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Strongly starlike SS*(β)

Strongly Starlike · parametric, arity 1

This function is certified starlike - the sharp, proven-exact coefficient bounds for the starlike class apply to it. See the certified proofs →
Canonical Key
strongly_starlike_extremal
Parameter Space
beta ∈ (0,1] (real)

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.

beta=0.2
beta=0.2 beta=1.0
Taylor series
r* ≈ 0.900
Radius of starlikeness r* ≈ 0.9000
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

→
30
Counterexamples
 →
35
Facts
 →
50
Verification runs
 →
20
Instances
0
Cited in papers

Facts (35)

Property Kind Params Holds Value / r Status Confidence
K membership {"beta": 0.2} ✗ -0.7094 unscreened -
K membership {"beta": 0.4} ✗ -5.0142 unscreened -
K membership {"beta": 0.5} ✗ -10.3086 unscreened -
K membership {"beta": 0.6} ✗ -23.0117 unscreened -
K membership {"beta": 0.8} ✗ -263.1290 unscreened -
r* radius {"beta": 0.2} - r ≤ 0.920 unscreened -
r* radius {"beta": 0.4} - r ≤ 0.920 unscreened -
r* radius {"beta": 0.5} - r ≤ 0.920 unscreened -
r* radius {"beta": 0.6} - r ≤ 0.920 unscreened -
r* radius {"beta": 0.8} - r ≤ 0.920 unscreened -
S* membership {"beta": 0.9} ✗ r ≤ 0.900 ✗ disproven proven
S* membership {"beta": 0.95} ✗ r ≤ 0.800 ✗ disproven proven
S* membership {"beta": 1.0} ✗ r ≤ 0.800 ✗ disproven proven
S* membership {"beta": 0.05} ✓ r ≤ 0.999 ✓ proven rigorous_verified
S* membership {"beta": 0.1} ✓ r ≤ 0.999 ✓ proven rigorous_verified
S* membership {"beta": 0.15} ✓ r ≤ 0.999 ✓ proven rigorous_verified
S* membership {"beta": 0.25} ✓ r ≤ 0.999 ✓ proven rigorous_verified
S* membership {"beta": 0.3} ✓ r ≤ 0.999 ✓ proven rigorous_verified
S* membership {"beta": 0.35} ✓ r ≤ 0.999 ✓ proven rigorous_verified
S* membership {"beta": 0.45} ✓ r ≤ 0.999 ✓ proven rigorous_verified
S* membership {"beta": 0.55} ✗ r ≤ 0.950 ✗ disproven proven
S* membership {"beta": 0.65} ✗ r ≤ 0.900 ✗ disproven proven
S* membership {"beta": 0.7} ✗ r ≤ 0.900 ✗ disproven proven
S* membership {"beta": 0.75} ✗ r ≤ 0.900 ✗ disproven proven
S* membership {"beta": 0.85} ✗ r ≤ 0.900 ✗ disproven proven
S* membership {"beta": 0.2} ✓ r ≤ 0.999 ✓ proven rigorous_verified
S* membership {"beta": 0.4} ✓ r ≤ 0.999 ✓ proven rigorous_verified
S* membership {"beta": 0.5} ✓ r ≤ 0.999 ✓ proven rigorous_verified
S* membership {"beta": 0.6} ✗ r ≤ 0.950 ✗ disproven proven
S* membership {"beta": 0.8} ✗ r ≤ 0.900 ✗ disproven proven
S membership {"beta": 0.2} ✗ - unscreened -
S membership {"beta": 0.4} ✗ - unscreened -
S membership {"beta": 0.5} ✗ - unscreened -
S membership {"beta": 0.6} ✗ - unscreened -
S membership {"beta": 0.8} ✗ - unscreened -

Transitively Implied Properties

Because starlikeness is proven, these properties follow by classical implications (unconditional edges in the lattice):

S C

Verification Runs (50)

Verifier Property Direction Outcome Domain r Witness Engine
boundary_scan_centered_v31 starlike disproves pass 0.800 0.9*exp(i*2.687534) boundary_v3.1
boundary_scan_centered_v31 starlike disproves pass 0.900 0.95*exp(i*2.785709) boundary_v3.1
boundary_scan_centered_v31 starlike disproves pass 0.800 0.9*exp(i*3.528156) boundary_v3.1
boundary_scan_centered_v31 starlike disproves pass 0.900 0.95*exp(i*3.144661) boundary_v3.1
boundary_scan_centered_v31 starlike disproves pass 0.900 0.95*exp(i*3.153864) boundary_v3.1
boundary_scan_centered_v31 starlike disproves pass 0.900 0.95*exp(i*3.153864) boundary_v3.1
boundary_scan_centered_v31 starlike disproves pass 0.900 0.95*exp(i*3.276583) boundary_v3.1
boundary_scan_centered_v31 starlike proves pass 0.999 - boundary_v3.1
boundary_scan_centered_v31 starlike proves pass 0.999 - boundary_v3.1
boundary_scan_centered_v31 starlike proves pass 0.999 - boundary_v3.1
boundary_scan_centered_v31 starlike proves pass 0.999 - boundary_v3.1
boundary_scan_centered_v31 starlike disproves pass 0.950 0.99*exp(i*3.147729) boundary_v3.1
boundary_scan_centered_v31 starlike proves pass 0.999 - boundary_v3.1
boundary_scan_centered_v31 starlike disproves pass 0.950 0.99*exp(i*3.166136) boundary_v3.1
boundary_scan_centered_v31 starlike disproves pass 0.900 0.95*exp(i*3.202952) boundary_v3.1
boundary_scan_centered_v31 starlike proves pass 0.999 - boundary_v3.1
boundary_scan_centered_v31 starlike proves pass 0.999 - boundary_v3.1
boundary_scan_centered_v31 starlike proves pass 0.999 - boundary_v3.1
boundary_scan_centered_v31 starlike proves pass 0.999 - boundary_v3.1
boundary_scan_centered_v31 starlike proves pass 0.999 - boundary_v3.1
pointwise_starlike_iv starlike disproves pass 0.990 [-0.99, 1.2124003311558797e-16] 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
pointwise_starlike_iv starlike disproves pass 0.990 [-0.99, 1.2124003311558797e-16] 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
pointwise_starlike_iv starlike disproves pass 0.990 [-0.974959675482086, 0.17191169589026098] gft-0.1.0-reaudit-20260610
pointwise_starlike_iv starlike disproves pass 0.990 [-0.9302956945780493, -0.338599941892412] gft-0.1.0-reaudit-20260610
pointwise_starlike_iv starlike disproves pass 0.990 [-0.9044100030661748, -0.4026692766450422] gft-0.1.0-reaudit-20260610
pointwise_starlike_iv starlike disproves pass 0.990 [-0.8898061058361754, 0.4339874353211865] 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
pointwise_starlike_iv starlike disproves pass 0.990 [-0.99, 1.2124003311558797e-16] gft-0.1.0-reaudit-20260610
boundary_scan_polynomial starlike disproves pass 0.900 0.95*exp(i*3.503612) boundary_v3.0
boundary_scan_polynomial starlike disproves pass 0.800 0.9*exp(i*2.764233) boundary_v3.0
boundary_scan_polynomial starlike disproves pass 0.800 0.9*exp(i*3.577243) boundary_v3.0
boundary_scan_polynomial starlike proves pass 0.999 - boundary_v3.0
boundary_scan_polynomial starlike proves pass 0.999 - boundary_v3.0
boundary_scan_polynomial starlike proves pass 0.999 - boundary_v3.0
boundary_scan_polynomial starlike proves pass 0.999 - boundary_v3.0
boundary_scan_polynomial starlike proves pass 0.999 - boundary_v3.0
boundary_scan_polynomial starlike proves pass 0.999 - boundary_v3.0
boundary_scan_polynomial starlike proves pass 0.999 - boundary_v3.0
boundary_scan_polynomial starlike disproves pass 0.950 0.99*exp(i*3.117049) boundary_v3.0
boundary_scan_polynomial starlike disproves pass 0.900 0.95*exp(i*3.153864) boundary_v3.0
boundary_scan_polynomial starlike disproves pass 0.900 0.95*exp(i*3.117049) boundary_v3.0
boundary_scan_polynomial starlike disproves pass 0.900 0.95*exp(i*3.117049) boundary_v3.0
boundary_scan_polynomial starlike disproves pass 0.900 0.95*exp(i*3.117049) boundary_v3.0
boundary_scan_polynomial starlike proves pass 0.999 - boundary_v3.0
boundary_scan_polynomial starlike proves pass 0.999 - boundary_v3.0
boundary_scan_polynomial starlike proves pass 0.999 - boundary_v3.0
boundary_scan_polynomial starlike disproves pass 0.950 0.99*exp(i*3.092505) boundary_v3.0
boundary_scan_polynomial starlike disproves pass 0.900 0.95*exp(i*3.117049) boundary_v3.0

Parameter Sweep (50 verified instances, starlikeness × disk radius vs parameter)

Proven on full disk (r ≥ 0.999)   ·   Proven on partial disk   ·   Disproven (witness)

Instances (20)

{"beta": 0.2} {"beta": 0.4} {"beta": 0.5} {"beta": 0.6} {"beta": 0.8} {"beta": 0.05} {"beta": 0.1} {"beta": 0.15} {"beta": 0.25} {"beta": 0.3} {"beta": 0.35} {"beta": 0.45} {"beta": 0.55} {"beta": 0.65} {"beta": 0.7} {"beta": 0.75} {"beta": 0.85} {"beta": 0.9} {"beta": 0.95} {"beta": 1.0}
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