Logarithmic coefficients problems in families related to starlike and convex functions

TL;DR

This paper investigates bounds on the logarithmic coefficients of certain subclasses of univalent functions, providing sharp estimates for initial coefficients and proposing conjectures for broader classes related to starlike and convex functions.

Contribution

It derives sharp bounds for the first three logarithmic coefficients in specific subclasses and introduces conjectures for general bounds in related families.

Findings

Sharp bounds for |gamma_n| for n=1,2,3 in classes F(c) and G()

Proposed conjectures for bounds in broader classes F(-1/2) and G()

Formulated inequalities involving logarithmic coefficients and special 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}