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 |
|---|---|---|---|---|---|---|
| Mittag-Leffler E_α |
{'alpha': 1.75}
|
starlike | 0.999*exp(i*3.153864) | Re ≤ -0.0116 | r ≤ 0.990 |
3 runs
|
| Mittag-Leffler E_α |
{'alpha': 0.2}
|
starlike | 0.99*exp(i*5.075942) | Re ≤ -0.2238 | r ≤ 0.950 |
2 runs
|
| Mittag-Leffler E_α |
{'alpha': 1.1}
|
starlike | 0.95*exp(i*3.153864) | Re ≤ -0.0226 | r ≤ 0.900 |
3 runs
|
| Mittag-Leffler E_α |
{'alpha': 1.15}
|
starlike | 0.95*exp(i*3.202952) | Re ≤ -0.0500 | r ≤ 0.900 |
3 runs
|
| Mittag-Leffler E_α |
{'alpha': 1.6}
|
starlike | 0.95*exp(i*3.117049) | Re ≤ -0.0383 | r ≤ 0.900 |
3 runs
|
| Mittag-Leffler E_α |
{'alpha': 0.15}
|
starlike | 0.95*exp(i*2.988195) | Re ≤ -0.6999 | r ≤ 0.900 |
2 runs
|
| Mittag-Leffler E_α |
{'alpha': 1.5}
|
starlike | 0.9*exp(i*3.150797) | Re ≤ -0.0009 | r ≤ 0.800 |
3 runs
|
| Mittag-Leffler E_α |
{'alpha': 1.25}
|
starlike | 0.9*exp(i*3.104777) | Re ≤ -0.0197 | r ≤ 0.800 |
3 runs
|
| Mittag-Leffler E_α |
{'alpha': 1.2}
|
starlike | 0.9*exp(i*3.16) | Re ≤ -0.0061 | r ≤ 0.800 |
3 runs
|
| Mittag-Leffler E_α |
{'alpha': 1.3}
|
starlike | 0.9*exp(i*3.202952) | Re ≤ -0.0249 | r ≤ 0.800 |
3 runs
|
| Mittag-Leffler E_α |
{'alpha': 1.4}
|
starlike | 0.9*exp(i*3.117049) | Re ≤ -0.0272 | r ≤ 0.800 |
3 runs
|
| Mittag-Leffler E_α |
{'alpha': 0.1}
|
starlike | 0.9900*exp(i*-2.09440) | Re ≤ -13.3170 | r ≤ - | 1 |
| Mittag-Leffler E_α |
{'alpha': 0.05}
|
starlike | 0.9900*exp(i*-2.09440) | Re ≤ -90.0891 | r ≤ - | 1 |
GET /api/v2/counterexamples