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# A Geometric Extension of Schwarz's Lemma and Applications

Published:2015-08-04
Printed: Mar 2016
• Galatia Cleanthous,
Department of Mathematics, Aristotle University of Thessaloniki, Thessaloniki 54124, Greece
 Format: LaTeX MathJax PDF

## Abstract

Let $f$ be a holomorphic function of the unit disc $\mathbb{D},$ preserving the origin. According to Schwarz's Lemma, $|f'(0)|\leq1,$ 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 $[0,e^{i\phi}],\;\phi\in[0,2\pi],$ i.e. does not contain any entire radius of the closed unit disc. Furthermore, we apply this result to the hyperbolic density and we give a covering theorem.
 Keywords: Schwarz's Lemma, polarization, hyperbolic density, covering theorems
 MSC Classifications: 30C80 - Maximum principle; Schwarz's lemma, Lindelof principle, analogues and generalizations; subordination 30C25 - Covering theorems in conformal mapping theory 30C99 - None of the above, but in this section

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