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A Geometric Extension of Schwarz’s Lemma and Applications

Published online by Cambridge University Press:  20 November 2018

Galatia Cleanthous
Galatia Cleanthous*
Affiliation:
Department of Mathematics, Aristotle University of Thessaloniki, Thessaloniki 54124, Greece e-mail: gkleanth@math.auth.gr
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Abstract

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Let $f$ be a holomorphic function of the unit disc $\mathbb{D}$ , preserving the origin. According to Schwarz’s Lemma, $\left| {{f}^{\prime }}(0) \right|\,\le \,1$ , provided that $f(\mathbb{D})\,\subset \,\mathbb{D}$ . We prove that this bound still holds, assuming only that $f(\mathbb{D})$ does not contain any closed rectilinear segment $\left[ 0,\,{{e}^{i\phi }} \right],\,\phi \,\in \,\left[ 0,\,2\pi\right]$ , i.e., does not contain any entire radius of the closed unit disc. Furthermore, we apply this result to the hyperbolic density and give a covering theorem.

Keywords

30C8030C2530C99Schwarz’s Lemmapolarizationhyperbolic densitycovering theorems

Information

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 59 , Issue 1 , 01 March 2016 , pp. 30 - 35
Copyright
Copyright © Canadian Mathematical Society 2016

References

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