Sufficiency for general hypergeometric transform associated with conic region

The main object of the present paper is to find conditions on a, b, c andλ such that the operator Hλ a,b,c f (z) maps certain sub classes of analytic functions in to some other classes of fun ctions that have geometric properties related to certain co nic regions.


Introduction
Let A denote the class of functions f (z) of the form f (z) = z + ∞ ∑ n=2 a n z n , which are analytic in the open disc U = {z : |z| < 1} and S denotes the sub classes of the function of A, which are univalent in U. A function f ∈ A is called star like of order α, denotes f ∈ S * (α), if A function f ∈ A is called convex of order α, if and only if, The class of all convex functions of order α, are denoted by K(α). The classes K(α) and S * (α), where introduced and studied by Robertson [13]. For α = 0, the classes S * (α) and K(α) reduced to the classes S * respectively. Let S * (λ > 0) denotes the class of functions in S such that A sufficient condition for f ∈ A of the form (1) to be S * 1 ⊂S * , the class of starlike functions in U, is given by ∑ ∞ n=2 n |a n | ≤ 1, and is proved by many authors for example (see [6]). A particular extension of this, due to [16] is We further note that when f (z) is of the form (2), the condition (2) is both necessary and sufficient for f ∈ S * λ .
This class generalizes various other classes which are worthy of mention here. The class k − UCV (0) called the k-Uniformly convex is to [8] and has a geometric characterization given in the following way.
Let 0 ≤ k < ∞, the function f ∈ A is said to to be k-Uniformly convex in U if f is convex in U and the image of every circular arc γ contained in U ,with center ξ where |ξ | ≤ k, in convex.
The class UCV (0) = UCV , [5] describes geometrically the domain of values of the expression Using the Alexander transformation, we can obtain the class k − S p (0) in the following way if The classes UCV and S P : (1 − S p (0)) are unified and studied using a certain fractional calculus operator in [16], we refer the reader to [9,7,14] and references there in for some interesting results in these directions.

Definition 2. The Gaussian Hypergeometric function
where (a) 0 = 1, (a) n+1 = (a + n)(a) n , n = 0, 1, 2, · · · has appeared in the literature in many situations and contributed to various including conformal mappings, quasi conformal theory, and continued fractions [3,4]. Here a,b,c are complex numbers and {C = 0, −1, −2, −3, · · ·}. In the case of a = −k, or b = −k, where J = 0, 1, 2, · · · ,and K ≤ m in this case F(a, b; c; z) becomes a polynomials of degree K, we refer to a hypergeometric polynomials. The hypergeometric functions satisfies numbers and we remark that the behavior of the hypergeometric functions F(a, b; c; z) near z=1, is classified in to three case according as R e (c − b − a) is positive, zero or negative. The case c = a + b is called zero balanced case and hypergeometric R e (c) ≤ R e (a + b) as the asymptotic behavior in two case a + b = c and a + b > c has been refined in [1] and [12] respectively.
If R e (c − a − b) > 0 (see [18]), then In this paper, we introduce the operator H λ a,b;c f (z) such that

Main results and preliminary lemmas
We state a few results obtained in the literature by various author which are useful in proving our results.
, which can be written as the following simple analytic characterization: Throughout this paper by P γ (β ), we mean P τ β with τ = e iη cosη where − π 2 < η < π 2 we need the following sufficient condition on the sufficient of the class k − UCV (α).
It was also found that the condition (10) is necessary, if f ∈ A is of the form n=2 a n z n , a n ≥ 0.
Further more, the condition Is sufficient for f to be in k − S p (α) and turns out to be also necessary if f ∈ A is of the form (10).

Corollary 1. Let f ∈ A be defined as in
, .