International Journal of Mathematics and Mathematical Sciences
Volume 32 (2002), Issue 1, Pages 29-40

A class of gap series with small growth in the unit disc

L. R. Sons and Zhuan Ye

Department of Mathematical Sciences, Northern Illinois University, DeKalb 60115, IL, USA

Received 14 November 2001

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.


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=0akzkα, limsupk(log+|ak|/logk)=α(1+β), then the first author has proved that f is unbounded in every sector {zD:Φϵ<argz<Φ+ϵ, for ϵ>0}. A natural conjecture concerning these functions is that limsupr1(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 limsupr1(logL(r)/logM(r))=1 and limsupr1(L(r)/M(r))=0 when ak=kα(1+β).