Paste a SymPy closed form like z/(1-z)^2 or a list of Taylor coefficients,
and the registry's three-tier verifier returns a verdict. Sandbox - nothing is saved.
A disproof verdict is a certified counterexample - useful mid-proof to
refute a claimed containment before you commit to it. Witnesses found this way are
collected below. Results also include the image domain
f(𝔻) and a check of your coefficients against the registry's certified
coefficient bounds for any subclass (Fekete–Szegő proven exact; Hankel/Zalcman
certified enclosures).
Every row is a certified disproof a run like the one above produced: a point z on |z| = r where interval arithmetic guarantees Re(zf′/f) < 0, so f cannot be starlike there. A sound witness, not a sampled point.
These are computational non-membership facts, not refutations of any published theorem - most are special functions that are starlike only on a smaller sub-disk, or synthetic test cases. We record them as honest negative results.
| Family | Closed form / Params | Property | Representative witness z | Re(zf′/f) ≤ | Proven r | Witnesses |
|---|---|---|---|---|---|---|
| z cos(z) |
z*cos(z)
|
starlike | 0.9*exp(i*0.024544) | Re ≤ -0.1305 | r ≤ 0.800 |
4 runs
|
| z cos(z) |
z*cos(z)
|
nehari_univalent | 0.8636*exp(i*0.00000) | ≥ 9014.3537 (threshold 2) | r ≤ - | 1 |
| z cos(z) |
z*cos(z)
|
becker_univalent | 0.8636*exp(i*0.00000) | ≥ 77.5740 (threshold 1) | r ≤ - | 1 |
GET /api/v2/counterexamples