Local and global subordination theorems for vector-valued analytic functions

Abstract

Introduction. We extend the basic result on the subordination of analytic functions to analytic functions with values in a Banach space. The fundamental form of this theorem is as follows [2, p. 421]: Let f and g be two analytic (scalarvalued) functions defined on the unit disc D with g univalent and the range of f contained in the range of g. Then f and g are analytically related, i.e. there is an analytic function co mapping D into itself with f(z) =g(cw(z)) for all z in D. This result extends to vector-valued analytic functions (for definitions see [1]) but there are subtleties: (1) the proof for scalar-valued functions consists of noting that w(z)=g-1(f(z)) is analytic; for vector-valued functions not only is the proper notion of analyticity for g-1 unavailable but, as we show below, g -1 need not even be continuous, and (2) the result for scalar-valued functions is true for arbitrary domains (usually stated for D only to avoid superfluous generality) while for vector-valued functions we cannot allow punctures in the domain of g. The basic theorem on subordination has as a simple consequence the interesting result that if f and g are nonconstant analytic (scalar-valued) functions on D with intersecting ranges then they are locally analytically related, i.e. there is a neighborhood V in D and an analytic function co mapping V into D with f(z) =g(cv(z)) for z in V. This result also extends to vector-valued analytic functions but in this case the appropriate hypothesis is thatf and g have ranges intersecting in an uncountable set. For f and g scalar-valued, if the ranges of f and g intersect at all, then they intersect in an uncountable set by the open mapping theorem; so, as in the basic theorem, it is the failure of the open mapping theorem for vector-valued analytic functions which lends interest to the generalization and makes the proof more delicate. We will show that for vector-valuedf and g their ranges may intersect in a set containing an infinite number of accumulation points and yet not be locally analytically related.