The Zalcman conjecture and related problems

Abstract

At the end of 1960's, Lawrence Zalcman posed a conjecture that the coefficients of univalent functions $f(z)=z+\sum _2^{\infty }{a}_n{z}^n$ on the unit disk satisfy the sharp inequality $|a_n^2-a_{2n-1}|\le (n-1{)}^2$, with equality only for the Koebe function. This remarkable conjecture implies the Bieberbach conjecture, investigated by many mathematicians, and still remains a very difficult open problem for all n > 3; it was proved only in certain special cases. We provide a proof of Zalcman's conjecture based on results concerning the plurisubharmonic functionals and metrics on the universal Teichmüller space. As a corollary, this implies a new proof of the Bieberbach conjecture. Our method gives also other new sharp estimates for large coefficients.

arXiv commentThis paper has been replaced with a revised version arXiv:1109.4646v2

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