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 \[ \Re\left\{1+zf''(z)/f'(z)\right\}<1+λ/2\quad (z\in \mathbb{D}), \] 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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