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Bloch and normal functions on general planar regions. (English) Zbl 0654.30025

Holomorphic functions and moduli I, Proc. Workshop, Berkeley/Calif. 1986, Publ., Math. Sci. Res. Inst. 10, 101-110 (1988).
[For the entire collection see Zbl 0646.00004.]
Suppose \(\Omega\) is a hyperbolic region in the complex plane \({\mathbb{C}}\). Let \(\lambda_{\Omega}\) be the density of the hyperbolic metric on \(\Omega\). If \(\delta_{\Omega}(z)=dist(z,\partial \Omega)\), then \(1/\delta_{\Omega}\) is the quasihyperbolic density on \(\Omega\). In certain contexts the roles of \(\lambda_{\Omega}\) and \(1/\delta_{\Omega}\) are interchangeable. A holomorphic function f defined on \(\Omega\) is a Bloch function if \[ \| f\|_{B(\Omega)}=\sup \{| f'(z)| /\lambda_{\Omega}(z):\quad z\in \Omega \} \] is finite; while f is quasi-Bloch if \[ \| f\|_{QB(\Omega)}=\sup \{| f'(z)| \delta_{\Omega}(z):\quad z\in \Omega \} \] is finite. These classes are identical; more precisely, \[ \| f\|_{QB(\Omega)}\leq \| f\|_{B(\Omega)}\leq (16/\sqrt{3})\| f\|_{QB(\Omega)}. \] The equivalence of Bloch and quasi-Bloch functions leads to a new characterization of non-Bloch functions: f is not a Bloch function if and only if there exist a sequence \(\{z_ n\}\) in \(\Omega\), a sequence \(\{\rho_ n\}\) of positive numbers with \(\rho_ n/\delta_{\Omega}(z_ n)\to 0\), and a unimodular constant a such that \(f(z_ n+\rho_ n\zeta)-f(z_ n)\to a\zeta\) locally uniformly on \({\mathbb{C}}\). For meromorphic functions the situation concerning the interchangeability of the two densities is different. A meromorphic function f on \(\Omega\) is normal if \[ \| f\|_{N(\Omega)}=\sup \{f^{\#}(z)/\lambda_{\Omega}(z):\quad z\in \Omega \} \] is finite \((f^{\#}(z)\) is the spherical derivative of f) and quasi-normal if \(\| f\|_{QN(\Omega)}=\sup \{f^{\#}(z)\delta_{\Omega}(z):\) \(z\in \Omega \}\) is finite. Every normal function is quasi-normal and \(\| f\|_{QN(\Omega)}\leq \| f\|_{N(\Omega)}\) but a quasi-normal function need not be normal. In the context of meromorphic functions the two densities are interchangeable in the sense that there is a constant n(\(\Omega)\) such that \(\| f\|_{N(\Omega)}\leq n(\Omega)\| f\|_{QN(\Omega)}\) for all meromorphic functions f defined on \(\Omega\) if and only if \(\Omega\) is uniformly perfect.
Reviewer: D.Minda

MSC:

30D45 Normal functions of one complex variable, normal families
30C99 Geometric function theory

Citations:

Zbl 0646.00004