Some application of Grunsky coefficients in the theory of univalent functions

Abstract Let function f be normalized, analytic and univalent in the unit disk 𝔻 = { z : | z | &lt; 1} and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>f</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mi>z</m:mi> <m:mo>)</m:mo> </m:mrow> <m:mo>=</m:mo> <m:mi>z</m:mi> <m:mo>+</m:mo> <m:msubsup> <m:mo>∑</m:mo> <m:mrow> <m:mi>n</m:mi> <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>a</m:mi> </m:mrow> <m:mi>n</m:mi> </m:msub> <m:msup> <m:mrow> <m:mi>z</m:mi> </m:mrow> <m:mi>n</m:mi> </m:msup> </m:mrow> </m:mrow> </m:math> f\left( z \right) = z + \sum\nolimits_{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 generalized Zalcman conjecture | a 2 a 3 − a 4 |, of the third logarithmic coefficient, and of the second Hankel determinant for the logarithmic coefficients.