Classes of meromorphic functions characterizable by growth of the spherical derivative
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).
scientific article; zbMATH DE number 794618