Some application of Grunsky coefficients in the theory of univalent functions
Milutin Obradović, Nikola Tuneski
TL;DR
The paper develops a Grunsky-coefficient framework to study coefficient problems in the univalent class S, focusing on a special case of the generalized Zalcman conjecture, the third logarithmic coefficient γ3, and the second Hankel determinant for logarithmic coefficients. By expressing low-order Taylor and logarithmic coefficients in terms of Grunsky data and applying Lebedev-type Grunsky inequalities, it derives explicit bounds: |a2a3−a4| ≤ 2.10064… for (n,m)=(2,3); |γ3| ≤ 0.5566178…; and |H_{2,1}(F_f/2)| ≤ 1/3 for f ∈ S. The results tighten known bounds in some cases and highlight the effectiveness of Grunsky-based techniques for classical univalent-function coefficient problems.
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 $|a_2a_3-a_4|$, of the third logarithmic coefficient, and of the second Hankel determinant for the logarithmic coefficients.