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 |
|---|---|---|---|---|---|---|
| Convex of order α — K(α) |
{'alpha': 0.5}
|
starlike | 0.99*exp(i*0.261927) | Re ≤ -0.3146 | r ≤ 0.950 |
3 runs
|
| Convex of order α — K(α) |
{'alpha': 0.5211}
|
starlike | 0.99*exp(i*6.024038) | Re ≤ -0.1079 | r ≤ 0.950 |
3 runs
|
| Convex of order α — K(α) |
{'alpha': 0.4168}
|
starlike | 0.99*exp(i*0.266913) | Re ≤ -1.4374 | r ≤ 0.950 |
2 runs
|
| Convex of order α — K(α) |
{'alpha': 0.4689}
|
starlike | 0.99*exp(i*6.022409) | Re ≤ -0.6742 | r ≤ 0.950 |
2 runs
|
| Convex of order α — K(α) |
{'alpha': 0.25}
|
starlike | 0.95*exp(i*6.004001) | Re ≤ -1.2921 | r ≤ 0.900 |
2 runs
|
| Convex of order α — K(α) |
{'alpha': 0.2605}
|
starlike | 0.95*exp(i*6.004001) | Re ≤ -1.1413 | r ≤ 0.900 |
2 runs
|
| Convex of order α — K(α) |
{'alpha': 0.3126}
|
starlike | 0.95*exp(i*6.005535) | Re ≤ -0.4973 | r ≤ 0.900 |
2 runs
|
| Convex of order α — K(α) |
{'alpha': 0.3647}
|
starlike | 0.95*exp(i*5.706792) | Re ≤ -0.0735 | r ≤ 0.900 |
2 runs
|
| Convex of order α — K(α) |
{'alpha': 0.1}
|
starlike | 0.9900*exp(i*-0.59341) | Re ≤ -18.9797 | r ≤ - | 1 |
| Convex of order α — K(α) |
{'alpha': 0.0}
|
starlike | 0.9900*exp(i*-2.09440) | Re ≤ -88.8649 | r ≤ - | 1 |
| Convex of order α — K(α) |
{'alpha': 0.0521}
|
starlike | 0.9900*exp(i*0.59341) | Re ≤ -33.5871 | r ≤ - | 1 |
| Convex of order α — K(α) |
{'alpha': 0.1042}
|
starlike | 0.9900*exp(i*-0.59341) | Re ≤ -18.1419 | r ≤ - | 1 |
| Convex of order α — K(α) |
{'alpha': 0.1563}
|
starlike | 0.9900*exp(i*0.27925) | Re ≤ -12.9686 | r ≤ - | 1 |
| Convex of order α — K(α) |
{'alpha': 0.2084}
|
starlike | 0.9900*exp(i*0.27925) | Re ≤ -8.8757 | r ≤ - | 1 |
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