A Note on Fractional Differential Subordination Based on the Srivastava-owa Fractional Operator

In this work, we consider a definition for the concept of fractional differential subordination in sense of Srivastava-Owa fractional operators. By employing some types of admissible functions involving differential operator of fractional order, we illustrate applications.


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 C as follows: Definition 1.1.The fractional derivative of order α is defined, for a function Further properties of these operators with applications can be found in [2][3][4][5][6].

PRELIMINARIES
Let H be the class of functions analytic in the unit disk

A function
Furthermore, Let P be the subclass of analytic functions in the unit disk and take the formula p is called a solution of the differential superordination.An analytic function q is called subordinate of the solution of the differential superordination if .p q  For details (see [7]).
Analogs to this definition, we impose the concept of fractional differential subordination.
 and let h be univalent in .
then p is called a solution of the fractional differential subordination.The univalent function q is called a dominant of the solutions of the fractional differential subordination, .
then p is called a solution of the fractional differential superordination.An analytic function q is called subordinate of the solution of the fractional differential superordination if .p q  It is clear that when 0, =


we have the differential subordination and the differential superordination of the first order.In the following sequel, we will assume that h is an analytic convex function in U with We will denote the class consisting of all solutions [7] We denote by Q the set of all functions ) (z f that are analytic and univalent on

FRACTIONAL DIFFERENTIAL SUBORDINATION
In this section, we establish some results which are related to the subordination of two functions.This will lead to develop and generalize the theory of differential subordinations.In addition, we show that the problem of finding best dominants of fractional subordination reduces to finding univalent solutions of fractional differential equations.
where m is a positive real number satisfying . n m  Consequently, by using Leibniz rule for fractional differentiation of analytic functions [8] yields This completes the proof.
We next consider the subordination of two functions.This will lead to suggest the concept of the fractional differential subordinations.If p is not subordination to , q then there exist and a real number where m is real.

Proof. Since
(0) = (0) q p and q p, are analytic in U we define

and this completes conclusion (i). Conclusion (ii) follows by applying Theorem 3.1.
We shall define the class of generalized admissible functions.This class plays an important role in the theory of fractional differential subordinations.The proof follows by applying Theorem 3.2.
and m is real.
Note that when 0 =  and n m  in Definition 3.1, we have the normal admissible functions.
and m real that satisfy (i)-(ii).Thus by Definition 3.1, we have  which contradicts (2); hence .q p  From the above result we pose dominants of fractional differential subordination (2) by using the generalized admissible function .
Proof.Since q is univalent in U then its univalent in ; U thus the set be simply connected domain and a conformal mapping Proof.Conditions ( 2) and ( 3) are equivalent.Thus in virtue of Theorem 3.3, we have .q p  Corollary 3.4.Let h and q be univalent in , U with a q = (0) and set Proof.By applying Theorem 3.3, we have .
has a univalent solution q and one of the following conditions is satisfied is analytic in U satisfies (4) then q p  and q is the best dominant.
Proof.In view of Corollaries 3.3, 3.4 and 3.5, we have q as the dominant of (4).Since q is a solution of (5), it implies that q is a solution for (4) and hence it is the best dominant.

APPLICATIONS
In this section, we deduce some applications of Theorem 3.3 and its corollaries.

 
q and h respectively.We will assume two cases; the first case corresponds to a point on the boundary rays on the sectors ) (U q and the second case corresponds to corner of the sector.that   (10)

Definition 1 . 2 .Remark 1 . 1 .
simply-connected region of the complex z-plane C containing the origin and the multiplicity of The fractional integral of order α is defined, for a function is analytic in simply-connected region of the complex z-plane ) (C containing the origin and the multiplicity of From Definitions 1.1 and 1.2, we have a positive real number

Definition 3 . 1 .
Let  be a set in

4 .
Let h be univalent in U and let

Theorem 4 . 1 .
Let n be a positive integer, a corner without itself has a corner.Hence this case does not hold.

Corollary 4 . 1 .
assumptions of the theorem.This completes the proof.Next we introduce some special cases of Theorem 4.we have the following result which can be found in[9]: Let n be a positive integer,