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 con...

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Bibliographic Details
Published in	arXiv.org
Main Authors	Kanas, Stanislawa, Vali Soltani Masih
Format	Paper
Language	English
Published	Ithaca Cornell University Library, arXiv.org 29.01.2020
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.
ISSN:	2331-8422

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.
ISSN:	2331-8422