A Problem of Bombieri on Univalent Functions

The famous Bieberbach Conjecture from 1916 on the coefficients of normalized univalent functions defined in the unit disk (Bieberbach, S.-B. Preuss Akad Wiss 138:940–955, 1916 ) that was finally proved by de Branges (Acta Math 154:137–152, 1985 ) some 70 years later, diverted the attention of many c...

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Veröffentlicht in:Computational methods and function theory Jg. 16; H. 4; S. 677 - 688
Hauptverfasser: Aharonov, D., Bshouty, D.
Format: Journal Article
Sprache:Englisch
Veröffentlicht: Berlin/Heidelberg Springer Berlin Heidelberg 01.12.2016
Springer Nature B.V
Schlagworte:
Analysis
Computational Mathematics and Numerical Analysis
Functions of a Complex Variable
Mathematics
Mathematics and Statistics
30C75
30C50
Univalent functions
De Branges weights system
Löwner chains
Variational method
30C70
Coefficient estimates
ISSN:1617-9447, 2195-3724
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Zusammenfassung:The famous Bieberbach Conjecture from 1916 on the coefficients of normalized univalent functions defined in the unit disk (Bieberbach, S.-B. Preuss Akad Wiss 138:940–955, 1916 ) that was finally proved by de Branges (Acta Math 154:137–152, 1985 ) some 70 years later, diverted the attention of many complex analysts to other subjects. Those who continued to explore de Branges method and push it as far as possible were not aware of where it may lead. Surprisingly enough, a paper that fell in our hands (Dong Acta Sci Nat Univ Norm Hunan 14:193–197, 1991 ) contained a way to tackle one of the problems of Bombieri (Research problems in function theory, The Athlone Press, University of London, London, 1967 ) on the behavior of the coefficients of univalent functions. We shall give an account of the history of the problem and a revised version of it.
Bibliographie:ObjectType-Article-1
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ISSN:1617-9447
2195-3724
DOI:10.1007/s40315-016-0165-z