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 |
|---|---|---|---|---|---|---|
| Lommel s_{μ,ν}(z) normalized (ν=1/2) |
{'mu': 4, 'nu': 0.5}
|
starlike | 0.3*exp(i*2.304614) | Re ≤ -12.8331 | r ≤ - |
2 runs
|
| Lommel s_{μ,ν}(z) normalized (ν=1/2) |
{'mu': 1, 'nu': 0.5}
|
starlike | 0.4845*exp(i*1.08210) | Re ≤ -121.4465 | r ≤ - | 1 |
| Lommel s_{μ,ν}(z) normalized (ν=1/2) |
{'mu': 2, 'nu': 0.5}
|
starlike | 0.3976*exp(i*2.96706) | Re ≤ -61.2408 | r ≤ - | 1 |
| Lommel s_{μ,ν}(z) normalized (ν=1/2) |
{'mu': 3, 'nu': 0.5}
|
starlike | 0.3502*exp(i*-2.96706) | Re ≤ -312.7242 | r ≤ - | 1 |
| Lommel s_{μ,ν}(z) normalized (ν=1/2) |
{'mu': 5, 'nu': 0.5}
|
starlike | 0.2791*exp(i*0.45379) | Re ≤ -951.1695 | r ≤ - | 1 |
| Lommel s_{μ,ν}(z) normalized (ν=1/2) |
{'mu': 6, 'nu': 0.5}
|
starlike | 0.2475*exp(i*0.45379) | Re ≤ -76.4059 | r ≤ - | 1 |
| Lommel s_{μ,ν}(z) normalized (ν=1/2) |
{'mu': 7, 'nu': 0.5}
|
starlike | 0.2238*exp(i*0.45379) | Re ≤ -74.1619 | r ≤ - | 1 |
| Lommel s_{μ,ν}(z) normalized (ν=1/2) |
{'mu': 8, 'nu': 0.5}
|
starlike | 0.2080*exp(i*1.95477) | Re ≤ -139.3103 | r ≤ - | 1 |
GET /api/v2/counterexamples