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 |
|---|---|---|---|---|---|---|
| Koebe Kα: z/(1-z)^(2α) |
z/(1-z)^(2*alpha)
{'alpha': 0.1}
|
starlike | 0.99*exp(i*0.241218) | Re ≤ -0.1215 | r ≤ 0.950 |
3 runs
|
| Koebe Kα: z/(1-z)^(2α) |
z/(1-z)^(2*alpha)
{'alpha': 0.25}
|
starlike | 0.95*exp(i*6.00937) | Re ≤ -0.7356 | r ≤ 0.900 |
3 runs
|
| Koebe Kα: z/(1-z)^(2α) |
z/(1-z)^(2*alpha)
{'alpha': 0.5}
|
starlike | 0.95*exp(i*0.266913) | Re ≤ -0.7458 | r ≤ 0.900 |
2 runs
|
| Koebe Kα: z/(1-z)^(2α) |
z/(1-z)^(2*alpha)
{'alpha': 1.5}
|
starlike | 0.6*exp(i*2.847068) | Re ≤ -0.1173 | r ≤ - |
2 runs
|
| Koebe Kα: z/(1-z)^(2α) |
z/(1-z)^(2*alpha)
{'alpha': 2.5}
|
starlike | 0.3*exp(i*2.945243) | Re ≤ -0.1446 | r ≤ - |
2 runs
|
| Koebe Kα: z/(1-z)^(2α) |
z/(1-z)^(2*alpha)
{'alpha': 2.25}
|
starlike | 0.3*exp(i*3.203719) | Re ≤ -0.0376 | r ≤ - |
2 runs
|
| Koebe Kα: z/(1-z)^(2α) |
z/(1-z)^(2*alpha)
{'alpha': 3.0}
|
starlike | 0.3*exp(i*3.005835) | Re ≤ -0.3794 | r ≤ - |
2 runs
|
| Koebe Kα: z/(1-z)^(2α) |
z/(1-z)^(2*alpha)
{'alpha': 0.4}
|
starlike | 0.9900*exp(i*0.27925) | Re ≤ -17.8377 | r ≤ - | 1 |
| Koebe Kα: z/(1-z)^(2α) |
z/(1-z)^(2*alpha)
{'alpha': 0.6}
|
starlike | 0.9663*exp(i*-0.90757) | Re ≤ -383.7491 | r ≤ - | 1 |
| Koebe Kα: z/(1-z)^(2α) |
z/(1-z)^(2*alpha)
{'alpha': 0.75}
|
starlike | 0.9189*exp(i*-1.22173) | Re ≤ -107.0447 | r ≤ - | 1 |
| Koebe Kα: z/(1-z)^(2α) |
z/(1-z)^(2*alpha)
{'alpha': 0.9}
|
starlike | 0.8715*exp(i*1.81514) | Re ≤ -182.2540 | r ≤ - | 1 |
| Koebe Kα: z/(1-z)^(2α) |
z/(1-z)^(2*alpha)
{'alpha': 1.0}
|
starlike | 0.8399*exp(i*-2.40855) | Re ≤ -388.8166 | r ≤ - | 1 |
| Koebe Kα: z/(1-z)^(2α) |
z/(1-z)^(2*alpha)
{'alpha': 1.1}
|
starlike | 0.8162*exp(i*2.40855) | Re ≤ -117.0306 | r ≤ - | 1 |
| Koebe Kα: z/(1-z)^(2α) |
z/(1-z)^(2*alpha)
{'alpha': 1.25}
|
starlike | 0.8399*exp(i*0.66323) | Re ≤ -244.5972 | r ≤ - | 1 |
| Koebe Kα: z/(1-z)^(2α) |
z/(1-z)^(2*alpha)
{'alpha': 1.75}
|
starlike | 0.7688*exp(i*0.69813) | Re ≤ -261.5592 | r ≤ - | 1 |
| Koebe Kα: z/(1-z)^(2α) |
z/(1-z)^(2*alpha)
{'alpha': 2.0}
|
starlike | 0.6898*exp(i*1.29154) | Re ≤ -244.5257 | r ≤ - | 1 |
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