On Zalcman's and Bieberbach conjectures

Abstract

The well-known Zalcman conjecture, which implies the Bieberbach conjecture, states that the coefficients of univalent functions $f(z)=z+\sum _2^{\infty }{a}_n{z}^n$ on the unit disk satisfy $|a_n^2-a_{2n-1}|\le (n-1{)}^2$ for all $n>2$, with equality only for the Koebe function and its rotations. The conjecture was proved by the author for $n\le 6$ (using geometric arguments related to the Ahlfors-Schwarz lemma) and remains open for $n\ge 7$. The main theorem of this paper states that these conjectures are equivalent and provides their simultaneous proof for all $n\ge 3$ combining the indicated geometric arguments with a new author's approach to extremal problems for holomorphic functions based on lifting the rotationally homogeneous coefficient functionals to the Bers fiber space over universal Teichmuller space.

arXiv admin note: text overlap with arXiv:2507.19767