Moduli difference of inverse logarithmic coefficients of univalent functions

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