Univalent Functions Defined by a Generalized Multipler Differential Operator

In this paper, we investigate a new subclass of univalent functions defined by a generalized differential operator, and obtain some interesting properties of functions belonging to the class Rλ,μ(α, β, γ, θ).


Introduction
Let A denote the class of the functions f of the form which are analytic in the open unit disc U = {z ∈ C : |z| < 1}.Let H(U) be the space of holomorphic functions in U.By S and K we denote the subclasses of functions in A which are univalent and convex in U, respectively.Let P be the well-known Caratheodory class of normalized functions with positive real part in U.
The Hadamard product or convolution of the functions is given by We now define a new generalized multiplier differential operator.
The main object of this paper is to present a systematic investigation for the class R m λ,µ (α, β, γ, ϑ).In particular, for this function class, we derive an inclusion result, structural formula, extreme points and other interesting properties.

Preliminaries
In order to prove our results, we will make use of the following lemmas.Lemma 1. [13] Let h ∈ K, and Let A ≥ 0. Suppose that B(z) and D(z) are analytic in U, with D(0) = 0 and If an analytic function p with p(0) = h(0) satisfies Lemma 2. [14] Let q be a convex function in U and let h(z) = q(z) + zq (z), p(z) ≺ q(z), z ∈ U, and this result is sharp.
Lemma 3. [15] If p(z) is analytic in U, p(0) = 1 and −1 (z) and , (p(z)) > 1 2 , then for any function F analytic in U , the function F * p takes values in the convex hull of F(U).
Note that the symbol " ≺ " stands for subordination throughout this paper.
Proof.Let the functions Using ( 9) in ( 8), we obtain Theorem 3. Let q be convex function with q(0) = 1 and let h be a function of the form h λ,µ (α, β, γ, ϑ)/z ≺ q(z) and the result is sharp.
The result is sharp.

Structural Formula
In this section, a structural formula, extreme points and coefficient bounds for functions in R m λ,µ (α, β, γ, ϑ) are obtained.
Using Hergoltz integral representation of functions in Caratheodory class P (see [19] and [20]), there exists a positive Borel probability measure σ such that Integrating (11), we obtain that is Equality ( 10) follows now, from (4), ( 5) and (12).Since the converse of this deductive process is also true, we have proved our theorem.
The result is sharp.
Proof.The coefficient bounds are maximized at an extreme point.Therefore, the result follows from (13).

Convolution Property
in this part, we prove the analogue of the Polya-Schoenberg conjecture for the class R m λ,µ (α, β, γ, ϑ).