On a logarithmic coefficients inequality for the class of close-to-convex functions

Abstract

In \emph{Logarithmic coefficients problems in families related to starlike and convex functions}, J. Aust. Math. Soc., 109, pp. 230--249, 2020, Ponnusamy et al.\! stated the conjecture for the sharp bounds of the logarithmic coefficients $\gamma_n$ for $f\in\mathcal{F}(3)$ as follows \[ |\gamma_n|\le\frac{1}{n}\left(1-\frac{1}{2^{n+1}}\right),\quad n\in\mathbb{N}, \] and \[ \sum\limits_{n=1}^{\infty}|\gamma_n|^2\le\dfrac{\pi^2}{6}+\frac{1}{4}\mathrm{Li}_2\left(\frac{1}{4}\right)-\mathrm{Li}_2\left(\frac{1}{2}\right), \] where $\mathrm{Li}_2$ is the Spence's (or dilogarithm) function. In this research we confirm that the conjecture for the above second inequality is true under some additional conditions.