Abstract
The purpose of the present paper is to determine the necessary and sufficient conditions for the power series B_{\mu} whose coefficients are probabilities of the Borel distribution to be in the family H(\lambda, \sigma, \delta, \mu) of analytic functions which defined in the open unit disk. We derive a number of important geometric properties, such as, coefficient estimates, integral representation, radii of starlikeness and convexity. Also we discuss the extreme points and neighborhood property for functions belongs to this family.
Highlights
Indicate by A the family of all functions f of the form ∞f (z) = z + anzn, (1.1)n=2 which are analytic and univalent in the open unit disk U = {z ∈ C : z < 1}.let W denote the subfamily of A consisting of functions of the form: Received: March 7, 2019; Accepted: April 2, 20202010 Mathematics Subject Classification: 30C45.Keywords and phrases: analytic function, Borel distribution, probability, coefficient estimates, integral representation, neighborhood.Abbas Kareem Wanas and Jubran Abdulameer Khuttar f (z) = z − anzn. (1.2) n=2
We introduce a power series whose coefficients are probabilities of the Borel distribution, that is
We begin this section by defining the family H(γ, δ, τ, μ) as follows: Definition 2.1
Summary
Indicate by A the family of all functions f of the form Let W denote the subfamily of A consisting of functions of the form: Received: March 7, 2019; Accepted: April 2, 2020 Keywords and phrases: analytic function, Borel distribution, probability, coefficient estimates, integral representation, neighborhood. The function f ∈ W is said to be starlike of order α (0 ≤ α < 1) if it satisfies the condition:
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