A problem of Bombieri on univalent functions. (English) Zbl 1361.30033
Summary: The famous Bieberbach Conjecture from 1916 on the coefficients of normalized univalent functions defined in the unit disk [L. Bieberbach, Berl. Ber. 1916, 940–955 (1916; JFM 46.0552.01)] that was finally proved by L. de Branges [Acta Math. 154, 137–152 (1985; Zbl 0573.30014)] 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 [X. Dong, Acta Sci. Nat. Univ. Norm. Hunanensis 14, No. 3, 193–197 (1991; Zbl 0742.30015)] contained a way to tackle one of the problems of Bombieri [W. K. Hayman, Research problems in function theory. London: University of London (1967; Zbl 0158.06301)] 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.
MSC:
| 30C50 | Coefficient problems for univalent and multivalent functions of one complex variable |
| 30C55 | General theory of univalent and multivalent functions of one complex variable |
| 30C70 | Extremal problems for conformal and quasiconformal mappings, variational methods |
| 30C75 | Extremal problems for conformal and quasiconformal mappings, other methods |
Keywords:
univalent functions; coefficient estimates; Löwner chains; variational method; de Branges weights systemReferences:
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