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

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.