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 |
|---|---|---|---|---|---|---|
| Wright φ(α,β; z) |
{'alpha': 1.25, 'beta': 1.5}
|
starlike | 0.99*exp(i*3.153864) | Re ≤ -0.0078 | r ≤ 0.950 |
3 runs
|
| Wright φ(α,β; z) |
{'alpha': 0.1, 'beta': 1.0}
|
starlike | 0.95*exp(i*3.095573) | Re ≤ -0.0129 | r ≤ 0.900 |
3 runs
|
| Wright φ(α,β; z) |
{'alpha': 1.5, 'beta': 1.25}
|
starlike | 0.95*exp(i*3.092505) | Re ≤ -0.0477 | r ≤ 0.900 |
3 runs
|
| Wright φ(α,β; z) |
{'alpha': 0.25, 'beta': 1.0}
|
starlike | 0.9*exp(i*3.202952) | Re ≤ -0.0805 | r ≤ 0.800 |
3 runs
|
| Wright φ(α,β; z) |
{'alpha': 0.5, 'beta': 0.75}
|
starlike | 0.9*exp(i*3.104777) | Re ≤ -0.0512 | r ≤ 0.800 |
3 runs
|
| Wright φ(α,β; z) |
{'alpha': 0.75, 'beta': 0.5}
|
starlike | 0.9*exp(i*3.117049) | Re ≤ -0.1024 | r ≤ 0.800 |
3 runs
|
| Wright φ(α,β; z) |
{'alpha': 2.0, 'beta': 0.75}
|
starlike | 0.9*exp(i*3.19068) | Re ≤ -0.0930 | r ≤ 0.800 |
3 runs
|
| Wright φ(α,β; z) |
{'alpha': 0.5, 'beta': 1.5}
|
starlike | 0.8*exp(i*3.117049) | Re ≤ -0.0621 | r ≤ 0.600 |
3 runs
|
| Wright φ(α,β; z) |
{'alpha': 1.0, 'beta': 1.0}
|
starlike | 0.8*exp(i*2.99433) | Re ≤ -0.3988 | r ≤ 0.600 |
3 runs
|
| Wright φ(α,β; z) |
{'alpha': 0.1, 'beta': 1.25}
|
starlike | 0.8*exp(i*3.104777) | Re ≤ -0.0406 | r ≤ 0.600 |
3 runs
|
| Wright φ(α,β; z) |
{'alpha': 0.1, 'beta': 1.5}
|
starlike | 0.8*exp(i*3.202952) | Re ≤ -0.0643 | r ≤ 0.600 |
3 runs
|
| Wright φ(α,β; z) |
{'alpha': 0.25, 'beta': 1.25}
|
starlike | 0.8*exp(i*3.117049) | Re ≤ -0.1004 | r ≤ 0.600 |
3 runs
|
| Wright φ(α,β; z) |
{'alpha': 0.25, 'beta': 1.5}
|
starlike | 0.8*exp(i*3.117049) | Re ≤ -0.0693 | r ≤ 0.600 |
3 runs
|
| Wright φ(α,β; z) |
{'alpha': 0.5, 'beta': 1.0}
|
starlike | 0.8*exp(i*3.215224) | Re ≤ -0.1693 | r ≤ 0.600 |
3 runs
|
| Wright φ(α,β; z) |
{'alpha': 0.5, 'beta': 1.25}
|
starlike | 0.8*exp(i*3.018874) | Re ≤ -0.1788 | r ≤ 0.600 |
3 runs
|
| Wright φ(α,β; z) |
{'alpha': 0.75, 'beta': 0.75}
|
starlike | 0.8*exp(i*3.19068) | Re ≤ -0.3197 | r ≤ 0.600 |
3 runs
|
| Wright φ(α,β; z) |
{'alpha': 0.75, 'beta': 1.0}
|
starlike | 0.8*exp(i*3.19068) | Re ≤ -0.4024 | r ≤ 0.600 |
3 runs
|
| Wright φ(α,β; z) |
{'alpha': 0.75, 'beta': 1.25}
|
starlike | 0.8*exp(i*3.19068) | Re ≤ -0.2498 | r ≤ 0.600 |
3 runs
|
| Wright φ(α,β; z) |
{'alpha': 0.75, 'beta': 1.5}
|
starlike | 0.8*exp(i*3.153864) | Re ≤ -0.0054 | r ≤ 0.600 |
3 runs
|
| Wright φ(α,β; z) |
{'alpha': 1.25, 'beta': 1.0}
|
starlike | 0.8*exp(i*3.239767) | Re ≤ -0.3284 | r ≤ 0.600 |
3 runs
|
| Wright φ(α,β; z) |
{'alpha': 1.25, 'beta': 1.25}
|
starlike | 0.8*exp(i*3.144661) | Re ≤ -0.0007 | r ≤ 0.600 |
3 runs
|
| Wright φ(α,β; z) |
{'alpha': 1.5, 'beta': 0.75}
|
starlike | 0.8*exp(i*3.239767) | Re ≤ -0.4981 | r ≤ 0.600 |
3 runs
|
| Wright φ(α,β; z) |
{'alpha': 1.5, 'beta': 1.0}
|
starlike | 0.8*exp(i*3.19068) | Re ≤ -0.1037 | r ≤ 0.600 |
3 runs
|
| Wright φ(α,β; z) |
{'alpha': 2.0, 'beta': 0.5}
|
starlike | 0.8*exp(i*3.239767) | Re ≤ -0.2420 | r ≤ 0.600 |
3 runs
|
| Wright φ(α,β; z) |
{'alpha': 1.25, 'beta': 0.5}
|
starlike | 0.6*exp(i*3.19068) | Re ≤ -0.1148 | r ≤ 0.300 |
3 runs
|
| Wright φ(α,β; z) |
{'alpha': 1.25, 'beta': 0.75}
|
starlike | 0.6*exp(i*3.144661) | Re ≤ -0.0004 | r ≤ 0.300 |
3 runs
|
| Wright φ(α,β; z) |
{'alpha': 1.5, 'beta': 0.5}
|
starlike | 0.6*exp(i*3.19068) | Re ≤ -0.0888 | r ≤ 0.300 |
3 runs
|
GET /api/v2/counterexamples