Vol. 43, No. 1, 1972

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On the univalence of some analytic functions

Ghulam M. Shah

Vol. 43 (1972), No. 1, 239–250
Abstract

Let

           ∑∞     k
f (z) = z +     akz
k=n+1

and

           ∞∑
g(z) = z +     akzk
k=n+1

be analytic and satisfy

Re(f(z)∕[λf(z)+ (1− λ)g(z)]) > 0
(a)

or

|f(z)∕[λf(z)+ (1− λ)g(z)]− 1| < 1 for |z| < 1,0 ≦ λ < 1.
(b)

We propose to determine the values of R such that f(z) is univalent and starlike for |z| < R under the assumption (i) Re(g(z)∕z) > 0, or (ii) Re(zg(z)∕g(z)) > α,0 α < 1.

We also consider the case when n = 1 and Re(g(z)∕z) > 12 and show that under condition (a) f(z) is univalent and starlike for |z| < (1 λ)(3 + λ).

Mathematical Subject Classification
Primary: 30A32
Milestones
Received: 1 July 1971
Revised: 28 January 1972
Published: 1 October 1972
Authors
Ghulam M. Shah