Solution of the logarithmic coefficients conjecture in some families of univalent functions
For univalent and normalized functions $f$ the logarithmic coefficients $\gamma_n(f)$ are determined by the formula $\log(f(z)/z)=\sum_{n=1}^{\infty}2\gamma_n(f)z^n$. In the paper \cite{Pon} the authors posed the conjecture that a locally univalent function in the unit disk, satisfying the condition...
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Main Authors | , |
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Format | Journal Article |
Language | English |
Published |
29.01.2020
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Subjects | |
Online Access | Get full text |
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Summary: | For univalent and normalized functions $f$ the logarithmic coefficients
$\gamma_n(f)$ are determined by the formula
$\log(f(z)/z)=\sum_{n=1}^{\infty}2\gamma_n(f)z^n$. In the paper \cite{Pon} the
authors posed the conjecture that a locally univalent function in the unit
disk, satisfying the condition \[
\Re\left\{1+zf''(z)/f'(z)\right\}<1+\lambda/2\quad (z\in \mathbb{D}), \]
fulfill also the following inequality: $$|\gamma_n(f)|\le \lambda/(2n(n+1)).$$
Here $\lambda$ is a real number such that $0<\lambda\le 1$. In the paper we
confirm that the conjecture is true, and sharp. |
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DOI: | 10.48550/arxiv.2001.11098 |