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 |
|---|---|---|---|---|---|---|
| Ruscheweyh family R(λ) |
{'lambda': 0.1}
|
starlike | 0.99*exp(i*6.022409) | Re ≤ -0.9232 | r ≤ 0.950 |
2 runs
|
| Ruscheweyh family R(λ) |
{'lambda': 0.5}
|
starlike | 0.95*exp(i*6.004001) | Re ≤ -1.2921 | r ≤ 0.900 |
2 runs
|
| Ruscheweyh family R(λ) |
{'lambda': 0.25}
|
starlike | 0.95*exp(i*5.707415) | Re ≤ -0.0030 | r ≤ 0.900 |
2 runs
|
| Ruscheweyh family R(λ) |
{'lambda': 0.4}
|
starlike | 0.95*exp(i*0.581379) | Re ≤ -0.6236 | r ≤ 0.900 |
2 runs
|
| Ruscheweyh family R(λ) |
{'lambda': 2.5}
|
starlike | 0.8*exp(i*4.176886) | Re ≤ -15.3090 | r ≤ - |
2 runs
|
| Ruscheweyh family R(λ) |
{'lambda': 1.0}
|
starlike | 0.9900*exp(i*-2.09440) | Re ≤ -88.8649 | r ≤ - | 1 |
| Ruscheweyh family R(λ) |
{'lambda': 1.5}
|
starlike | 0.9347*exp(i*-1.50098) | Re ≤ -176.9168 | r ≤ - | 1 |
| Ruscheweyh family R(λ) |
{'lambda': 2.0}
|
starlike | 0.8952*exp(i*-0.62832) | Re ≤ -528.2005 | r ≤ - | 1 |
| Ruscheweyh family R(λ) |
{'lambda': 0.6}
|
starlike | 0.9900*exp(i*0.27925) | Re ≤ -9.4526 | r ≤ - | 1 |
| Ruscheweyh family R(λ) |
{'lambda': 0.75}
|
starlike | 0.9900*exp(i*0.27925) | Re ≤ -15.9485 | r ≤ - | 1 |
| Ruscheweyh family R(λ) |
{'lambda': 0.8}
|
starlike | 0.9900*exp(i*-0.59341) | Re ≤ -18.9797 | r ≤ - | 1 |
| Ruscheweyh family R(λ) |
{'lambda': 0.9}
|
starlike | 0.9900*exp(i*0.59341) | Re ≤ -34.5376 | r ≤ - | 1 |
| Ruscheweyh family R(λ) |
{'lambda': 1.1}
|
starlike | 0.9821*exp(i*-2.09440) | Re ≤ -215.3095 | r ≤ - | 1 |
| Ruscheweyh family R(λ) |
{'lambda': 1.25}
|
starlike | 0.9663*exp(i*-1.50098) | Re ≤ -471.6441 | r ≤ - | 1 |
| Ruscheweyh family R(λ) |
{'lambda': 1.4}
|
starlike | 0.9505*exp(i*0.90757) | Re ≤ -790.2280 | r ≤ - | 1 |
| Ruscheweyh family R(λ) |
{'lambda': 1.6}
|
starlike | 0.9189*exp(i*-1.50098) | Re ≤ -94.3174 | r ≤ - | 1 |
| Ruscheweyh family R(λ) |
{'lambda': 1.75}
|
starlike | 0.9031*exp(i*-1.50098) | Re ≤ -70.9235 | r ≤ - | 1 |
| Ruscheweyh family R(λ) |
{'lambda': 1.9}
|
starlike | 0.9031*exp(i*0.62832) | Re ≤ -178.1950 | r ≤ - | 1 |
| Ruscheweyh family R(λ) |
{'lambda': 2.25}
|
starlike | 0.8557*exp(i*-1.22173) | Re ≤ -128.6963 | r ≤ - | 1 |
| Ruscheweyh family R(λ) |
{'lambda': 2.75}
|
starlike | 0.8083*exp(i*1.81514) | Re ≤ -147.0676 | r ≤ - | 1 |
| Ruscheweyh family R(λ) |
{'lambda': 3.0}
|
starlike | 0.8004*exp(i*0.94248) | Re ≤ -229.2260 | r ≤ - | 1 |
| Ruscheweyh family R(λ) |
{'lambda': 3.5}
|
starlike | 0.7530*exp(i*1.81514) | Re ≤ -170.8297 | r ≤ - | 1 |
| Ruscheweyh family R(λ) |
{'lambda': 4.0}
|
starlike | 0.7135*exp(i*2.40855) | Re ≤ -105.4944 | r ≤ - | 1 |
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