Subclass of p-valent Function with Negative Coefficients Applying Generalized Al-Oboudi Differential Operator

Abstract In this paper we introduce a new subclass R∗(p, g, ψ, %, β, φ, γ, ζ) of p-valent functions with negative coefficient defined by Hadamard product associated with a generalized differential operator. Radii of close-to-convexity, starlikeness and convexity of the class R∗(p, g, ψ, %, β, φ, γ, ζ) are obtained. Also, distortion theorem, growth theorem and coefficient inequalities are established.

Let T denote the subclass of G consisting of the form where l w ≥ 0 and w ∈ N. This class has introduced and studied by Silverman [9].
The Hadamard product of two power series and it is defined in T as follows: Let G p denote the class of functions of the form f (z) = z p + ∞ w=1 l p+w z p+w (1.6) that are holomorphic and p-valent in |z| < 1.
Also let T p denote the subclass of G p consisting of functions that can be expressed as (1.7) The Hadamard product of two power series and it is defined in T p as follows: l p+w j p+w z p+w .

Coefficient Inequalities
In the following theorem we obtain necessary and sufficient condition for a function f (z) to be in the class R * (p, g, ψ, , β, φ, γ, ζ). We have the following lemma useful for this work.    Then f (z) ∈ R * (p, g, ψ, , β, φ, γ, ζ) if and only if The result is sharp for the function Proof. If f (z) ∈ R * (p, g, ψ, , β, φ, γ, ζ) and |z| = 1, then by Definition 1.1 Using Lemma 2.2, it is sufficient to show that For convenience, let and That is equation (2) is equivalent to Also, It is easy to show that Conversely, suppose the inequality (2.5) holds, we need to show that Since |e iθ | = 1, hence (e iθ ) ≤ |e iθ | = 1, letting |z| −→ 1 −1 yields, we let H = which completes the proof.

Growth Theorem and Distortion Theorem
Theorem 3.1. If f (z) ∈ R * (p, g, ψ, , β, φ, γ, ζ) and j p+w ≥ j 2 , then The result is sharp for, (|z| = r < 1) Proof. Since Using Theorem 2.3, we have using the above equation in 3.2, we have The result is sharp for Similarly, since Using Theorem 2.3, we have using the above equation in 3.2, This completes the proof.
Remark 4.4. If we put p = 1 in Theorems 2.3, 3.1 and 4.1, we obtain the corresponding result studied by Godwin and Opoola [7].