Ostrowski type inequalities for p-convex functions

for all x ∈ [a,b] . In the literature, the inequality ( 1) is known as Ostrowski inequality (see [ 18]), which gives an upper bound for the approximation of the integral average 1 b−a ∫ b a f (t)dt by the valuef (x) at pointx∈ [a,b]. In [3,5,6,9,10,11], the reader can find generalizations, improvements and exten sions for the inequality ( 1). For p∈ R the power meanMp(a,b) of orderp of two positive numbers a andb is defined by


Introduction
Let f : I→ R, where I ⊆ R is an interval, be a mapping differentiable in I • (the interior of I) and let a, b ∈ I • with a < b.
, then the following inequality holds for all x ∈ [a, b] . In the literature, the inequality (1) is known as Ostrowski inequality (see [18]), which gives an upper bound for the approximation of the integral average 1 b−a b a f (t)dt by the value f (x) at point x ∈ [a, b]. In [3,5,6,9,10,11], the reader can find generalizations, improvements and extensions for the inequality (1). For p ∈ R the power mean M p (a, b) of order p of two positive numbers a and b is defined by It is well-known that M p (a, b) is continuous and strictly increasing with respect to p ∈ R for fixed a, b > 0 with a = b. Let M be the family of all mean values of two numbers in R + = (0, ∞) . Given M, N ∈ M, we say that a function f : R + → R + is (M, N)-convex if f (M(x, y)) ≤ N ( f (x), f (y)) for all x, y ∈ R + . The concept of (M, N)-convexity has been studied extensively in the literature from various points of view (see e.g. [1,4,12,15]).
and M p (a, b;t) = (ta p + (1 − t)b p ) 1/p be the weighted arithmetic, weighted geometric, weighted harmonic , weighted power of order p means of two positive real numbers a and b with a = b for t ∈ [0, 1], respectively. M p (a, b;t) is continuous and strictly increasing with respect to t ∈ R for fixed p ∈ R\ {0} and a, b > 0 with a > b. See [8,14] for some kinds of convexity obtained by using weighted means.
In [8], the author gave definition Harmonically convex and concave functions as follow.
for all x, y ∈ I and t ∈ [0, 1]. If the inequality (2) is reversed, then f is said to be harmonically concave.
The following result of the Hermite-Hadamard type holds for harmonically convex functions.
The above inequalities are sharp.

The Definition of p-convex Function
In [19], Zhang and Wan give the definition of p-convex function as follows: Definition 2. Let I be a p-convex set. A function f : I → R is said to be a p-convex function or belongs to the class PC(I), for all x, y ∈ I and t ∈ [0, 1].
Remark. If I ⊂ (0, ∞) be a real interval and p ∈ R\ {0}, then According to Remark 2, we can give a different version of the definition of p-convex function as follows: for all x, y ∈ I and t ∈ [0, 1]. If the inequality (3) is reversed, then f is said to be p-concave.
According to Definition 3, It can be easily seen that for p = 1 and p = −1, p-convexity reduces to ordinary convexity and harmonically convexity of functions defined on I ⊂ (0, ∞), respectively. In [7,Theorem 5], if we take I ⊂ (0, ∞) , h(t) = t and p ∈ R\ {0} , then we have the following theorem.
Remark. The inequalities (4) are sharp. Indeed we consider the function f : which shows us that the inequalities (4) are sharp.
For some results related to p-convex functions and its generalizations, we refer the reader to see [7,?,?,17,19].

Main Results
Proposition 1. Let I ⊂ (0, ∞) be a real interval, p ∈ R\ {0} and f : I → R is a function, then ; (1) If p ≤ 1 and f is convex and nondecreasing function then f is p-convex.
(2) If p ≥ 1 and f is p-convex and nondecreasing function then f is convex.
According to above Proposition, we can give the following examples for p-convex and p-concave functions.
The following proposition is obvious.
Remark. According to Proposition 2, as examples of p-convex functions we can take Thus, we can obtain the inequality (4) in a different manner as follows: If f is a is p-convex on [a, b] then we write the Hermite-Hadamard inequality for the convex function that is equivalent to Using the change of variable and we get the inequality (4) by using the inequality (5).
Proof. Integrating by part and changing variables of integration yields β is Euler Beta function defined by c 2016 BISKA Bilisim Technology and 2 F 1 is hypergeometric function defined by [2]).
Proof. (i) Let p > 0. Then By using Lemma 1 and Lemma 2, we obtained the following some new Ostrowski type inequalities for p-convex functions.

Proof. From Lemma 1, Power mean integral inequality and the p-convexity
Hence, If we use (7) and the equalities in Lemma 2 , we obtain the desired result. This completes the proof.

Proof. From Lemma 1 and Lemma 2, Power mean integral inequality and the p-convexity
This completes the proof.
For q ≥ 1, we can give the following result: This completes the proof.
This completes the proof.

Conclusion
The paper deals with Ostrowski type inequalities for p-convex functions. Firstly, we give a different version of the concept of p-convex functions and get some new properties of p-convex functions. Later, by using a new identity, we obtain several new Ostrowski type inequalities for this class of functions via hypergeometric functions.