What this is
Every analytic function f(z) = z + a2z2 + a3z3 + … normalized to the unit disk has geometric properties: is it starlike (maps the disk onto a star-shaped domain)? convex? univalent? What is its radius of starlikeness? Do its Taylor coefficients satisfy known bounds?
This registry collects function families (Bessel, Mittag-Leffler, hypergeometric, Janowski, Koebe, and more). For each one it stores facts about these properties — proven, disproven, or computationally screened — with the verification runs behind them, cross-referenced to the literature.
How to navigate
- Registry — the master table of every function family, filterable by search and status.
- Applications — cross-tabulation of families × application areas (image processing, medical imaging, machine learning, etc.).
- Counterexamples — recorded starlikeness witnesses, including legacy rows that still require contract replay.
- Proofs — machine-certified coefficient-bound enclosures over Ma–Minda starlike classes.
- Verify — sandbox: paste a closed form or coefficients and run the three-tier verifier to check starlikeness yourself.
- Papers — indexed papers with extracted GFT claims (starlike/convex/univalent mentions, coefficient bounds, radius results).
- FAQ — methodology, known failure modes, and how to interpret the verification tiers.
- Architecture — request flow, verifier internals, data model, and deployment details.
Verification tiers
Every fact in the registry carries a confidence label:
- proven — interval-arithmetic certification on the true closed-form function. Gold standard.
- rigorous — interval arithmetic on the Taylor truncation (sound for the polynomial, approximate for the full function).
- screened — float-grid numerical evidence. Fast but not a proof.
- disproven — a certified witness point where Re(zf′/f) < 0. Terminal — sound counterexample.
See the FAQ or the architecture page for a full explanation of methodology and known limitations.