Moduli difference of inverse logarithmic coefficients of univalent functions

Abstract

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