A Subclass of Harmonic Functions Associated with Wright ’ s Hypergeometric Functions

A continuous function f = u + iv is a complex-valued harmonic function in a complex domain G if both u and v are real and harmonic in G. In any simply-connected domain D ⊂ G, we can write g h f   , where h and g are analytic in D. We call h the analytic part and g the co-analytic part of f. A necessary and sufficient condition for f to be locally univalent and orientation preserving in D is that | ) ( ' | | ) ( ' | z g z h  in D (see [1]). Denote by H the family of functions g h f   (1) which are harmonic, univalent and orientation preserving in the open unit disc } 1 | :| {   z z U so that f is normalized by 0 1 ) 0 ( ) 0 ( ) 0 (     z f h f . Thus, for g h f   ∈ H, we may express


Introduction
A continuous function f = u + iv is a complex-valued harmonic function in a complex domain G if both u and v are real and harmonic in G.In any simply-connected domain D ⊂ G, we can write , where h and g are analytic in D. We call h the analytic part and g the co-analytic part of f.A necessary and sufficient condition for f to be locally univalent and orientation preserving in in D (see [1]).Denote by H the family of functions which are harmonic, univalent and orientation preserving in the open unit disc where the analytic functions h and g are in the forms

 
We note that the family H of orientation preserving, normalized harmonic univalent functions reduces to the well known class S of normalized univalent functions if the co-analytic part of f is identically zero, that is g ≡ 0.
Let the Hadamard product (or convolution) of two power series  , B 1 ,..., q  q B ( p, q  N = 1, 2, 3, ...) such that ., 0 , 1 is the generalized hypergeometric function (see for details [3]) where N denotes the set of all positive integers and n ) ( is the Pochhammer symbol and By using the generalized hypergeometric function Dziok and Srivastava [3] introduced the linear operator.In [4] Dziok and Raina extended the linear operator by using Wright generalized hypergeometric function.First we define a function   be a linear operator defined by , , ; * () We observe that, for f(z) of the form (1), we have , where If, for convenience, we write introduced by Dziok and Raina [4].
It is of interest to note that, if A t = 1 (t = 1, 2, ..., p), B t = 1 (t = 1, 2, ...,q) in view of the relationship (6) the linear operator (8) includes the Dziok-Srivastava operator (see [3]), for more details on these operators see [3,4,6,7] and [8].It is interesting to note that Wright generalized hypergeometric function contains, Dziok-Srivastava operator as its special cases, further other linear operators the Hohlov operator, the Carlson-Shaffer operator [6], the Ruscheweyh derivative operator [7], the generalized Bernardi-Libera-Livingston operator, the fractional derivative operator [8], and so on.For example if p = 2 and q = 1 with From ( 8) now we define, Wright generalized hypergeometric harmonic function of the form (1), as and we call this as Wright generalized operator on harmonic function.

U z 
We also let where H V the class of harmonic functions with varying arguments introduced by Jahangiri and Silverman [10], consisting of functions f of the form (1) in H for which there exists a real number φ such that In this paper we obtain a sufficient coefficient condition for functions f given by (2) to be in the class ) ], . It is shown that this coefficient condition is necessary also for functions belonging to the class ) ], . Further, distortion results and extreme points for functions in ) ], are also obtained.

The Class , H 1 WS α γ   
We begin deriving a sufficient coefficient condition for the functions belonging to the class ) ], be given by ( 2).If . Proof.We first show that if the inequality (15) holds for the coefficients of , then the required condition ( 13) is satisfied.Using ( 11) and ( 13), we can write In view of the simple assertion that Substituting for A(z) and B(z) the appropriate expressions in (16), we get  by virtue of the inequality (15).This implies that ) ], . Now we obtain the necessary and sufficient condition for function be given with condition (14).
be given by ( 2) and for we only need to prove the necessary part of the theorem.Assume that ) ], , then by virture of ( 11) to (13), we obtain The above inequality is equivalent to This condition must hold for all values of z, such that |z| = r < 1. Upon choosing φ according to (14)

Distortion Bounds and Extreme Points
In this section we obtain the distortion bounds for the functions ) ], that lead to a covering result for the family ) ], Proof.We will only prove the right-hand inequality of the above theorem.The arguments for the left-hand inequality are similar and so we omit it.Let which establish the desired inequality.
As consequences of the above theorem and corollary 1, we state the following corollary.
and of the form (2) be so that ) ], For a compact family, the maximum or minimum of the real part of any continuous linear functional occurs at one of the extreme points of the closed convex hull.Unlike many other classes, characterized by necessary and sufficient coefficient conditions, the family ],  is not a convex family.Nevertheless, we may still apply the coefficient characterization of the ], ([ 1  to determine the extreme points.
then for 1 b fixed, the extreme points for clco where the coefficient satisfy the inequality (15).Set , ) ( In particular, putting , and so the proof is complete.

Inclusion Relation
Following Avici and Zlotkiewicz [9] (see also Ruscheweyh [14]), we refer to the δ-neighborhood of the function f(z) defined by (2) to be the set of functions F for which In our case, let us define the generalized δ−neighborhood of f to be the set .
Theorem 5. Let f be given by ( 2).If f satisfies the conditions