Moduli difference of inverse logarithmic coefficients of univalent functions

Let f be analytic in the unit disk and $S$ be the subclass of normalized univalent functions with $f\left(0\right)=0$, and $f^{\prime }\left(0\right)=1$. Let F be the inverse function of f, given by $F\left(w\right)=w+{\sum }_{n=2}^{\infty }{A}_n{w}^n$ defined on some disk $\left|w\right|\le r_0\left(f\right)$. The inverse logarithmic coefficients ${\Gamma }_n$, $n\in N$, of f are defined by the equation $\log \left(F\right(w)/w)=2{\sum }_{n=1}^{\infty }{\Gamma }_n{w}^n,\left|w\right|<1/4$. In this paper, we find the sharp upper and lower bounds for moduli difference of second and first inverse logarithmic coefficients, i.e., $\left|{\Gamma }_2\right|-\left|{\Gamma }_1\right|$ for functions in class $S$ and for functions in some important subclasses of univalent functions.