A class of gap series with small growth in the unit disc
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}....
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| Main Authors: | , |
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| Format: | Article |
| Language: | English |
| Published: |
Wiley
2002-01-01
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| Series: | International Journal of Mathematics and Mathematical Sciences |
| Online Access: | http://dx.doi.org/10.1155/S0161171202111136 |
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| Summary: | 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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| ISSN: | 0161-1712 1687-0425 |