On certain applications of grunsky coefficients in the theory of univalent functions
M. Obradovic, N. Tuneski
TL;DR
This work surveys Grunsky-coefficient based methods to bound coefficient-related quantities for the general class $\mathcal{S}$ of univalent functions, where analytic characterisation is unavailable. It derives sharp bounds for the third and fourth logarithmic coefficients $|\gamma_3|$ and $|\gamma_4|$, and proves bounds on differences $|\gamma_n|-|\gamma_{n-1}|$ using Grunsky inequalities and optimization over the Grunsky data. It also provides sharp bounds for Hankel determinants of the inverse function $f^{-1}$, notably $|H_{2,2}(f^{-1})|\le 3$ and $|H_{3,1}(f^{-1})|\le 2.36639\ldots$, along with differences between inverse and direct determinants, and collects coefficient inequalities for special cases with missing coefficients. The results strengthen the understanding of coefficient regimes in univalent function theory and demonstrate the effectiveness of the Grunsky-coefficient approach when explicit characterisations are lacking.
Abstract
In this paper a survey is given of application of a method based on Grunsky coefficients for obtaining different estimates (some sharp) for the general class of univalent functions where no analytical characterisation exists. More precisely, estimates are given for the modulus of the third and the fourth logarithmic coefficients, for the modulus of the second and the third Hankel determinant for the general class of univalent functions, and for the modulus of some coefficients of the inverse function, and some coefficient differences.