On Zalcman's and Bieberbach conjectures

Abstract

The well-known Zalcman conjecture, which implies the Bieberbach conjecture, states that the coefficients of univalent functions $f(z) = z + \sum\limits_2^{\infty} a_n z^n$ 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.