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 |
|---|---|---|---|---|---|---|
| Janowski S*[A,B] extremal |
{'A': -0.5, 'B': -1.0}
|
starlike | 0.95*exp(i*4.491496) | Re ≤ -0.1138 | r ≤ 0.900 |
3 runs
|
| Janowski S*[A,B] extremal |
{'A': 0.5, 'B': -0.5}
|
starlike | 0.9*exp(i*1.969631) | Re ≤ -0.0404 | r ≤ 0.800 |
3 runs
|
| Janowski S*[A,B] extremal |
{'A': 0.0, 'B': -1.0}
|
starlike | 0.8*exp(i*4.417865) | Re ≤ -0.8541 | r ≤ 0.600 |
3 runs
|
| Janowski S*[A,B] extremal |
{'A': 1.0, 'B': -0.5}
|
starlike | 0.8*exp(i*2.098486) | Re ≤ -0.1635 | r ≤ 0.600 |
3 runs
|
| Janowski S*[A,B] extremal |
{'A': 0.25, 'B': -1.0}
|
starlike | 0.8*exp(i*4.417865) | Re ≤ -1.7310 | r ≤ 0.600 |
3 runs
|
| Janowski S*[A,B] extremal |
{'A': 0.25, 'B': -0.75}
|
starlike | 0.8*exp(i*4.393321) | Re ≤ -0.2027 | r ≤ 0.600 |
3 runs
|
| Janowski S*[A,B] extremal |
{'A': 0.75, 'B': -0.75}
|
starlike | 0.8*exp(i*2.012583) | Re ≤ -1.1790 | r ≤ 0.600 |
3 runs
|
| Janowski S*[A,B] extremal |
{'A': 1.0, 'B': -0.75}
|
starlike | 0.6*exp(i*2.054) | Re ≤ -0.0098 | r ≤ 0.300 |
3 runs
|
| Janowski S*[A,B] extremal |
{'A': 1.0, 'B': -1.0}
|
starlike | 0.6*exp(i*2.012583) | Re ≤ -0.8652 | r ≤ - |
2 runs
|
| Janowski S*[A,B] extremal |
{'A': 0.75, 'B': -1.0}
|
starlike | 0.6*exp(i*2.012583) | Re ≤ -0.5348 | r ≤ - |
2 runs
|
GET /api/v2/counterexamples