Sharp Bounds for Initial Toeplitz and Vandermonde Determinants Associated with the Cusp-Domain Class M_tan

Abstract

Let $\A$ denote the family of normalized analytic functions
$f(z)=z+\sum_{n=2}^{\infty}a_nz^n$ in the unit disk $\D$. We investigate the
cusp-domain class
\[
\Mtan=\left\{f\in\A:
f'(z)\prec h(z):=\frac{2(1+\arctan z)}{1+e^{-z}}\right\}.
\]
Using an exact two-coefficient parametrization of the Schwarz function in the
subordination, we obtain the sharp estimates
\[
|\Tdet_2(1)|\le \frac{25}{16},\qquad
|\Tdet_2(2)|\le \frac{85}{144},\qquad
|\Tdet_3(1)|\le \frac{329}{144},
\]
for the initial symmetric Toeplitz determinants, and
\[
|\Vdet_2(1)|\le \frac74,\qquad
|\Vdet_2(2)|\le \frac{11}{12},
\]
for the initial Vandermonde determinants. Every constant is best possible.
The extremal functions are given explicitly by
$F_\lambda(z)=\int_0^z h(\lambda t)\,dt$, with $\lambda=i$ for the Toeplitz
bounds and $\lambda=-1$ for the Vandermonde bounds. The proofs retain the
dependence between $a_2$ and $a_3$; this is essential because independent
coefficient estimates do not, in general, establish sharpness of determinant
functionals.