Two applications of Grunsky coefficients in the theory of univalent functions

Abstract

<jats:title>Abstract</jats:title> <jats:p>Let <jats:italic>S</jats:italic> denote the class of functions f which are analytic and univalent in the unit disk 𝔻 = {z : |z| < 1} and normalized with <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_ausm-2023-0017_eq_001.png" /> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="inline"> <m:mrow> <m:mtext>f</m:mtext> <m:mrow> <m:mo>(</m:mo> <m:mtext>z</m:mtext> <m:mo>)</m:mo> </m:mrow> <m:mo>=</m:mo> <m:mtext>z</m:mtext> <m:mo>+</m:mo> <m:msubsup> <m:mo>∑</m:mo> <m:mrow> <m:mtext>n</m:mtext> <m:mo>=</m:mo> <m:mn>2</m:mn> </m:mrow> <m:mo>∞</m:mo> </m:msubsup> <m:mrow> <m:msub> <m:mrow> <m:mi>α</m:mi> </m:mrow> <m:mtext>n</m:mtext> </m:msub> <m:msup> <m:mrow> <m:mtext>z</m:mtext> </m:mrow> <m:mtext>n</m:mtext> </m:msup> </m:mrow> </m:mrow> </m:math> <jats:tex-math>{\rm{f}}\left( {\rm{z}} \right) = {\rm{z}} + \sum\nolimits_{{\rm{n = 2}}}^\infty {{\alpha _{\rm{n}}}{{\rm{z}}^{\rm{n}}}}</jats:tex-math> </jats:alternatives> </jats:inline-formula>. Using a method based on Grusky coefficients we study two problems over the class <jats:italic>S</jats:italic>: estimate of the fourth logarithmic coefficient and upper bound of the coefficient difference |α<jats:sub>5</jats:sub>| − |α<jats:sub>4</jats:sub>|.</jats:p>