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Canonical Key
z*exp(2*z)
Closed Form
f(z) = z*exp(2*z)

Image domain \(f(\mathbb{D})\)

The image of the unit disk under \(f\): concentric circles \(|z|=r\) and radial spokes mapped through \(f(z)=z+a_2z^2+\dots\); the bold curve is the boundary \(f(e^{i\theta})\). Rendered in your browser from the certified Taylor coefficients - nothing is computed server-side.

Taylor series
r* ≈ 0.300
Radius of starlikeness r* ≈ 0.3000
disproven f is starlike on |z| < r*; the colored disk shows the certified region.

Unit disk explorer - hover to evaluate

Hover over the disk

Related

Univalence Criteria

Schwarzian norm ||S(f)||
5.667
✗ exceeds Nehari bound 2
Pre-Schwarzian norm ||T(f)||
2.958
✗ exceeds Becker bound 1

Facts (5)

Property Kind Params Holds Value / r Status Confidence
B membership {} 2.9575 ✗ disproven rigorous_verified
N membership {} 5.6673 ✗ disproven rigorous_verified
||T(f)|| bound {} 2.9575 ✓ proven rigorous_verified
||S(f)|| bound {} 5.6673 ✓ proven rigorous_verified
S* membership {} r ≤ 0.300 ✗ disproven proven

Verification Runs (6)

Verifier Property Direction Outcome Domain r Witness Engine
boundary_scan_closed_form_v31 starlike disproves pass 0.300 0.6*exp(i*2.945243) boundary_v3.1
pointwise_nehari_univalent_iv nehari_univalent disproves pass 0.500 [-0.5002521008403361, 6.126321340608566e-17] gft-0.1.0-reaudit-20260610
pointwise_becker_univalent_iv becker_univalent disproves pass 0.500 [-0.5002521008403361, 6.126321340608566e-17] gft-0.1.0-reaudit-20260610
pointwise_starlike_iv starlike disproves pass 0.990 [-0.99, 1.2124003311558797e-16] gft-0.1.0-reaudit-20260610
boundary_scan_centered_v31 starlike disproves pass 0.300 0.6*exp(i*3.362486) boundary_v3.1
boundary_scan_closed_form starlike disproves pass 0.300 0.6*exp(i*2.945243) boundary_v3.0

Instances (1)

{}
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