Moduli difference of initial inverse logarithmic coefficients for starlike and convex functions

TL;DR

This paper establishes sharp bounds on the difference of the moduli of the first two inverse logarithmic coefficients for various subclasses of univalent functions, enhancing understanding of their geometric properties.

Contribution

It provides the first sharp bounds for the difference of inverse logarithmic coefficients in several important subclasses of univalent functions, with explicit extremal functions identified.

Findings

Sharp bounds for $| ext{Gamma}_2| - | ext{Gamma}_1|$ for starlike functions with respect to symmetric points.

Sharp bounds for convex functions with respect to symmetric points.

Explicit extremal functions attaining these bounds.

Abstract

Let $\mathcal{A}$ denote the class of functions $f$ that are analytic in the open unit disk $\mathbb{D}$ and satisfy the normalization conditions $f(0) = 0$ and $f'(0) = 1$. This paper investigates the inverse logarithmic coefficients $\Gamma_n$, which are defined by the expansion $\log(f^{-1}(w)/w) = 2\sum_{n=1}^{\infty} \Gamma_n w^n$. We establish sharp upper and lower bounds for the difference of the moduli of the first two inverse logarithmic coefficients, $|\Gamma_2| - |\Gamma_1|$, for several significant subclasses of univalent functions. Specifically, we derive sharp estimates for functions belonging to the class of starlike functions with respect to symmetric points ($\mathcal{S}_S^*$), convex functions with respect to symmetric points ($\mathcal{K}_S$), and functions associated with the lune domain ($\mathcal{S}_{\leftmoon}^*$ and $\mathcal{C}_{\leftmoon}$). The results are obtained by employing subordination techniques and utilizing sharp estimates for the coefficients of Schwarz functions. In each case, the extremal functions that attain these bounds are explicitly identified. Our findings provide further insights into the geometric properties of inverse mappings and extend recent research on coefficient functionals in geometric function theory.