On spherical invariance

Abstract

In 1964 Pommerenke introduced the notion of linear invariant family for locally injective analytic functions defined in the unit disk of the complex plane. Following Ma and Minda (who extended this notion to spherical geometry), we consider in this paper locally injective meromorphic functions in the unit disk. More precisely, we study families of such functions for which a certain invariant, called spherical order, is finite. Several consequences on the finiteness of the spherical order are explored, in particular the connection with the Schwarzian and normal orders, and with uniform perfectness.