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arXiv · 2606.10186

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

Pradip Das·Nabadwip Sarkar
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Abstract

Let $\mathcal{S}_{ex}^{\ast}$ denote the class of normalized analytic functions $f$ in the unit disk satisfying \[ \frac{zf'(z)}{f(z)} \prec e^{αz},\qquad 0<α\le 1. \] We obtain sharp bounds for the initial inverse logarithmic coefficients $Γ_1$, $Γ_2$, and $Γ_3$. In particular, the bound for $|Γ_3|$ has a four-branch structure with transition values \[ α_\star \approx 0.542712,\qquad α_b \approx 0.679103,\qquad α_c \approx 0.691095. \] We also establish sharp upper and lower bounds for the successive coefficient difference \[ |Γ_2|-|Γ_1|, \] 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 \[ α_H=\frac{-1+\sqrt{241}}{15}\approx0.9683. \] Furthermore, we completely solve the sharp generalized Fekete--Szegő problem for the functional \[ |a_3-λa_2^2|-μ|a_2|. \] All estimates are sharp, and the corresponding extremal functions are explicitly constructed by using Carathéodory function representations.

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Pradip Das, Nabadwip Sarkar. 2026-08-15. Sharp Coefficient Estimates for the Exponential Starlike class $\mathcal{S}_{ex}^{\ast}$. https://arxiv.org/abs/2606.10186

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