields of integral inequalities, approximation theory, special means theory, optimization theory, information theory, and numerical analysis, as done before by a number of mathematicians.
4. Acknowledgements
The first two authors were partially supported by the Science Research Funding of Inner Mongolia University for Nationalities under Grant No. NMD1103.