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Let $\es$ be the class of analytic and univalent functions in the unit disk$|z|<1$, that have a series of the form $f(z)=z+ \sum_{n=2}^{\infty}a_nz^n$.Let $F$ be the inverse of the function $f\in\es$ with the series expansion %ina disk of radius at least $1/4$ $F(w)=f^{-1}(w)=w+ \sum_{n=2}^{\infty}A_nw^n$for $|w|<1/4$. The logarithmic inverse coefficients $\Gamma_n$ of $F$ aredefined by the formula$\log\left(F(w)/w\right)\,=\,2\sum_{n=1}^{\infty}\Gamma_n(F)w^n$. % In thispaper, we determine the logarithmic inverse coefficients bound of $F$ for theclass In this paper, we first determine the sharp bound for the absolute valueof $\Gamma_n(F)$ when $f$ belongs to $\es$ and for all $n \geq 1$. This resultmotivates us to carry forward similar problems for some of its importantgeometric subclasses. In some cases, we have managed to solve this questioncompletely but in some other cases it is difficult to handle for $n\geq 4$. Forexample, in the case of convex functions $f$, we show that the logarithmicinverse coefficients $\Gamma_n(F)$ of $F$ satisfy the inequality \[|\Gamma_n(F)|\,\le \, \frac{1}{2n} \mbox{ for } n\geq 1,2,3 \] and theestimates are sharp for the function $l(z)=z/(1-z)$. Although this cannot betrue for $n\ge 10$, it is not clear whether this inequality could still be truefor $4\leq n\leq 9$.

Logarithmic coefficients of the inverse of univalent functions