A version of the Lohwater-Pommerenke theorem for strongly normal functions
A meromorphic function \(f\) in the unit disk is said to be normal if its spherical derivative is \(O(1-|z|^2)^{-1}\) as \(|z|\rightarrow 1.\) When the spherical derivative is \(o(1-|z|^2)^{-1},\) the authors call \(f\) strongly normal. A theorem of Lohwater and Pommerenke from 1973 gives a necessary and sufficient condition for normality: \(f\) is not normal if and only if there exists a sequence \(\{z_n\}\) in the disk and a rapidly decreasing positive sequence \(\{\rho_n\}\)such that the sequence of functions \(g_n(w) = f(z_n + \rho_n w)\) converges locally uniformly in the spherical metric to a function \(g(w)\) which is meromorphic and nonconstant in the whole plane. In the paper under review the authors provide a corresponding characterization for strongly normal functions: \(f\) is not strongly normal if and only if there is a sequence \(\{g_n\}\) as above which converges locally uniformly in the spherical metric to a function \(g\) which is meromorphic and nonconstant in some disk \(|w|< R.\) As application, the authors show that the solutions to certain algebraic differential equations are strongly normal. They also note that their methods prove a corresponding characterization for functions in the little Bloch space \({\mathcal B}_o.\) One need only substitute ``locally uniformly in the Euclidean metric'' for ``locally uniformly in the spherical metric''.
scientific article; zbMATH DE number 1833059