On the Failure of Bombieri’s Conjecture for Univalent Functions
A conjecture of Bombieri (Invent Math 4:26–67, 1967) states that the coefficients of a normalized univalent function f should satisfy lim inff→Kn-Reanm-Ream=mint∈Rnsint-sin(nt)msint-sin(mt),when f approaches the Koebe function K(z)=z(1-z)2. Recently, Leung [10] disproved this conjecture for n= 2 and...
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| Formato: | artículo |
| Fecha de publicación: | 2018 |
| País: | España |
| Recursos: | Universidad Autónoma de Madrid |
| Repositorio: | Biblos-e Archivo. Repositorio Institucional de la UAM |
| Idioma: | inglés |
| OAI Identifier: | oai:repositorio.uam.es:10486/710717 |
| Acesso em linha: | http://hdl.handle.net/10486/710717 https://dx.doi.org/10.1007/s40315-017-0222-2 |
| Access Level: | acceso abierto |
| Palavra-chave: | Bombieri conjecture Dieudonné criterion univalent functions Matemáticas |
| Resumo: | A conjecture of Bombieri (Invent Math 4:26–67, 1967) states that the coefficients of a normalized univalent function f should satisfy lim inff→Kn-Reanm-Ream=mint∈Rnsint-sin(nt)msint-sin(mt),when f approaches the Koebe function K(z)=z(1-z)2. Recently, Leung [10] disproved this conjecture for n= 2 and for all m≥ 3 and, also, for n= 3 and for all odd m≥ 5. Complementing his work, we disprove it for all m> n≥ 2 which are simultaneously odd or even and, also, for the case when m is odd, n is even and n≤m+12. We mostly not only make use of trigonometry but also employ Dieudonné’s criterion for the univalence of polynomials |
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