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 $\log (f(z)\mathrm{/}z)={\sum }_{n=1}^{\mathrm{\infty }}2{\gamma }_n(f)z^n$. In the paper \cite{Pon} the authors posed the conjecture that a locally univalent function in the unit disk, satisfying the condition $$\mathrm{\Re }\{1+z{f}^{\mathrm{\prime }\mathrm{\prime }}(z)\mathrm{/}{f}^{\mathrm{\prime }}(z)\}<1+\lambda \mathrm{/}2(z\in \mathbb{D}),$$ fulfill also the following inequality: $$\mathrm{\mid }{\gamma }_n(f)\mathrm{\mid }\le \lambda \mathrm{/}(2n(n+1))\mathrm{.}$$ Here $\lambda$ is a real number such that $0<\lambda \le 1$. In the paper we confirm that the conjecture is true, and sharp.