On the possible rate of growth of an analytic function

Abstract

Introduction('). The present paper deals with a number of diverse topics, ranging from purely topological considerations, through a general theory of possible distributions of values of an analytic function, to more special theorems on simultaneous expansions of an infinity of analytic functions. No unifying principle is presented as an excuse for treating such a variety of subjects; but there is a slight sequence of argument running throughout. The parts of the paper were actually written in reverse order. The initial investigation (Part III) was started as an attempt to generalize, from n to infinity, a known theorem(2) on simultaneous expansions of n analytic functions. This generalization was found to depend on an affirmative answer to the following question on level curves of an analytic function: Given any sequence of points a1, a2, . , in the complex plane, which has the point at infinity as its only limit point, does there exist an analytic function with a level curve C such that C contains a distinct branch about each given point which separates that point from all the other points? This was, in turn, made to depend on a certain problem relative to the possible rate of growth of an integral function. It was shown long, ago by Poincare(3), Borel(4), and others that an integral function may be made to grow arbitrarily fast along the real axis or along other lines or curves extending to infinity. Our problem was to obtain an affirmative answer to the following related question: Does there exist a sequence of regions Si, S2, . , with ai interior to Si, such that, no matter how fast the sequence of numbers m1, M2, increases there will be an integraL function f(z) for which