Some application of Grunsky coefficients in the theory of univalent functions

Abstract

Let function $f$ be normalized, analytic and univalent in the unit disk $\mathbb{D}=\{z:|z|<1\}$ and $f(z)=z+{\sum }_{n=2}^{\infty }{a}_n{z}^n$. Using a method based on Grusky coefficients we study several problems over that class of univalent functions: upper bounds of the special case of the generalised Zalcman conjecture $|a_2{a}_3-a_4|$, of the third logarithmic coefficient, and of the second Hankel determinant for the logarithmic coefficients.