Integral Means of Univalent Solution for Fractional Differential Equation

By employing the Srivastava-Owa fractional operators, we consider a class of fractional differential equation in the unit disk. The existence of the univalent solution is founded by using the Schauder fixed point theorem while the uniqueness is obtained by using the Banach fixed point theorem. Moreover, the integral mean of these solutions is studied by applying the concept of the subordination.


Introduction
Recently, the theory of fractional calculus has found interesting applications in the theory of analytic functions.The classical definitions of fractional operators and their generalizations have fruitfully been applied in obtaining, for example, the characterization properties, coefficient estimates [1], distortion inequalities [2] and convolution structures for various subclasses of analytic functions and the works in the research monographs.In [3], Srivastava and Owa, gave definitions for fractional operators (derivative and integral) in the complex z-plane as follows:  Definition 1.1.The fractional derivative of order  is defined, for a function   where the function   f z is analytic in simply-connected region of the complex z-plane containing the origin and the multiplicity of


is analytic in simply-connected region of the complex z-plane () containing the origin and the multiplicity of is removed by requiring Further properties of these operators can be found in [4,5].

Let be the class of all normalized analytic functions
Let be the class of analytic functions in U and for any and be the subclass of consisting of functions of the form For given two functions F and G, which are analytic in U, the function F is said to be subordinate to G in U if there exists a function h analytic in U with Section 2, we w 0 , 1 . In Section 3, we study the existence of locally univale subject to the initial condition , where In other words, every solution of the Volterra Equation ( 2wing assumptions are needed in the next th here exists a continuous function ) is also a solution of the initial value problem (1) and vice versa.
The follo eorem: with the property that is the Banach space of all contin s function p in U, such th uous positi s. (H2) There exists a continuou at Remark 3.1.By using fractional calculus we observe that Equation ( 2) is equivalent to the integral equation of the form that is, the existence of Equation ( 2 , 1

that is
Then P mapped into itself.: .
(H4) Assume that there exists a positive n ber such that for each Proof.Assum operat we only need t w e the or P defined in Equation ( 4), o sho that P is a contraction mapping that is P has a unique fixed point which is corresponding to the unique solution of the Equation (1).Let 1 2 , u u   , then for all , z U  we obtain that Thus by the assumption of the theorem we have that P is a contraction mapping.Then in view of Banach fixed point theorem, P has a unique fixed point which corresponds to the univalent solution (Theorem 3.1) of Equation (1).Hence the proof.
The next result shows the integral means of univalent solutions of problem (1).
) is the existence of the assumptions (H1) and (H2) hold.Th the Equation (3).Theorem 3.1.Let en Equation (1) has a univalent solution   u z on U. Proof.We need only to show that : P  has a   fix ed point by using Theorem 1.2 where Copyright © 2012 SciRes.AM work is organized as follows: I Ou n ill derive the integral means for normalized analytic functions involving fractional integral in the open unit disk U If the function h is analytic, then the initial value problem (1) is equivalent to the nonlinear Volterra integral equation