Normal families and the Nevanlinna theory

Abstract

1. Let ~ be a family of nonconstant holomorphic functions defined in the disc A = {Izl < 1}. :~ is said to be normal if every sequence of functions in :~ either contains a subuniformly convergent subsequence, or contains a subsequence which converges subuniformly to the constant co. A family :~ of meromorphic functions is normal when every sequence of functions of :~ has a subsequence which is subuniformly convergent with respect to the chordal metric. P. Montel [15] first realized the scope and coherence of these families, and used them to give a particularly unified t reatment of Picard's great theorems, and Schottky's and Landau's theorems. The fact that these results were so intimately related led A. Bloch to the hypothesis tha t precisely those properties which reduce a function meromorphic in C (= { I z I < ~ }) to a constant, make normal a family of functions meromorphic in A.