Two applications of Grunsky coefficients in the theory of univalent functions

Abstract

Abstract Let S denote the class of functions f which are analytic and univalent in the unit disk 𝔻 = {z : |z| < 1} and normalized with f ( z ) = z + ∑ n = 2 ∞ α n z n {\rm{f}}\left( {\rm{z}} \right) = {\rm{z}} + \sum\nolimits_{{\rm{n = 2}}}^\infty {{\alpha _{\rm{n}}}{{\rm{z}}^{\rm{n}}}} . Using a method based on Grusky coefficients we study two problems over the class S: estimate of the fourth logarithmic coefficient and upper bound of the coefficient difference |α5| − |α4|.