Neighborhoods of Univalent Functions

Abstract

Abstract.For an analytic function f(z) = z + 2Zk0-2akzk m ^ unit disc E conditions are established such that all functions g(z) = z + 2£-2bkzk e ^«(Z)» i.e. X'k^2^\ak ~ *<tl < ^> are m some class of univalent functions in E. For instance, we prove that every g e Nl/4(f) is starlike univalent in E if / is convex univalent.Let A denote the class of analytic functions/in the unit disc E = {z\ \z\ < 1} with /(0) = 0, /'(0) = 1. For /(z) = z -f-2k°.2akzk G A and S > 0 we define the neighborhood Ns(f) as follows.

New Results Subclasses of Analytic Functions Define by Opooladifferential Operator

“…The concept of neighbourhoods was first introduced by Goodman in [1] and then generalized byRuschewey in [6].…”

A Certain Subclass of Meromorphically Multivalent Analytic Functions with Negative Coefficients

“…Depending on the earlier works by (Goodman, 1957;Ruscheweyh, 1981;Liu and Srivastava, 2004;Aouf and El-Ashwah, 2009) that based upon the familiar concept of neighborhood of analytic functions, we introduce the definition of the δ-neighborhood of a function …”

Some subclasses of analytic functions of complex order defined by new differential operator

“…By following the earlier investigations by Goodman [2] and Ruscheweyh [30], for any f (z) ∈ A (n) and δ ≥ 0, we define the (n, δ)-neighborhood of f by We say that a function f (z) ∈ A (n) is said to be starlike functions of complex order γ or f (z) ∈ S * n (γ) if it satisfies the inequality…”