Fourth coefficient estimate in the class of univalent functions with quasiconformal extensions

Abstract

We denote by \( S(k) \) the class of all univalent conformal maps \( f \) defined in the unit disk \( \Delta \) normalized by \[f(z) = z + \sum_{n=2}^{\infty} a_n z^n, \]  such that all \( f \) admit \( k \)-quasiconformal homeomorphic extension to the whole Riemann sphere \( \hat{\mathbb{C}} \), and \( f(\infty) = \infty \). In our note we give a new estimate for \( |a_4| \) in \( S(k) \) making use of the Area Principle.