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

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.