Analytic / CONCEPT

Exact two-node regions for self-conjugate Schur-ratio and hook-Schur bounds

On two-node self-conjugate grids in the closed unit disk, both inequalities in Conjecture 6.2 of Ostrovskii and Shcherbakov collapse across all degrees t >= 2 to their first nontrivial degree t = 3. The shifted Schur-ratio bound holds precisely in a closed sector of opening 2*pi/3 with vertex -1, whose largest inscribed origin-centred disk has radius sqrt(3)/2; the nonnegative-real-part hook-Schur bound holds precisely in a closed hyperbolic region inside the right half-disk.