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Some application of Grunsky coefficients in the theory of univalent functions

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arxiv 2009.11945 v3 pith:G6L2YT4L submitted 2020-09-24 math.CV

Milutin Obradović , Nikola Tuneski This is my paper · ORCID
classification math.CV
keywords coefficientsunivalentfunctionslogarithmicanalyticapplicationboundscase
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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.

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  1. Symmetric Toeplitz determinants of some classes of univalent functions

    math.CV 2025-09 reject novelty 4.0 of 10

    Estimates for Toeplitz determinants of univalent functions are derived, but the claimed sharp bound 3/16 for T3,2 in class U with a2=0 is false; the true value is 1/4.

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