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UNIVALENCE CRITERIA AND ANALOGUES OF THE JOHN CONSTANT

Part of: Geometric function theory

Published online by Cambridge University Press:  12 December 2012

YONG CHAN KIM  and
TOSHIYUKI SUGAWA
YONG CHAN KIM
Affiliation:
Department of Mathematics Education, Yeungnam University, 214-1 Daedong Gyongsan 712-749, Korea email kimyc@ynu.ac.kr
TOSHIYUKI SUGAWA*
Affiliation:
Graduate School of Information Sciences, Tohoku University, Aoba-ku, Sendai 980-8579, Japan
*
✉sugawa@math.is.tohoku.ac.jp
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Abstract

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Let $p(z)= z{f}^{\prime } (z)/ f(z)$ for a function $f(z)$ analytic on the unit disc $\mid z\mid \lt 1$ in the complex plane and normalised by $f(0)= 0, {f}^{\prime } (0)= 1$. We provide lower and upper bounds for the best constants ${\delta }_{0} $ and ${\delta }_{1} $ such that the conditions ${e}^{- {\delta }_{0} / 2} \lt \mid p(z)\mid \lt {e}^{{\delta }_{0} / 2} $ and $\mid p(w)/ p(z)\mid \lt {e}^{{\delta }_{1} } $ for $\mid z\mid , \mid w\mid \lt 1$ respectively imply univalence of $f$ on the unit disc.

Keywords

univalent functionunivalence criterionGrunsky coefficients

MSC classification

Secondary: 30C55: General theory of univalent and multivalent functions
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 88 , Issue 3 , December 2013 , pp. 423 - 434
Copyright
Copyright ©2012 Australian Mathematical Publishing Association Inc. 

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