International Journal of Mathematics and Mathematical Sciences

International Journal of Mathematics and Mathematical Sciences / 2002 / Article

Open Access

Volume 32 |Article ID 914873 | 12 pages | https://doi.org/10.1155/S0161171202111136

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

Received14 Nov 2001

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=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).

Copyright © 2002 Hindawi Publishing Corporation. 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.


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