A general coefficient theorem for univalent functions

Abstract

Using the Bers isomorphism theorem for Teichmuller 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 Teichmuller 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 Teichmuller 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 Teichm\"{u}ller space theory and geometric complex analysis. In particular, it provides an alternate and direct proof of the Bieberbach conjecture.