Holomorphic Functions with Spiral asymptotic Paths

Abstract

Let f(z) be a holomorphic and unbounded function in | z | < 1, with the property that it remains bounded on some spiral S in | z | < 1 which approaches | z | = 1 asymptotically. The existence of such functions was first established by G. Valiron. Accordingly, we shall refer to such functions as functions of class (V) relative to S. More recently, F. Bagemihl and W. Seidel obtained examples of functions holomorphic and unbounded in | z | < 1 which approach prescribed finite or infinite values as | z | → 1 on any given enumerable set of disjunct spirals which approach | z | =1 asymptotically, as well as on certain sets of such spirals having the power of the continuum.