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Extremal problems for a class of functions of positive real part and applications

Part of: Geometric function theory

Published online by Cambridge University Press:  09 April 2009

V. V. Anh  and
P. D. Tuan
V. V. Anh
Affiliation:
Department of Mathematics, Queensland Institute of Technology, P. O. Box 2434 Brisbane, Queensland 4001, Australia
P. D. Tuan
Affiliation:
First Interstate Bank of California, 600 South Spring Street, Los Angeles, California 90014, U.S.A.
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Abstract

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In this paper we determine the lower bound on |z| = r < 1 for the functional Re{αp(z) + β zp′(z)/p(z)}, α ≧0, β ≧ 0, over the class Pk (A, B). By means of this result, sharp bounds for |F(z)|, |F',(z)| in the family and the radius of convexity for are obtained. Furthermore, we establish the radius of starlikness of order β, 0 ≦ β < 1, for the functions F(z) = λf(Z) + (1-λ) zf′ (Z), |z| < 1, where ∞ < λ <1, and .

MSC classification

Secondary: 30C45: Special classes of univalent and multivalent functions (starlike, convex, bounded rotation, etc.)
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 41 , Issue 2 , October 1986 , pp. 152 - 164
Copyright
Copyright © Australian Mathematical Society 1986

References

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