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 |
|---|---|---|---|---|---|---|
| Two-parameter Mittag-Leffler E_{a,b} |
{'alpha': 0.25, 'beta': 0.5}
|
starlike | 0.95*exp(i*1.23102) | Re ≤ -0.0906 | r ≤ 0.900 | 1 |
| Two-parameter Mittag-Leffler E_{a,b} |
{'alpha': 1.5, 'beta': 1.0}
|
starlike | 0.9*exp(i*3.150797) | Re ≤ -0.0009 | r ≤ 0.800 |
3 runs
|
| Two-parameter Mittag-Leffler E_{a,b} |
{'alpha': 0.5, 'beta': 0.5}
|
starlike | 0.9*exp(i*2.853204) | Re ≤ -0.0562 | r ≤ 0.800 | 1 |
| Two-parameter Mittag-Leffler E_{a,b} |
{'alpha': 0.75, 'beta': 0.5}
|
starlike | 0.6*exp(i*3.166136) | Re ≤ -0.4199 | r ≤ 0.300 |
3 runs
|
| Two-parameter Mittag-Leffler E_{a,b} |
{'alpha': 1.0, 'beta': 0.5}
|
starlike | 0.6*exp(i*3.313399) | Re ≤ -1.2959 | r ≤ - |
2 runs
|
| Two-parameter Mittag-Leffler E_{a,b} |
{'alpha': 1.5, 'beta': 0.5}
|
starlike | 0.6*exp(i*3.19068) | Re ≤ -4.7443 | r ≤ - |
2 runs
|
| Two-parameter Mittag-Leffler E_{a,b} |
{'alpha': 2.0, 'beta': 0.5}
|
starlike | 0.6*exp(i*3.043418) | Re ≤ -1.4660 | r ≤ - |
2 runs
|
GET /api/v2/counterexamples