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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
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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 Schwarz's Lemma, polarization, hyperbolic density, covering theorems
MSC Classifications: 30C80, 30C25, 30C99 show english descriptions Maximum principle; Schwarz's lemma, Lindelof principle, analogues and generalizations; subordination
Covering theorems in conformal mapping theory
None of the above, but in this section
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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