Some application of Grunsky coefficients in the theory of univalent functions

Obradović, Milutin; Tuneski, Nikola

doi:10.2478/aupcsm-2025-0005

Abstract

Let function f be normalized, analytic and univalent in the unit disk 𝔻 = {z : |z| < 1} and $f (z) = z + \sum_{n = 2}^{\infty} a_{n} z^{n}$ 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 |a₂a₃ − a₄|, of the third logarithmic coefficient, and of the second Hankel determinant for the logarithmic coefficients.

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Some application of Grunsky coefficients in the theory of univalent functions

Abstract

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