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math.CV — Complex Variables

The Möbius pseudodistance of the annulus and an improved lower bound for Korenblum's constant

Contributed by David Niedbala Giraudin

We show that Korenblum's constant of the Bergman space satisfies $c_2\ge 0.446$ . The best published lower bound was $c_2\ge 0.4263$ (Wikström, 2026). The new ingredient is a closed form, in Jacobi theta functions, of the Möbius pseudodistance of the annulus $\{c<|z|<1\}$ between two points of the same centred circle. It separates the variables and yields three monotonicity properties, one of which, a profile inequality, combined with Kirszbraun's theorem in the hyperbolic plane, improves the constant in Wang's annular estimate. Wikström's dual moment criterion is then verified at $c=223/500$ for an explicit measure with fourteen rational atoms, in rigorous arithmetic, by a self-contained verifier. The French version of this note is dated 21 October 2026.

Primarily AI-generated textHuman understanding: no partsBergman spaceKirszbraun theoremKorenblum's constantMöbius pseudodistancecomputer-assisted proofmaximum principlemoment dualitytheta functions

Nov 02, 2026PDF ↗View work →

hexagon:2610.00721v1

math.CA — Classical Analysis and ODEs

Sharp order-four sign regularity of a Jacobi theta kernel

Contributed by Suvrit Sra

We prove that the Hankel kernel $\Phi(\sqrt{x+y})$ , where $\Phi$ is the Jacobi theta kernel whose cosine transform is the Riemann $\Xi$ -function, is strictly reverse-regular through order four. The order is sharp: an interval enclosure of the fifth Hankel–Wronskian at $t=10^{-2}$ shows failure at order five. The third-order condition is Csordas's conjecture that $s'^2-ss''$ is strictly log-concave for $s(t)=\Phi(\sqrt t)$ . The proof reduces the kernel minors to Wronskian signs and establishes the signs by exact polynomial estimates for $t\ge1$ and interval arithmetic on $[0,1]$ . The fourth-order sign also yields three positive iterates of $s\mapsto s'^2-ss''$ , the triple Turán inequalities for the normalized Taylor coefficients of $\Xi$ , and quartic integral inequalities for $\Xi$ .

A mix of human-written and AI-generated textHuman understanding: some partsCsordas-VargaJacobi theta functionRiemann Xisign regularitytotal positivity

Nov 02, 2026PDF ↗View work →

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The Möbius pseudodistance of the annulus and an improved lower bound for Korenblum's constant

Sharp order-four sign regularity of a Jacobi theta kernel

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