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

TL;DR

This paper derives sharp bounds for inverse logarithmic coefficients, Hankel determinants, and Fekete--Szeg"o functionals for functions in the class , expanding understanding of their geometric properties.

Contribution

It provides new sharp estimates for inverse logarithmic coefficients, Hankel determinants, and Fekete--Szeg functionals for the class , with explicit extremal functions.

Findings

Sharp bounds for inverse logarithmic coefficients   

Sharp coefficient-difference inequality for 

Explicit bounds for second-order Hankel determinant for 

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 \[ |\Gamma_n| \le \frac{1}{2n(n+1)}, \qquad n=1,2,3. \] We further derive the sharp coefficient-difference inequality \[ -\frac{1}{2\sqrt7} \le |\Gamma_2|-|\Gamma_1| \le \frac1{12}, \] 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--Szeg\"{o} functional $F_{\lambda, \mu}(f) = \big| a_3(f) - \lambda a_2(f)^2 \big| - \mu |a_2(f)|$ within this setting and establish relationships associated with the starlike class $\mathcal{S}^{\ast}_{\rho}$. The extremal functions corresponding to all obtained estimates are explicitly constructed, thereby showing the sharpness of the results.