Covering properties of meromorphic functions, negative curvature and spherical geometry

M. Bonk and A. Eremenko

Review from Zentralblatt MATH:

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.

Reviewed by Anatoly Filip Grishin

Keywords: five islands theorem; Gaussian curvature; spherical triangle; asymptotic value

Classification (MSC2000): 30D35

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