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

Abstract

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 \[ 1+\frac{z f''(z)}{f'(z)} \prec e^z, \quad z\in\mathbb{D}. \] If $F_{f^{-1}}(w)=\log \left(\frac{f^{-1}(w)}{w}\right)=2{\sum }_{n=1}^{\infty }{\Gamma }_n{w}^n$ 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 \[ |Γ_n| \le \frac{1}{2n(n+1)}, \qquad n=1,2,3. \] We further derive the sharp coefficient-difference inequality \[ -\frac{1}{2\sqrt7} \le |Γ_2|-|Γ_1| \le \frac1{12}, \] and obtain the sharp bound for the second-order Hankel determinant associated with the inverse logarithmic coefficients: \[ \left| H_{2,1}\!\left(F_{f^{-1}}/2\right) \right| \le \frac{85}{12096}. \] Additionally, we evaluate the sharp lower and upper bounds of the generalized Fekete--Szegö functional $F_{\lambda ,\mu }(f)=|a_3(f)-\lambda a_2(f{)}^2|-\mu |a_2(f)|$ 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.