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

Sons, L. R.; Ye, Zhuan

doi:https://doi.org/10.1155/S0161171202111136

International Journal of Mathematics and Mathematical Sciences

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

L. R. Sons, Zhuan Ye, "A class of gap series with small growth in the unit disc", International Journal of Mathematics and Mathematical Sciences, vol. 32, Article ID 914873, 12 pages, 2002. https://doi.org/10.1155/S0161171202111136

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

L. R. Sons¹ and Zhuan Ye¹

¹Department of Mathematical Sciences, Northern Illinois University, USA

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=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).

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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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