On a Question of Seidel Concerning Holomorphic Functions Bounded on a Spiral

Abstract

Let S be a spiral contained in D = {| z | < 1} such that S tends to C = {| z | = 1}. For the sake of brevity, by “ f is bounded on S ” we shall mean that f is holomorphic in D , unbounded, and bounded on S . The existence of such functions was first discussed by Valiron ( 9; 10 ); see also ( 1; 3; 8 ). Valiron also proved that any function that is “bounded on a spiral” must have the asymptotic value ∞ ( 10 , p. 432). Functions that are bounded on a spiral may also have finite asymptotic values ( 1 , p. 1254). In view of the above, Seidel has raised the question (oral communication): “Does there exist a function bounded on a spiral that has only the asymptotic value ∞?”. The following theorem answers this question affirmatively.

Locations

• Canadian Journal of Mathematics - View - PDF