Solution of the logarithmic coefficients conjecture in some families of univalent functions

TL;DR

This paper proves a conjecture about logarithmic coefficients for certain univalent functions, confirming a specific inequality holds with sharp bounds for functions satisfying a given real part condition.

Contribution

It confirms the conjecture that a bound on logarithmic coefficients holds for functions meeting a particular real part condition, establishing the result as sharp.

Findings

The conjecture is true and sharp.

The inequality for logarithmic coefficients is validated.

The result applies to functions satisfying the specified real part condition.

Abstract

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.