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 $f(0)=f^{\prime }(0)-1=0$, 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 $\G(δ)$ of functions $f\in {\es}$ defined by $$ {\rm Re} \left ( 1+\frac{zf''(z)}{f'(z)}\right )>1-\frac{c}{2}\, \mbox{ and } \, {\rm Re} \left ( 1+\frac{zf''(z)}{f'(z)}\right )<1+\fracδ{2},\quad z\in \D $$ 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 $\G(δ)$, respectively. The paper concludes with the following two conjectures: \begin{itemize} \item If $f\in\F (-1/2)$, then $|{\gamma }_n|\le \frac{1}{n}\left(1-\frac{1}{2^{n+1}}\right)$ for $n\ge 1$, and $$\sum _{n=1}^{\infty }|{\gamma }_n{|}^2\le \frac{{\pi }^2}{6}+\frac{1}{4}{\mathrm{Li}}_2\left(\frac{1}{4}\right)-{\mathrm{Li}}_2\left(\frac{1}{2}\right),$$ where ${\mathrm{Li}}_2(x)$ denotes the dilogarithm function. \item If $f\in \G(δ)$, then $|{\gamma }_n|\le \frac{\delta }{2n(n+1)}$ for $n\ge 1$. \end{itemize}

22 pages; To appear in Journal of Australian Mathematical Society