Abstract
In this paper we introduce a new subclass $\mathcal{R}^*(p,g,\psi,\varrho,\beta,\phi,\gamma,\zeta)$ 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 $\mathcal{R}^*(p,g,\psi,\varrho,\beta,\phi,\gamma,\zeta)$ are obtained. Also, distortion theorem, growth theorem and coefficient inequalities are established.
Highlights
Introduction and DefinitionsLet G be class of functions f (z) of the form ∞f (z) = z + lwzw w=2 (1.1)Received: December 1, 2020; Accepted: January 2, 20212010 Mathematics Subject Classification: Primary 30C45; Secondary 30C50.Keywords and phrases: close to convexity, univalent functions, Hadamard product, convexity, Al-Oboudi differential operator, starlike function.Copyright c 2021 AuthorsT
G(z) = z − jwzw w=2 and it is defined in T as follows:
Let Gp denote the class of functions of the form
Summary
Keywords and phrases: close to convexity, univalent functions, Hadamard product, convexity, Al-Oboudi differential operator, starlike function. A. Ibrahim which are holomorphic in the open unit disk = {z ∈ C : |z| < 1}. The Hadamard product of two power series g(z) = z − jwzw w=2 and it is defined in T as follows:. Let Gp denote the class of functions of the form f (z) = zp + lp+wzp+w w=1 http://www.earthlinepublishers.com (1.6). Let Tp denote the subclass of Gp consisting of functions that can be expressed as f (z) = zp − lp+wzp+w. Motivated by [2], [10], [7], we define a new subclass R∗(p, g, ψ, , β, φ, γ, ζ) of the class Tp. 0, we let R∗(p, g, ψ, , β, φ, γ, ζ) be subclass of the class Tp consisting of functions of the form (1.7) and satisfying the analytic criterion zDφh,,ζγ,p(f Dφh,,ζγ,p(f. Of starlikeness, distortion properties, convexity and close to convexity for the class R∗(p, g, ψ, , β, φ, γ, ζ). [5], [9], [1], [4], [8], [3], study the univalent functions for different classes
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