On the Failure of Bombieri’s Conjecture for Univalent Functions
Author
Efraimidis, IasonEntity
UAM. Departamento de MatemáticasPublisher
SpringerDate
2018-09-01Citation
Computational Methods and Function Theory 18.3 (2018): 427-438ISSN
1617-9447 (print); 2195-3724 (online)Funded by
Acknowledgements The author has been supported by a fellowship of the International Excellence Graduate Program in Mathematics at Universidad Autónoma de Madrid (422Q101) and also partially supported by Grant MTM2015-65792-P by MINECO/FEDER-EU. This work forms part of his Ph.D. thesis at UAM under the supervision of professor Dragan Vukotić. The author would like to thank him for his encouragement and help. The author would also like to thank professor Yuk J. Leung for providing him with a copy of [10] and suggesting that formula (6) should be true under the hypothesis (a) and (b) of Theorem 1.Project
Gobierno de España. MTM2015-65792-PEditor's Version
https://doi.org/10.1007/s40315-017-0222-2Subjects
Bombieri conjecture; Dieudonné criterion; univalent functions; MatemáticasRights
© 2017, Springer-Verlag GmbH GermanyAbstract
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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