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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| Format: | Article |
| Language: | English |
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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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| author | L. R. Sons Zhuan Ye |
| author_facet | L. R. Sons Zhuan Ye |
| author_sort | L. R. Sons |
| collection | DOAJ |
| description | 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). |
| format | Article |
| id | doaj-art-dd1c5119cb1b4fc1985b3e196ec8206a |
| institution | Kabale University |
| issn | 0161-1712 1687-0425 |
| language | English |
| publishDate | 2002-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | International Journal of Mathematics and Mathematical Sciences |
| spelling | doaj-art-dd1c5119cb1b4fc1985b3e196ec8206a2025-02-03T01:22:55ZengWileyInternational Journal of Mathematics and Mathematical Sciences0161-17121687-04252002-01-01321294010.1155/S0161171202111136A class of gap series with small growth in the unit discL. R. Sons0Zhuan Ye1Department of Mathematical Sciences, Northern Illinois University, DeKalb 60115, IL, USADepartment of Mathematical Sciences, Northern Illinois University, DeKalb 60115, IL, USALet 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).http://dx.doi.org/10.1155/S0161171202111136 |
| spellingShingle | L. R. Sons Zhuan Ye A class of gap series with small growth in the unit disc International Journal of Mathematics and Mathematical Sciences |
| title | A class of gap series with small growth in the unit disc |
| title_full | A class of gap series with small growth in the unit disc |
| title_fullStr | A class of gap series with small growth in the unit disc |
| title_full_unstemmed | A class of gap series with small growth in the unit disc |
| title_short | A class of gap series with small growth in the unit disc |
| title_sort | class of gap series with small growth in the unit disc |
| url | http://dx.doi.org/10.1155/S0161171202111136 |
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