FRACTIONAL OSTROWSKI TYPE INEQUALITIES FOR FUNCTIONS WHOSE MIXED DERIVATIVES ARE PREQUASIINVEX AND α-PREQUASIINVEX FUNCTIONS

The aim of this paper is to establish a new fractional Ostrowski type inequalities involving functions of two independent variable whose mixed derivatives are prequasiinvex and α-prequasiinvex functions which are two novel classes of generalized convex functions. These estimates are relying on a new integral identity.

Also the concept of convexity has been extended and generalized in several directions.One of the most significant generalization is that introduced by Hanson [5] where he introduced the concept of invexity, In [3] the authors gave the notion of preinvex functions which is special case of invexity.Many authors have study the basic properties of invex set and preinvex functions, and their role in optimization, variational inequalities and equilibrium problems, see [19,20,30,35,38].
It is important to remember that the credit goes to Professor Noor, who was the first to have had the opportunity to study the integral inequalities in the context of the preinvex functions see [21][22][23][24][25][26].
Barnett et al. [2] gave the following Ostrowski's inequality involving functions of two independent variable .
Latif et al. [8] established the following fractional Ostrowski's inequality for double variables , where The aim of this paper is to establish a new fractional Ostrowski type inequalities involving functions of two independent variable whose mixed derivatives are prequasiinvex and α-prequasiinvex functions which are two novel classes of generalized convex functions.These estimates are relying on a new integral identity.

Preliminaries
In this sections we begin by giving some definitions We note that the set K 1 × K 2 is an invex set with respect to η 1 and η 2 , if In [9] Latif and Dragomir introduced the class of co-ordinated preinvex functions Using this new class they have established some Hermite-Hadamard type inequalities, of which certain results are recalled as follows: for any function twice partially differentiable on the invex set And if ∂ 2 f ∂t∂s q is co-ordinated preinvex with q ≥ 1 we have , where 1 p + 1 q = 1 and and and where Γ is the Gamma function, and ), and J α c + f (a, d) of order α, β > 0 where a, c ≥ 0 with a < b and c < d are defined by and where Γ is the Gamma function.

Main results
We first present these two new classes of generalized convex functions called co-ordinated prequasiinvex and co-ordinated α-prequasiinvex, Throughout this paper we assume that holds for all t, λ ∈ [0, 1] and (a, c), (b, d) ∈ ∆. where ))+f (a,y)+f (a+η1(b,a),y) 2 and Proof.Let where and k, h are defined by ( 12) and ( 13) respectively.
where O is defined as in (11).
Proof.From Lemma 2.3, properties of modulus, and prequasiinvexity on the coordinates of , which is the desired result.
Theorem 2.5.Let f : K → R be a partially differentiable function on K with is co-ordinated prequasiinvex function on K, where q > 1 with 1 p + 1 q = 1, then the following fractional inequality holds , where O is defined as in (11).
Proof.By Lemma 2.3, properties of modulus, Hölder inequality, and prequasiinvexity on the co-ordinates of ∂ 2 f ∂t∂λ q , we have , which is the desired result.
Proof.By Lemma 2.3, properties of modulus, and α-prequasiinvexity on the coordinates of ∂ 2 f ∂t∂λ , we have which is the desired result.
Proof.By Lemma 2.3, properties of modulus, Hölder inequality, and α-prequasiinvexity on the co-ordinates of , which is the desired result.