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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.

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