Verify & find counterexamples

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).

Closed form SymPy syntax LaTeX type math, not code Taylor coefficients a₂, a₃, a₄, …

f(z) - SymPy syntax (required: f(0) = 0 and f′(0) = 1)

Presets: Koebe · z/(1−z) · z·exp(z/2) · log(1+z) · arctan(z) · sin(z) · z+z² (not starlike)
Ma-Minda extremals: cardioid · lemniscate · ℘ cardioid · 3-leaf · 4-leaf · epi-3 · epi-6

Counterexamples found with this tool

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.

Filtered to z/(1-z). Show all →

1Counterexample facts

Family	Closed form / Params	Property	Representative witness z	Re(zf′/f) ≤	Proven r	Witnesses
z/(1-z)	z/(1-z)	becker_univalent	0.8715exp(i0.94248)	≥ 146.9651 (threshold 1)	r ≤ -	1

How to read a witness

z = r·exp(i·θ) is a point on the boundary circle of radius r in the unit disk. The verifier bounds Re(zf′(z)/f(z)) over a small arc around z with mpmath interval arithmetic; the upper bound shown is strictly negative, so Re(zf′/f) < 0 on the whole arc - f cannot be starlike on a disk containing z. Proven, not approximated. Programmatic access: GET /api/v2/counterexamples