L. R. Sons and Zhuan Ye

Copyright © 2002 L. R. Sons and Zhuan Ye. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Let β>0 and let α be an integer which is at least 2. If f is an analytic function in the unit disc D which has power series representation f(z)=∑k=0∞ak zkα, limsupk→∞ (log+|ak|/logk)=α(1+β), then the first author has proved that f is unbounded in every sector {z∈D:Φ−ϵ<argz<Φ+ϵ, for ϵ>0}. A natural conjecture concerning these functions is that limsupr→1−(logL(r)/logM(r))>0, where L(r) is the minimum of |f(z)| on |z|=r and M(r) is the maximum of |f(z)| on |z|=r. In this paper, investigations concerning this conjecture are discussed. For example, we prove that limsupr→1−(logL(r)/logM(r))=1 and limsupr→1−(L(r)/M(r))=0 when ak=kα(1+β).