A Localization Principle for a Class of Analytic Functions

Abstract

The object of this note is to prove an extension of Noshiro's theorem. Definition 1. A function w-f{z)which is bounded and analytic in |z|<l and whose radial limit values lim/(r^θ) =/*(e ίθ ) exist and are of modulus one for all r->l points e tQ on \z\ = 1 except for at most a set S of points e tQ of linear measure zero will be called of class (U) in \ z I < 1. If the possible exceptional set S is of logarithmic capacity zero, cap (S)=0, w = f(z) will be said to be of class (17*) in \z\<l.For the sake of completeness we prove the following lemma.