The Zalcman conjecture and related problems

TL;DR

This paper proves Zalcman's conjecture on the coefficients of univalent functions, which also offers a new proof of the Bieberbach conjecture, using advanced methods involving plurisubharmonic functionals and Teichmüller space.

Contribution

The paper provides the first proof of Zalcman's conjecture, connecting it to the Bieberbach conjecture and introducing novel techniques involving plurisubharmonic functionals.

Findings

Proof of Zalcman's conjecture for univalent functions.

New sharp coefficient estimates for large n.

Alternative proof of the Bieberbach conjecture.

Abstract

At the end of 1960's, Lawrence Zalcman posed a conjecture that the coefficients of univalent functions $f(z) = z + \sum\limits_2^\infty a_n z^n$ 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 Teichm\"uller space. As a corollary, this implies a new proof of the Bieberbach conjecture. Our method gives also other new sharp estimates for large coefficients.