A Janowski positivity threshold for the sixth-order symmetric Toeplitz determinant of odd starlike functions

Abstract

Abstract We establish a sharp positivity threshold for the sixth-order symmetric Toeplitz determinant $T_{6}(1)$ T 6 ( 1 ) over the Janowski family of odd starlike functions with real coefficients. Writing $\Phi (A,B)$ Φ ( A , B ) for an explicit nonnegative quantity built from the Janowski parameters, we prove that $\Phi (A,B)^{2}\le T_{6}(1)\le 1$ Φ ( A , B ) 2 ≤ T 6 ( 1 ) ≤ 1 for every $f\in \mathcal{S}^{*}_{\mathbb{R}}[A,B]$ f ∈ S R ∗ [ A , B ] , with both bounds sharp, and that $\Phi (A,B)$ Φ ( A , B ) vanishes precisely at the full odd starlike endpoint $(A,B)=(1,-1)$ ( A , B ) = ( 1 , − 1 ) . Consequently every proper Janowski subclass imposes a strictly positive floor on $T_{6}(1)$ T 6 ( 1 ) , while that endpoint is the unique class in which $T_{6}(1)$ T 6 ( 1 ) may vanish. The bounds rest on a structural device of independent interest: for an odd function the Toeplitz matrix splits, under a parity permutation, into two equal reduced blocks, so that the sixth-order determinant becomes the square of a determinant in only the two coefficients $a_{3}$ a 3 and $a_{5}$ a 5 . For the exponential class we obtain the sharp two-sided bound between $81/256$ 81 / 256 and 1, and we show that this real bound does not persist in the complex exponential class, where the sharp estimate rises to $625/256$ 625 / 256 and is attained by a purely imaginary linear Schwarz function; the real-coefficient hypothesis thus changes the extremal value, not merely the method. The factorisation underlying the threshold is established by an explicit algebraic identity. Such a sharp positivity threshold for