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Search: All articles in the CMB digital archive with keyword Schwarz's lemma

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1. CMB 2015 (vol 59 pp. 30)

Cleanthous, Galatia
 A Geometric Extension of Schwarz's Lemma and Applications 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 theoremsCategories:30C80, 30C25, 30C99

2. CMB 2012 (vol 56 pp. 241)

Betsakos, Dimitrios; Pouliasis, Stamatis
 Versions of Schwarz's Lemma for Condenser Capacity and Inner Radius We prove variants of Schwarz's lemma involving monotonicity properties of condenser capacity and inner radius. Also, we examine when a similar monotonicity property holds for the hyperbolic metric. Keywords:condenser capacity, inner radius, hyperbolic metric, Schwarz's lemmaCategories:30C80, 30F45, 31A15
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