Two applications of Grunsky coefficients in the theory of univalent functions

Abstract

Let $\mathcal{S}$ denote the class of functions $f$ which are analytic and univalent in the unit disk ${\mathbb D}=\{z:|z|<1\}$ and normalized with $f(z)=z+\sum_{n=2}^{\infty} a_n z^n$. Using a method based on Grusky coefficients we study two problems over the class $\mathcal{S}$: estimate of the fourth logarithmic coefficient and upper bound of the coefficient difference $|a_5|-|a_4|$.