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Covering properties of meromorphic functions, negative curvature and spherical geometry. (English) Zbl 0970.30016
Let \(f\) be meromorphic in a domain \(G\), \(z_0\in G\). Let \(\varphi\) be the branch of \(f^{-1}\) such that \(\varphi(f(z_0)) =z_0\). If the function \(\varphi\) has an analytic continuation as a meromorphic function in the open disc \(D(f(z_0), b_f(z_0))\) of spherical radius \(b_f(z_0)\) centered at \(f(z_0)\) and does not continue in a grater disc, then the spherical Bloch radius of \(f\) is defined as \(b_f=\sup_{z\in G}b_f(z)\). If we have the family \(M\) of meromorphic functions in \(G\), then the spherical Bloch radius of \(M\) is defined as \(\inf_{f\in M}b_f\). The first results in this direction are due to Valiron (1923), Bloch (1926). The authors find \(b_{M_1}=\arctan\sqrt{8}\) for the class \(M_1\) of meromorphic functions in the whole plane and \(b_{M_2}=\pi/2\) for the family \(M_2\) of those functions from \(M_1\) which have only multiple critical points.

30D35 Value distribution of meromorphic functions of one complex variable, Nevanlinna theory
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