Vol. 69, No. 1, 1977

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On starlikeness and convexity of certain analytic functions

V. V. Anh and P. D. Tuan

Vol. 69 (1977), No. 1, 1–9
Abstract

Let N be the class of normalised regular functions

         ∑∞    k
f(z) = z +   akz,  |z| < 1.
k=2

For 0 λ < 1, γ 1, let f(z),g(z) N be such that

|f(z)∕[λf(z)+ (1− λ)g(z)]− γ| < γ, |z| < 1.

We establish the radius of starlikeness of f(z) under the assumption Re {g(z)∕z} > 0, or Re {g(z)∕z} > 12, or Re {zg(z)∕g(z)} > α, 0 α < 1, or Re {1 + zg′′(z)∕g(z)} > 0 for |z| < 1. The analysis may be extended to the problem of finding the radius of convexity for certain subclasses of N.

Mathematical Subject Classification
Primary: 30A32, 30A32
Milestones
Received: 15 October 1976
Published: 1 March 1977
Authors
V. V. Anh
P. D. Tuan