Geodesic
A Geometric Extension of Schwarz’s Lemma and Applications
Canadian mathematical bulletin, Tome 59 (2016) no. 1, pp. 30-35

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DOI

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.
DOI : 10.4153/CMB-2015-054-0
Mots-clés : 30C80, 30C25, 30C99, Schwarz’s Lemma, polarization, hyperbolic density, covering theorems 
Cleanthous, Galatia. A Geometric Extension of Schwarz’s Lemma and Applications. Canadian mathematical bulletin, Tome 59 (2016) no. 1, pp. 30-35. doi: 10.4153/CMB-2015-054-0
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