Logarithmic coefficients problems in families related to starlike and convex functions

Abstract: Let $\es$ be the family of analytic and univalent functions $f$ in the unit disk $\D$ with the normalization $f(0)=f'(0)-1=0$, and let $\gamma_n(f)=\gamma_n$ denote the logarithmic coefficients of $f\in {\es}$. In this paper, we study bounds for the logarithmic coefficients for certain subfamilies of univalent functions. Also, we consider the families $\F(c)$ and $\G(\delta)$ of functions $f\in {\es}$ defined by $$ {\rm Re} \left ( 1+\frac{zf''(z)}{f'(z)}\right )>1-\frac{c}{2}\, \mbox{ and } \, {\rm Re} \left ( 1+\frac{zf''(z)}{f'(z)}\right )<1+\frac{\delta}{2},\quad z\in \D $$ for some $c\in(0,3]$ and $\delta\in (0,1]$, respectively. We obtain the sharp upper bound for $|\gamma_n|$ when $n=1,2,3$ and $f$ belongs to the classes $\F(c)$ and $\G(\delta)$, respectively. The paper concludes with the following two conjectures: \begin{itemize} \item If $f\in\F (-1/2)$, then $ \displaystyle |\gamma_n|\le \frac{1}{n}\left(1-\frac{1}{2^{n+1}}\right)$ for $n\ge 1$, and $$ \sum_{n=1}^{\infty}|\gamma_{n}|^{2} \leq \frac{\pi^2}{6}+\frac{1}{4} ~{\rm Li\,}{2}\left(\frac{1}{4}\right) -{\rm Li\,}{2}\left(\frac{1}{2}\right), $$ where ${\rm Li}_2(x)$ denotes the dilogarithm function. \item If $f\in \G(\delta)$, then $ \displaystyle |\gamma_n|\,\leq \,\frac{\delta}{2n(n+1)}$ for $n\ge 1$. \end{itemize}