Teichmüller spaces and coefficient problems for univalent holomorphic functions

Abstract

We show that the features of Teichmüller spaces give rise to a completely new powerful tool for solving the classical coefficient problems in geometric function theory. Here we concern the univalent functions. Using the Bers isomorphism theorem for Teichmüller spaces of punctured Riemann surfaces and some of their other complex geometric features, we prove a general theorem on maximization of homogeneous polynomial (in fact, more general holomorphic) coefficient functionals $J(f) = J(a_{m_1}, a_{m_2},\dots , a_{m_n})$ on some classes of univalent functions in the unit disk naturally connected with the canonical class S. The given functional J is lifted to the Teichmüller space $\mathbf{T}_1$ of the punctured disk $\mathbb {D}_{*} = \{0< |z| < 1\}$ which is biholomorphically equivalent to the Bers fiber space over the universal Teichmüller space. This generates a positive subharmonic function on the disk $\{|t| < 4\}$ with $\sup _{|t|<4} u(t) = \max _{{\mathbf {T}}_1} |J|$ attaining this maximal value only on the boundary circle, which correspond to rotations of the Koebe function. This theorem implies new sharp distortion estimates for univalent functions giving explicitly the extremal functions, and creates a new bridge between Teichmüller space theory and geometric complex analysis. In particular, it provides an alternate and direct proof of the Bieberbach conjecture.