A link between covering and coefficient theorems for holomorphic functions

Abstract

Recently the author presented a new approach to solving the coefficient problems for various classes of holomorphic functions $f(z)=\sum _0^{\infty }{c}_n{z}^n$, not necessarily univalent. This approach is based on lifting the given polynomial coefficient functionals $J(f)=J(c_{m_1},\dots ,c_{m_s}),2<c_{m_1}<\cdots <c_{m_s}<\infty$, onto the Bers fiber space over universal Teichmuller space and applying the analytic and geometric features of Teichmüller spaces, especially the Bers isomorphism theorem for Teichmuller spaces of punctured Riemann surfaces. In this paper, we extend this approach to more general classes of functions. In particular, this provides a strengthening of de Branges' theorem solving the Bieberbach conjecture.