Inverse Logarithmic Coefficients, Differences, Hankel Determinant, and Fekete--Szegö Functionals for the Class ${\mathcal{C}}_e$

In this paper, we investigate the inverse logarithmic coefficients associated with the class ${\mathcal{C}}_e$ of analytic and univalent functions satisfying the subordination condition $$1+\frac{z{f}^{\mathrm{\prime }\mathrm{\prime }}(z)}{f^{\mathrm{\prime }}(z)}\prec e^z,z\in \mathbb{D}\mathrm{.}$$ If $F_{f^{-1}}(w)=\log \text{\hspace{0.17em}⁣}\left(\frac{f^{-1}(w)}{w}\right)=2{\sum }_{n=1}^{\mathrm{\infty }}{\mathrm{\Gamma }}_n{w}^n$ denotes the logarithmic expansion corresponding to the inverse function $f^{-1}$, then we establish sharp estimates for the initial inverse logarithmic coefficients and prove that $$\mathrm{\mid }{\mathrm{\Gamma }}_n\mathrm{\mid }\le \frac{1}{2n(n+1)},n=1,2,3.$$ We further derive the sharp coefficient-difference inequality $$-\frac{1}{2\sqrt{7}}\le \mathrm{\mid }{\mathrm{\Gamma }}_2\mathrm{\mid }-\mathrm{\mid }{\mathrm{\Gamma }}_1\mathrm{\mid }\le \frac{1}{12},$$ and obtain the sharp bound for the second-order Hankel determinant associated with the inverse logarithmic coefficients: $$\mid H_{2,1}\text{\hspace{0.17em}⁣}\left(F_{f^{-1}}\mathrm{/}2\right)\mid \le \frac{85}{12096}\mathrm{.}$$ Additionally, we evaluate the sharp lower and upper bounds of the generalized Fekete–Szegö functional $F_{\lambda ,\mu }(f)=\mid a_3(f)-\lambda a_2(f{)}^2\mid -\mu \mathrm{\mid }{a}_2(f)\mathrm{\mid }$ within this setting and establish relationships associated with the starlike class ${\mathcal{S}}_{\rho }^{\ast }$. The extremal functions corresponding to all obtained estimates are explicitly constructed, thereby showing the sharpness of the results.