Classes of meromorphic functions characterizable by growth of the spherical derivative
type: Item
instance of: scholarly article
title: Classes of meromorphic functions characterizable by growth of the spherical derivative
zbMATH Open document ID: 0830.30017
DOI: 10.1007/BF02104943
published in: Siberian Mathematical Journal
publication date: 1995-09-18T00:00:00Z
review text: Let \(\rho (f) = |f' |/1 + |f |^2\) be the spherical derivative of a meromorphic function \(f\), \(T(r,f)\) be the Nevanlinna characteristic function. A set \(E\) is called a Picard set for a given class \(B\) of meromorphic functions if the Picrd theorem is valid on \(\mathbb{C} \backslash E\) for every function \(f \in B\). Let \(\varphi\) be a piecewise continuous function, \(\varliminf_{r \to \infty} r \varphi (r) > 0\), and let \(M (\varphi)\) be the class of meromorphic functions such that \(\varlimsup_{z \to \infty} \rho (f(z))/ \varphi (|z |) = \infty\). The main results are the following. Theorem 1.1. Let \(\Phi\) be a logarithmically convex function on \([1, \infty) \) and \(\Phi (2r) = O (\Phi (r))\), \(r \to \infty\). If \(f\) is a transcendental meromorphic function satisfying the conditions \(\rho (f(z)) = O (\Phi' (z))\) and \(T(r,f) \neq O (\Phi (r))\), \(|z |= r \to \infty\) then \(f\) has no Nevanlinna exceptional values. Theorem 1.2. Let \(E = (a_n)\) be a sequence of numbers in \(\mathbb{C}\) with unique accumulation point at infinity. If \((\exists \varepsilon > 0)\) \((\exists n_0 \in \mathbb{N})\) \((\forall n \geq n_0)\) \([\{z : |z - a_n |< \varepsilon / \varphi (|a_n |)\} \cap E = \{a_n\}]\), then \(E\) is a Picard set for the function class \(M(\varphi)\). The theorem 1.1 generalizes the assertion of J. M. Anderson and J. Clunie (1966).
Mathematics Subject Classification ID: 30D30
zbMATH DE Number: 794618
zbMATH Keywords: slowly growing meromorhpic functions
zbMATH Keywords: spherical derivative
author: N. V. Zabolotskii
MaRDI profile type: Publication
cites work: Q3825006
cites work: Q5719719
cites work: Q4711784
cites work: Reduction of nonholonomic variation problems to isoperimetric ones and connections in principal bundles
cites work: On a constrained variational problem and the space of horizontal paths
full work available at URL: https://doi.org/10.1007/bf02104943
OpenAlex ID: W2046838141
description: scientific article; zbMATH DE number 794618