Sharp estimates for inverse logarithmic coefficients Γ1–Γ3, |Γ2|−|Γ1|, the Hankel determinant H2,1, the Hermitian–Toeplitz determinant T3,1, and the generalized Fekete–Szegő functional are obtained for the exponential starlike class S_ex^*.
Logarithmic coefficients for exponential classes of starlike and convex functions
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abstract
In this paper, we investigate two subclasses of analytic and univalent functions associated with the exponential mapping $\varphi(z)=e^{\alpha z},\qquad 0<\alpha\le1,$ defined via the subordination conditions $\frac{zf'(z)}{f(z)}\prec e^{\alpha z} \quad \text{and} \quad 1+\frac{zf''(z)}{f'(z)}\prec e^{\alpha z}$. These classes provide a natural exponential analogue of several classical subclasses arising in geometric function theory. We obtain sharp coefficient estimates, logarithmic coefficient inequalities and sharp bounds for the associated Hankel and upper bounds for Toeplitz determinants. In particular, explicit estimates are derived for $$ |H_{2,1}(F_f/2)|, \quad |T_{2,1}(F_f/2)|, $$ for functions belonging to the introduced exponential subclasses of starlike and convex functions. Our results extend and unify several earlier works on exponential subclasses and highlight connections with logarithmic coefficients and determinant functionals.
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2026 1verdicts
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Sharp Coefficient Estimates for the Exponential Starlike class $\mathcal{S}_{ex}^{\ast}$
Sharp estimates for inverse logarithmic coefficients Γ1–Γ3, |Γ2|−|Γ1|, the Hankel determinant H2,1, the Hermitian–Toeplitz determinant T3,1, and the generalized Fekete–Szegő functional are obtained for the exponential starlike class S_ex^*.