Estimates logarithmic coefficients for certain classes of univalent functions

Abstract: Let $\mathcal{S}$ be the family of analytic and univalent functions $f$ in the unit disk $\mathbb{D}$ with the normalization $f(0)=f'(0)-1=0$. The logarithmic coefficients $\gamma_n$ of $f\in \mathcal{S}$ are defined by the formula $$\log\left(\frac{f(z)}{z}\right)\,=\,2\sum_{n=1}^{\infty}\gamma_n(f)z^n. $$ In this talk, we will discuss bounds for the logarithmic coefficients for certain geometric subfamilies of univalent functions as starlike, convex, close-to-convex and Janowski starlike functions. Also, we consider the families $\mathcal{F}(c)$ and $\mathcal{G}(\delta)$ of functions $f\in \mathcal{S}$ defined by $$ {\rm Re} \left ( 1+\frac{zf''(z)}{f'(z)}\right )>1-\frac{c}{2}\, \mbox{ and } \, {\rm Re} \left ( 1+\frac{zf''(z)}{f'(z)}\right )<1+\frac{\delta}{2},\quad z\in \mathbb{D} $$ 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 $\mathcal{F}(c)$ and $\mathcal{G}(\delta)$, respectively. We conclude with the following two conjectures:

* If $f\in\mathcal{F}(-1/2)$, then $ \displaystyle |\gamma_n|\le \frac{1}{n}\left(1-\frac{1}{2^{n+1}}\right)$ for $n\ge 4$, 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.

* If $f\in \mathcal{G}(\delta)$, then $ \displaystyle |\gamma_n|\,\leq \,\frac{\delta}{2n(n+1)}$ for $n\ge 4$.

This talk is based on the following article.

S. Ponnusamy, N. L. Sharma and K.-J. Wirths, Logarithmic coefficients problems in families related to starlike and convex functions, . Aust. Math. Soc., 109(2) (2019), 230--249.

complex variables

Audience: researchers in the topic

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