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
|
| z - z^2/2 + z^3/3 |
z - z**2/2 + z**3/3
|
starlike | 0.999*exp(i*1.119806) | Re ≤ -0.0254 | r ≤ 0.990 |
3 runs
|
| z + z^2/2 + z^3/3 |
z + z**2/2 + z**3/3
|
starlike | 0.999*exp(i*2.021787) | Re ≤ -0.0254 | r ≤ 0.990 |
3 runs
|
| Alternating signs Σ(-1)^(n-1)/n z^n | starlike | 0.99*exp(i*3.40352) | Re ≤ -0.3146 | r ≤ 0.950 |
3 runs
|
|
| 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
|
| ₂F₁(a, b; c; z) — Gauss hypergeometric |
{'a': 1.0, 'b': 1.0, 'c': 2.0}
|
starlike | 0.99*exp(i*0.261927) | Re ≤ -0.3146 | r ≤ 0.950 |
3 runs
|
| ₂F₁(a, b; c; z) — Gauss hypergeometric |
{'a': 1.0, 'b': 2.0, 'c': 3.0}
|
starlike | 0.99*exp(i*0.266913) | Re ≤ -1.0720 | r ≤ 0.950 |
2 runs
|
| 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
|
| Mittag-Leffler E_α |
{'alpha': 0.2}
|
starlike | 0.99*exp(i*5.075942) | Re ≤ -0.2238 | r ≤ 0.950 |
2 runs
|
| Mocanu α-convex |
{'alpha': 0.3}
|
starlike | 0.99*exp(i*2.40835) | Re ≤ -0.0486 | r ≤ 0.950 |
3 runs
|
| Nephroid starlike | starlike | 0.99*exp(i*4.089593) | Re ≤ -0.0319 | r ≤ 0.950 |
3 runs
|
|
| q-exponential e_q(z) |
{'q': 0.15}
|
starlike | 0.99*exp(i*2.988195) | Re ≤ -0.3983 | r ≤ 0.950 |
2 runs
|
| q-exponential e_q(z) |
{'q': 0.175}
|
starlike | 0.99*exp(i*2.992797) | Re ≤ -0.0586 | r ≤ 0.950 |
2 runs
|
| Ruscheweyh family R(λ) |
{'lambda': 0.1}
|
starlike | 0.99*exp(i*6.022409) | Re ≤ -0.9232 | r ≤ 0.950 |
2 runs
|
| Slow decay Σ 1/n z^n | starlike | 0.99*exp(i*0.261927) | Re ≤ -0.3146 | r ≤ 0.950 |
3 runs
|
|
| Strongly starlike SS*(β) |
{'beta': 0.6}
|
starlike | 0.99*exp(i*3.092505) | Re ≤ -0.0575 | r ≤ 0.950 |
3 runs
|
| Strongly starlike SS*(β) |
{'beta': 0.55}
|
starlike | 0.99*exp(i*3.117049) | Re ≤ -0.0262 | r ≤ 0.950 |
3 runs
|
| Wright φ(α,β; z) |
{'alpha': 1.25, 'beta': 1.5}
|
starlike | 0.99*exp(i*3.153864) | Re ≤ -0.0078 | r ≤ 0.950 |
3 runs
|
| Normalized Bessel J_ν |
{'nu': -0.6}
|
starlike | 0.95*exp(i*0.049087) | Re ≤ -0.0870 | r ≤ 0.900 |
3 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
|
| (e^(3z)-1)/3 |
(exp(3*z) - 1)/3
|
starlike | 0.95*exp(i*1.922078) | Re ≤ -0.0088 | r ≤ 0.900 |
3 runs
|
| ₂F₁(a, b; c; z) — Gauss hypergeometric |
{'a': 1.0, 'b': 1.0, 'c': 1.5}
|
starlike | 0.95*exp(i*6.004001) | Re ≤ -1.0509 | r ≤ 0.900 |
2 runs
|
| Janowski S*[A,B] extremal |
{'A': -0.5, 'B': -1.0}
|
starlike | 0.95*exp(i*4.491496) | Re ≤ -0.1138 | r ≤ 0.900 |
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
|
| 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
|
| 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 |
| Mocanu α-convex |
{'alpha': 0.25}
|
starlike | 0.95*exp(i*2.402214) | Re ≤ -0.0571 | r ≤ 0.900 |
3 runs
|
| q-exponential e_q(z) |
{'q': 0.1}
|
starlike | 0.95*exp(i*2.988195) | Re ≤ -0.5560 | r ≤ 0.900 |
2 runs
|
| q-exponential e_q(z) |
{'q': 0.125}
|
starlike | 0.95*exp(i*3.291923) | Re ≤ -0.1437 | r ≤ 0.900 |
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
|
| Strongly starlike SS*(β) |
{'beta': 0.8}
|
starlike | 0.95*exp(i*3.117049) | Re ≤ -0.0353 | r ≤ 0.900 |
3 runs
|
| Strongly starlike SS*(β) |
{'beta': 0.65}
|
starlike | 0.95*exp(i*3.153864) | Re ≤ -0.0107 | r ≤ 0.900 |
3 runs
|
| Strongly starlike SS*(β) |
{'beta': 0.7}
|
starlike | 0.95*exp(i*3.117049) | Re ≤ -0.0271 | r ≤ 0.900 |
3 runs
|
| Strongly starlike SS*(β) |
{'beta': 0.75}
|
starlike | 0.95*exp(i*3.117049) | Re ≤ -0.0353 | r ≤ 0.900 |
3 runs
|
| Strongly starlike SS*(β) |
{'beta': 0.85}
|
starlike | 0.95*exp(i*3.117049) | Re ≤ -0.0289 | r ≤ 0.900 |
3 runs
|
| Strongly starlike SS*(β) |
{'beta': 0.9}
|
starlike | 0.95*exp(i*3.503612) | Re ≤ -0.0509 | r ≤ 0.900 |
3 runs
|
| Wright φ(α,β; z) |
{'alpha': 0.1, 'beta': 1.0}
|
starlike | 0.95*exp(i*3.095573) | Re ≤ -0.0129 | r ≤ 0.900 |
3 runs
|
| Wright φ(α,β; z) |
{'alpha': 1.5, 'beta': 1.25}
|
starlike | 0.95*exp(i*3.092505) | Re ≤ -0.0477 | r ≤ 0.900 |
3 runs
|
| z(1-z/2)/(1-z)^2 |
z*(1 - z/2)/(1-z)**2
|
starlike | 0.95*exp(i*0.546097) | Re ≤ -0.0860 | r ≤ 0.900 |
3 runs
|
| z(1-z/3)/(1-z)^2 |
z*(1 - z/3)/(1-z)**2
|
starlike | 0.95*exp(i*5.548409) | Re ≤ -0.0158 | r ≤ 0.900 |
3 runs
|
| (z - z^2/2)/(1-z)^2 |
(z - z**2/2)/(1-z)**2
|
starlike | 0.95*exp(i*0.546097) | Re ≤ -0.0860 | r ≤ 0.900 |
3 runs
|
| ₂F₁(a, b; c; z) — Gauss hypergeometric |
{'a': 5.0, 'b': 5.0, 'c': 15.0}
|
starlike | 0.9*exp(i*2.988195) | Re ≤ -0.3101 | r ≤ 0.800 |
2 runs
|
| Janowski S*[A,B] extremal |
{'A': 0.5, 'B': -0.5}
|
starlike | 0.9*exp(i*1.969631) | Re ≤ -0.0404 | r ≤ 0.800 |
3 runs
|
| Ma-Minda cosh | starlike | 0.9*exp(i*1.546253) | Re ≤ -0.1305 | r ≤ 0.800 |
3 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
|
| 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 |
| Mocanu α-convex |
{'alpha': 0.05}
|
starlike | 0.9*exp(i*2.380738) | Re ≤ -0.3401 | r ≤ 0.800 |
3 runs
|
| Mocanu α-convex |
{'alpha': 0.1}
|
starlike | 0.9*exp(i*2.368466) | Re ≤ -0.2263 | r ≤ 0.800 |
3 runs
|
| Mocanu α-convex |
{'alpha': 0.15}
|
starlike | 0.9*exp(i*2.386874) | Re ≤ -0.1219 | r ≤ 0.800 |
3 runs
|
| Mocanu α-convex |
{'alpha': 0.2}
|
starlike | 0.9*exp(i*2.40068) | Re ≤ -0.0242 | r ≤ 0.800 |
3 runs
|
| q-exponential e_q(z) |
{'q': 0.075}
|
starlike | 0.9*exp(i*2.992797) | Re ≤ -0.0508 | r ≤ 0.800 |
2 runs
|
| Strongly starlike SS*(β) |
{'beta': 0.95}
|
starlike | 0.9*exp(i*2.764233) | Re ≤ -0.0320 | r ≤ 0.800 |
3 runs
|
| Strongly starlike SS*(β) |
{'beta': 1.0}
|
starlike | 0.9*exp(i*3.577243) | Re ≤ -0.0967 | r ≤ 0.800 |
3 runs
|
| Wright φ(α,β; z) |
{'alpha': 0.25, 'beta': 1.0}
|
starlike | 0.9*exp(i*3.202952) | Re ≤ -0.0805 | r ≤ 0.800 |
3 runs
|
| Wright φ(α,β; z) |
{'alpha': 0.5, 'beta': 0.75}
|
starlike | 0.9*exp(i*3.104777) | Re ≤ -0.0512 | r ≤ 0.800 |
3 runs
|
| Wright φ(α,β; z) |
{'alpha': 0.75, 'beta': 0.5}
|
starlike | 0.9*exp(i*3.117049) | Re ≤ -0.1024 | r ≤ 0.800 |
3 runs
|
| Wright φ(α,β; z) |
{'alpha': 2.0, 'beta': 0.75}
|
starlike | 0.9*exp(i*3.19068) | Re ≤ -0.0930 | r ≤ 0.800 |
3 runs
|
| z cos(z) |
z*cos(z)
|
starlike | 0.9*exp(i*0.024544) | Re ≤ -0.1305 | r ≤ 0.800 |
4 runs
|
| z cosh(z) |
z*cosh(z)
|
starlike | 0.9*exp(i*1.546253) | Re ≤ -0.1305 | r ≤ 0.800 |
4 runs
|
| Normalized Bessel J_ν |
{'nu': -0.7}
|
starlike | 0.8*exp(i*6.185011) | Re ≤ -0.4239 | r ≤ 0.600 |
3 runs
|
| z + z² + z³/2 |
z + z^2 + z^3/2
|
starlike | 0.8*exp(i*2.374602) | Re ≤ -0.0624 | r ≤ 0.600 |
4 runs
|
| Janowski S*[A,B] extremal |
{'A': 0.0, 'B': -1.0}
|
starlike | 0.8*exp(i*4.417865) | Re ≤ -0.8541 | r ≤ 0.600 |
3 runs
|
| Janowski S*[A,B] extremal |
{'A': 1.0, 'B': -0.5}
|
starlike | 0.8*exp(i*2.098486) | Re ≤ -0.1635 | r ≤ 0.600 |
3 runs
|
| Janowski S*[A,B] extremal |
{'A': 0.25, 'B': -1.0}
|
starlike | 0.8*exp(i*4.417865) | Re ≤ -1.7310 | r ≤ 0.600 |
3 runs
|
| Janowski S*[A,B] extremal |
{'A': 0.25, 'B': -0.75}
|
starlike | 0.8*exp(i*4.393321) | Re ≤ -0.2027 | r ≤ 0.600 |
3 runs
|
| Janowski S*[A,B] extremal |
{'A': 0.75, 'B': -0.75}
|
starlike | 0.8*exp(i*2.012583) | Re ≤ -1.1790 | r ≤ 0.600 |
3 runs
|
| Laguerre L_n |
{'n': 5}
|
starlike | 0.8*exp(i*0.073631) | Re ≤ -0.0991 | r ≤ 0.600 |
3 runs
|
| Laguerre L_n |
{'n': 6}
|
starlike | 0.8*exp(i*0.049087) | Re ≤ -0.4210 | r ≤ 0.600 |
3 runs
|
| Mocanu α-convex |
{'alpha': 0.0}
|
starlike | 0.8*exp(i*2.374602) | Re ≤ -0.0624 | r ≤ 0.600 |
3 runs
|
| Silverman polynomial test |
{'variant': 2}
|
starlike | 0.8*exp(i*6.258642) | Re ≤ -0.0203 | r ≤ 0.600 |
3 runs
|
| Wright φ(α,β; z) |
{'alpha': 0.5, 'beta': 1.5}
|
starlike | 0.8*exp(i*3.117049) | Re ≤ -0.0621 | r ≤ 0.600 |
3 runs
|
| Wright φ(α,β; z) |
{'alpha': 1.0, 'beta': 1.0}
|
starlike | 0.8*exp(i*2.99433) | Re ≤ -0.3988 | r ≤ 0.600 |
3 runs
|
| Wright φ(α,β; z) |
{'alpha': 0.1, 'beta': 1.25}
|
starlike | 0.8*exp(i*3.104777) | Re ≤ -0.0406 | r ≤ 0.600 |
3 runs
|
| Wright φ(α,β; z) |
{'alpha': 0.1, 'beta': 1.5}
|
starlike | 0.8*exp(i*3.202952) | Re ≤ -0.0643 | r ≤ 0.600 |
3 runs
|
| Wright φ(α,β; z) |
{'alpha': 0.25, 'beta': 1.25}
|
starlike | 0.8*exp(i*3.117049) | Re ≤ -0.1004 | r ≤ 0.600 |
3 runs
|
| Wright φ(α,β; z) |
{'alpha': 0.25, 'beta': 1.5}
|
starlike | 0.8*exp(i*3.117049) | Re ≤ -0.0693 | r ≤ 0.600 |
3 runs
|
| Wright φ(α,β; z) |
{'alpha': 0.5, 'beta': 1.0}
|
starlike | 0.8*exp(i*3.215224) | Re ≤ -0.1693 | r ≤ 0.600 |
3 runs
|
| Wright φ(α,β; z) |
{'alpha': 0.5, 'beta': 1.25}
|
starlike | 0.8*exp(i*3.018874) | Re ≤ -0.1788 | r ≤ 0.600 |
3 runs
|
| Wright φ(α,β; z) |
{'alpha': 0.75, 'beta': 0.75}
|
starlike | 0.8*exp(i*3.19068) | Re ≤ -0.3197 | r ≤ 0.600 |
3 runs
|
| Wright φ(α,β; z) |
{'alpha': 0.75, 'beta': 1.0}
|
starlike | 0.8*exp(i*3.19068) | Re ≤ -0.4024 | r ≤ 0.600 |
3 runs
|
| Wright φ(α,β; z) |
{'alpha': 0.75, 'beta': 1.25}
|
starlike | 0.8*exp(i*3.19068) | Re ≤ -0.2498 | r ≤ 0.600 |
3 runs
|
| Wright φ(α,β; z) |
{'alpha': 0.75, 'beta': 1.5}
|
starlike | 0.8*exp(i*3.153864) | Re ≤ -0.0054 | r ≤ 0.600 |
3 runs
|
| Wright φ(α,β; z) |
{'alpha': 1.25, 'beta': 1.0}
|
starlike | 0.8*exp(i*3.239767) | Re ≤ -0.3284 | r ≤ 0.600 |
3 runs
|
| Wright φ(α,β; z) |
{'alpha': 1.25, 'beta': 1.25}
|
starlike | 0.8*exp(i*3.144661) | Re ≤ -0.0007 | r ≤ 0.600 |
3 runs
|
| Wright φ(α,β; z) |
{'alpha': 1.5, 'beta': 0.75}
|
starlike | 0.8*exp(i*3.239767) | Re ≤ -0.4981 | r ≤ 0.600 |
3 runs
|
| Wright φ(α,β; z) |
{'alpha': 1.5, 'beta': 1.0}
|
starlike | 0.8*exp(i*3.19068) | Re ≤ -0.1037 | r ≤ 0.600 |
3 runs
|
| Wright φ(α,β; z) |
{'alpha': 2.0, 'beta': 0.5}
|
starlike | 0.8*exp(i*3.239767) | Re ≤ -0.2420 | r ≤ 0.600 |
3 runs
|
| z(1+z/2)/(1-z) |
z*(1 + z/2)/(1 - z)
|
starlike | 0.8*exp(i*3.043418) | Re ≤ -0.0986 | r ≤ 0.600 |
3 runs
|
| Modified Bessel I_ν |
{'nu': -0.75}
|
starlike | 0.6*exp(i*3.043418) | Re ≤ -0.1797 | r ≤ 0.300 |
3 runs
|
| Bi-univalent test: a₂=1, a₃=1 | starlike | 0.6*exp(i*4.1847) | Re ≤ -0.1635 | r ≤ 0.300 |
3 runs
|
|
| z + z² |
z + z^2
|
starlike | 0.6*exp(i*3.239767) | Re ≤ -0.4303 | r ≤ 0.300 |
4 runs
|
| Janowski S*[A,B] extremal |
{'A': 1.0, 'B': -0.75}
|
starlike | 0.6*exp(i*2.054) | Re ≤ -0.0098 | r ≤ 0.300 |
3 runs
|
| ₁F₁(a; c; z) — Kummer confluent hypergeometric |
{'a': 2.5, 'c': 1.5}
|
starlike | 0.6*exp(i*3.215224) | Re ≤ -0.2581 | r ≤ 0.300 |
3 runs
|
| Laguerre L_n |
{'n': 7}
|
starlike | 0.6*exp(i*0.122718) | Re ≤ -0.1612 | r ≤ 0.300 |
3 runs
|
| Laguerre L_n |
{'n': 8}
|
starlike | 0.6*exp(i*0.122718) | Re ≤ -0.3781 | r ≤ 0.300 |
3 runs
|
| Laguerre L_n |
{'n': 9}
|
starlike | 0.6*exp(i*0.122718) | Re ≤ -0.5975 | r ≤ 0.300 |
3 runs
|
| Laguerre L_n |
{'n': 10}
|
starlike | 0.6*exp(i*6.025477) | Re ≤ -0.7332 | r ≤ 0.300 |
3 runs
|
| 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
|
| z + z² + z³ | starlike | 0.6*exp(i*4.1847) | Re ≤ -0.1635 | r ≤ 0.300 |
3 runs
|
|
| Wright φ(α,β; z) |
{'alpha': 1.25, 'beta': 0.5}
|
starlike | 0.6*exp(i*3.19068) | Re ≤ -0.1148 | r ≤ 0.300 |
3 runs
|
| Wright φ(α,β; z) |
{'alpha': 1.25, 'beta': 0.75}
|
starlike | 0.6*exp(i*3.144661) | Re ≤ -0.0004 | r ≤ 0.300 |
3 runs
|
| Wright φ(α,β; z) |
{'alpha': 1.5, 'beta': 0.5}
|
starlike | 0.6*exp(i*3.19068) | Re ≤ -0.0888 | r ≤ 0.300 |
3 runs
|
| z(1+z)/(1-z) |
z*(1 + z)/(1 - z)
|
starlike | 0.6*exp(i*2.945243) | Re ≤ -0.6219 | r ≤ 0.300 |
3 runs
|
| z·exp(2z) — convex of order 1/2 |
z*exp(2*z)
|
starlike | 0.6*exp(i*2.945243) | Re ≤ -0.1769 | r ≤ 0.300 |
4 runs
|
| z/(1-z)^4 |
z/(1-z)**4
|
starlike | 0.6*exp(i*2.748894) | Re ≤ -0.4815 | r ≤ 0.300 |
3 runs
|
| z/(1-z)^3 |
z/(1-z)**3
|
starlike | 0.6*exp(i*2.945243) | Re ≤ -0.1216 | r ≤ 0.300 |
3 runs
|
| z/(1+z)^3 |
z/(1+z)**3
|
starlike | 0.6*exp(i*6.086836) | Re ≤ -0.1216 | r ≤ 0.300 |
3 runs
|
| arctanh(z) |
atanh(z)
|
nehari_univalent | 0.9821*exp(i*1.43117) | ≥ 5.1150 (threshold 2) | r ≤ - | 1 |
| arctanh(z) |
atanh(z)
|
becker_univalent | 0.9900*exp(i*1.43117) | ≥ 1.7223 (threshold 1) | r ≤ - | 1 |
| Modified Bessel I_ν |
{'nu': -0.9}
|
starlike | 0.3*exp(i*3.043418) | Re ≤ -1.3073 | r ≤ - |
2 runs
|
| Normalized Bessel J_ν |
{'nu': -0.9}
|
starlike | 0.3*exp(i*6.185011) | Re ≤ -1.3073 | r ≤ - |
2 runs
|
| Normalized Bessel J_ν |
{'nu': -0.8}
|
starlike | 0.6*exp(i*6.185011) | Re ≤ -1.0644 | r ≤ - |
2 runs
|
| Bi-univalent √2 extremal | starlike | 0.6*exp(i*3.043418) | Re ≤ -2.9994 | r ≤ - |
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 |
| z + 2z² |
z + 2z^2
|
starlike | 0.3*exp(i*3.239767) | Re ≤ -0.4303 | r ≤ - |
2 runs
|
| z + z² |
z + z^2
|
becker_univalent | 0.5003*exp(i*3.14159) | ≥ 2973.9997 (threshold 1) | r ≤ - | 1 |
| z + z² + z³/2 |
z + z^2 + z^3/2
|
nehari_univalent | 0.8162*exp(i*-2.51327) | ≥ 1529.2850 (threshold 2) | r ≤ - | 1 |
| z + z² + z³/2 |
z + z^2 + z^3/2
|
becker_univalent | 0.8162*exp(i*-2.51327) | ≥ 32.1175 (threshold 1) | r ≤ - | 1 |
| de Branges boundary, a₂=2 | starlike | 0.3*exp(i*3.239767) | Re ≤ -0.4303 | r ≤ - |
2 runs
|
|
| Janowski S*[A,B] extremal |
{'A': 1.0, 'B': -1.0}
|
starlike | 0.6*exp(i*2.012583) | Re ≤ -0.8652 | r ≤ - |
2 runs
|
| Janowski S*[A,B] extremal |
{'A': 0.75, 'B': -1.0}
|
starlike | 0.6*exp(i*2.012583) | Re ≤ -0.5348 | r ≤ - |
2 runs
|
| Koebe z/(1-z)² |
z/(1-z)^2
|
nehari_univalent | 0.7530*exp(i*-1.29154) | ≥ 281712.3568 (threshold 2) | r ≤ - | 1 |
| Koebe z/(1-z)² |
z/(1-z)^2
|
becker_univalent | 0.7530*exp(i*-1.29154) | ≥ 432.0025 (threshold 1) | r ≤ - | 1 |
| 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 |
| ₁F₁(a; c; z) — Kummer confluent hypergeometric |
{'a': 5.0, 'c': 2.0}
|
starlike | 0.6*exp(i*3.276583) | Re ≤ -1.4924 | r ≤ - |
2 runs
|
| Large a₃ test | starlike | 0.5713*exp(i*-1.60570) | Re ≤ -69.7404 | r ≤ - | 1 | |
| −log(1−z) |
-log(1-z)
|
becker_univalent | 0.9900*exp(i*-2.09440) | ≥ 1.8064 (threshold 1) | r ≤ - | 1 |
| Lommel s_{μ,ν}(z) normalized (ν=1/2) |
{'mu': 4, 'nu': 0.5}
|
starlike | 0.3*exp(i*2.304614) | Re ≤ -12.8331 | r ≤ - |
2 runs
|
| Lommel s_{μ,ν}(z) normalized (ν=1/2) |
{'mu': 1, 'nu': 0.5}
|
starlike | 0.4845*exp(i*1.08210) | Re ≤ -121.4465 | r ≤ - | 1 |
| Lommel s_{μ,ν}(z) normalized (ν=1/2) |
{'mu': 2, 'nu': 0.5}
|
starlike | 0.3976*exp(i*2.96706) | Re ≤ -61.2408 | r ≤ - | 1 |
| Lommel s_{μ,ν}(z) normalized (ν=1/2) |
{'mu': 3, 'nu': 0.5}
|
starlike | 0.3502*exp(i*-2.96706) | Re ≤ -312.7242 | r ≤ - | 1 |
| Lommel s_{μ,ν}(z) normalized (ν=1/2) |
{'mu': 5, 'nu': 0.5}
|
starlike | 0.2791*exp(i*0.45379) | Re ≤ -951.1695 | r ≤ - | 1 |
| Lommel s_{μ,ν}(z) normalized (ν=1/2) |
{'mu': 6, 'nu': 0.5}
|
starlike | 0.2475*exp(i*0.45379) | Re ≤ -76.4059 | r ≤ - | 1 |
| Lommel s_{μ,ν}(z) normalized (ν=1/2) |
{'mu': 7, 'nu': 0.5}
|
starlike | 0.2238*exp(i*0.45379) | Re ≤ -74.1619 | r ≤ - | 1 |
| Lommel s_{μ,ν}(z) normalized (ν=1/2) |
{'mu': 8, 'nu': 0.5}
|
starlike | 0.2080*exp(i*1.95477) | Re ≤ -139.3103 | r ≤ - | 1 |
| Ma-Minda 1+z+z²/2 | starlike | 0.6*exp(i*3.043418) | Re ≤ -2.3794 | r ≤ - |
2 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 |
| 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
|
| z + 3z² (truly disproven) |
z + 3*z**2
|
starlike | 0.3*exp(i*3.043418) | Re ≤ -3.5891 | r ≤ - |
3 runs
|
| z + 3z² (truly disproven) |
z + 3*z**2
|
nehari_univalent | 0.1685*exp(i*3.14159) | ≥ 427156.2015 (threshold 2) | r ≤ - | 1 |
| z + 3z² (truly disproven) |
z + 3*z**2
|
becker_univalent | 0.1685*exp(i*3.14159) | ≥ 533.6392 (threshold 1) | r ≤ - | 1 |
| z + 3z³ (boundary test) |
z + 3*z**3
|
nehari_univalent | 0.3344*exp(i*1.57080) | ≥ 1102790.3191 (threshold 2) | r ≤ - | 1 |
| z + 3z³ (boundary test) |
z + 3*z**3
|
becker_univalent | 0.3344*exp(i*1.57080) | ≥ 858.3202 (threshold 1) | r ≤ - | 1 |
| q-exponential e_q(z) |
{'q': 0.05}
|
starlike | 0.9900*exp(i*2.68781) | Re ≤ -8.5270 | r ≤ - | 1 |
| q-exponential e_q(z) |
{'q': 0.025}
|
starlike | 0.9900*exp(i*-2.09440) | Re ≤ -20.0265 | r ≤ - | 1 |
| 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 |
| z cos(z) |
z*cos(z)
|
nehari_univalent | 0.8636*exp(i*0.00000) | ≥ 9014.3537 (threshold 2) | r ≤ - | 1 |
| z cos(z) |
z*cos(z)
|
becker_univalent | 0.8636*exp(i*0.00000) | ≥ 77.5740 (threshold 1) | r ≤ - | 1 |
| z·exp(2z) — convex of order 1/2 |
z*exp(2*z)
|
nehari_univalent | 0.5003*exp(i*3.14159) | ≥ 13262553.3751 (threshold 2) | r ≤ - | 1 |
| z·exp(2z) — convex of order 1/2 |
z*exp(2*z)
|
becker_univalent | 0.5003*exp(i*3.14159) | ≥ 2972.5003 (threshold 1) | r ≤ - | 1 |
| z e^(-z) |
z*exp(-z)
|
nehari_univalent | 0.9900*exp(i*0.00000) | ≥ 5.9799 (threshold 2) | r ≤ - | 1 |
| z e^(-z) |
z*exp(-z)
|
becker_univalent | 0.5003*exp(i*0.00000) | ≥ 2.2500 (threshold 1) | r ≤ - | 1 |
| z e^z |
z*exp(z)
|
becker_univalent | 0.5003*exp(i*3.14159) | ≥ 2.2500 (threshold 1) | r ≤ - | 1 |
| z/(1-z) |
z/(1-z)
|
becker_univalent | 0.8715*exp(i*0.94248) | ≥ 146.9651 (threshold 1) | r ≤ - | 1 |
| z/(1-z/2) |
z/(1 - z/2)
|
becker_univalent | 0.2712*exp(i*0.00000) | ≥ 1.0718 (threshold 1) | r ≤ - | 1 |
| z/(1-z/2)² |
z/(1 - z/2)^2
|
becker_univalent | 0.1527*exp(i*0.00000) | ≥ 2.0398 (threshold 1) | r ≤ - | 1 |
| z/(1+z) |
z/(1+z)
|
becker_univalent | 0.8715*exp(i*-2.19911) | ≥ 146.9651 (threshold 1) | r ≤ - | 1 |
| z/(1+z²) |
z/(1+z^2)
|
nehari_univalent | 0.8794*exp(i*-0.94248) | ≥ 56234.0327 (threshold 2) | r ≤ - | 1 |
| z/(1+z²) |
z/(1+z^2)
|
becker_univalent | 0.8794*exp(i*-0.94248) | ≥ 191.8831 (threshold 1) | r ≤ - | 1 |
GET /api/v2/counterexamples