Sharp Coefficient Estimates for the Exponential Starlike class $\mathcal{S}_{ex}^{\ast}$

Abstract

Let $\mathcal{S}_{ex}^{\ast}$ denote the class of normalized analytic functions $f$ in the unit disk satisfying \[ \frac{zf'(z)}{f(z)} \prec e^{\alpha z},\qquad 0<\alpha\le 1. \] We obtain sharp bounds for the initial inverse logarithmic coefficients $\Gamma_1$, $\Gamma_2$, and $\Gamma_3$. In particular, the bound for $|\Gamma_3|$ has a four-branch structure with transition values \[ \alpha_\star \approx 0.542712,\qquad \alpha_b \approx 0.679103,\qquad \alpha_c \approx 0.691095. \] We also establish sharp upper and lower bounds for the successive coefficient difference \[ |\Gamma_2|-|\Gamma_1|, \] for the inverse logarithmic Hankel determinant \[ H_{2,1}\bigl(F_{f^{-1}}/2\bigr), \] and for the third-order Hermitian--Toeplitz determinant $T_{3,1}(f)$, whose sharp lower bound changes at \[ \alpha_H=\frac{-1+\sqrt{241}}{15}\approx0.9683. \] Furthermore, we completely solve the sharp generalized Fekete--Szeg\H{o} problem for the functional \[ |a_3-\lambda a_2^2|-\mu|a_2|. \] All estimates are sharp, and the corresponding extremal functions are explicitly constructed by using Carath\'eodory function representations.