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Convex of order α — K(α)

Convex Order · 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
convex_order_extremal
Parameter Space
alpha ∈ [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.

alpha=0.1
alpha=0.1 alpha=0.99
Taylor series
r* ≈ 0.999
Radius of starlikeness r* ≈ 0.9990
proven f is starlike on |z| < r*; the colored disk shows the certified region.

Unit disk explorer - hover to evaluate

Hover over the disk

Related

→
24
Counterexamples
 →
40
Facts
 →
50
Verification runs
 →
25
Instances
0
Cited in papers

Facts (40)

Property Kind Params Holds Value / r Status Confidence
K membership {"alpha": 0.1} ✗ -40.7181 unscreened -
K membership {"alpha": 0.25} ✗ -71.1186 unscreened -
K membership {"alpha": 0.5} ✗ -3.8277 unscreened -
K membership {"alpha": 0.75} ✓ 0.3081 unscreened -
K membership {"alpha": 0.9} ✓ 0.8498 unscreened -
r* radius {"alpha": 0.1} - r ≤ 0.865 unscreened -
r* radius {"alpha": 0.25} - r ≤ 0.906 unscreened -
r* radius {"alpha": 0.5} - r ≤ 0.920 unscreened -
r* radius {"alpha": 0.75} - r ≤ 0.920 unscreened -
r* radius {"alpha": 0.9} - r ≤ 0.920 unscreened -
S* membership {"alpha": 0.9379} ✓ r ≤ 0.999 ✓ proven rigorous_verified
S* membership {"alpha": 0.99} ✓ r ≤ 0.999 ✓ proven rigorous_verified
S* membership {"alpha": 0.8858} ✓ r ≤ 0.999 ✓ proven rigorous_verified
S* membership {"alpha": 0.8337} ✓ r ≤ 0.999 ✓ proven rigorous_verified
S* membership {"alpha": 0.7816} ✓ r ≤ 0.999 ✓ proven rigorous_verified
S* membership {"alpha": 0.7295} ✓ r ≤ 0.999 ✓ proven rigorous_verified
S* membership {"alpha": 0.6774} ✓ r ≤ 0.999 ✓ proven rigorous_verified
S* membership {"alpha": 0.6253} ✓ r ≤ 0.999 ✓ proven rigorous_verified
S* membership {"alpha": 0.5732} ✓ r ≤ 0.999 ✓ proven rigorous_verified
S* membership {"alpha": 0.5211} ✗ r ≤ 0.950 ✗ disproven proven
S* membership {"alpha": 0.4689} ✗ r ≤ 0.950 ✗ disproven proven
S* membership {"alpha": 0.4168} ✗ r ≤ 0.950 ✗ disproven proven
S* membership {"alpha": 0.3647} ✗ r ≤ 0.900 ✗ disproven proven
S* membership {"alpha": 0.3126} ✗ r ≤ 0.900 ✗ disproven proven
S* membership {"alpha": 0.2605} ✗ r ≤ 0.900 ✗ disproven proven
S* membership {"alpha": 0.2084} ✗ r ≤ 0.000 ✗ disproven proven
S* membership {"alpha": 0.1563} ✗ r ≤ 0.000 ✗ disproven proven
S* membership {"alpha": 0.1042} ✗ r ≤ 0.000 ✗ disproven proven
S* membership {"alpha": 0.0521} ✗ r ≤ 0.000 ✗ disproven proven
S* membership {"alpha": 0.0} ✗ r ≤ 0.000 ✗ disproven proven
S* membership {"alpha": 0.1} ✗ r ≤ 0.000 ✗ disproven proven
S* membership {"alpha": 0.25} ✗ r ≤ 0.900 ✗ disproven proven
S* membership {"alpha": 0.5} ✗ r ≤ 0.950 ✗ disproven proven
S* membership {"alpha": 0.75} ✓ r ≤ 0.999 ✓ proven rigorous_verified
S* membership {"alpha": 0.9} ✓ r ≤ 0.999 ✓ proven rigorous_verified
S membership {"alpha": 0.1} ✗ - unscreened -
S membership {"alpha": 0.25} ✗ - unscreened -
S membership {"alpha": 0.5} ✗ - unscreened -
S membership {"alpha": 0.75} ✗ - unscreened -
S membership {"alpha": 0.9} ✗ - 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 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
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*6.02471) boundary_v3.1
boundary_scan_centered_v31 starlike disproves pass 0.950 0.99*exp(i*6.022409) boundary_v3.1
boundary_scan_centered_v31 starlike disproves pass 0.950 0.99*exp(i*0.266913) boundary_v3.1
boundary_scan_centered_v31 starlike disproves pass 0.900 0.95*exp(i*5.706792) boundary_v3.1
boundary_scan_centered_v31 starlike disproves pass 0.900 0.95*exp(i*6.005535) boundary_v3.1
boundary_scan_centered_v31 starlike disproves pass 0.900 0.95*exp(i*6.004001) boundary_v3.1
boundary_scan_centered_v31 starlike disproves pass 0.000 - boundary_v3.1
boundary_scan_centered_v31 starlike disproves pass 0.000 - boundary_v3.1
boundary_scan_centered_v31 starlike disproves pass 0.000 - boundary_v3.1
boundary_scan_centered_v31 starlike disproves pass 0.000 - boundary_v3.1
boundary_scan_centered_v31 starlike disproves pass 0.000 - 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*6.023943) boundary_v3.1
boundary_scan_centered_v31 starlike disproves pass 0.900 0.95*exp(i*6.004001) boundary_v3.1
boundary_scan_centered_v31 starlike disproves pass 0.000 - boundary_v3.1
pointwise_starlike_iv starlike disproves pass 0.990 [0.8207471968294912, 0.5536009744360394] gft-0.1.0-reaudit-20260610
pointwise_starlike_iv starlike disproves pass 0.990 [0.8207471968294914, -0.5536009744360391] gft-0.1.0-reaudit-20260610
pointwise_starlike_iv starlike disproves pass 0.990 [0.9516490789789357, 0.2728809822588292] gft-0.1.0-reaudit-20260610
pointwise_starlike_iv starlike disproves pass 0.990 [0.9516490789789357, 0.2728809822588292] gft-0.1.0-reaudit-20260610
pointwise_starlike_iv starlike disproves pass 0.990 [0.9516490789789357, 0.2728809822588292] gft-0.1.0-reaudit-20260610
pointwise_starlike_iv starlike disproves pass 0.990 [0.9516490789789357, 0.2728809822588292] gft-0.1.0-reaudit-20260610
pointwise_starlike_iv starlike disproves pass 0.990 [0.9516490789789357, 0.2728809822588292] gft-0.1.0-reaudit-20260610
pointwise_starlike_iv starlike disproves pass 0.990 [0.9516490789789357, -0.27288098225882895] gft-0.1.0-reaudit-20260610
pointwise_starlike_iv starlike disproves pass 0.990 [0.9516490789789357, -0.27288098225882895] gft-0.1.0-reaudit-20260610
pointwise_starlike_iv starlike disproves pass 0.990 [0.9605927690132365, -0.2395026766436712] gft-0.1.0-reaudit-20260610
pointwise_starlike_iv starlike disproves pass 0.990 [0.8207471968294914, -0.5536009744360391] gft-0.1.0-reaudit-20260610
pointwise_starlike_iv starlike disproves pass 0.990 [0.9516490789789357, 0.2728809822588292] gft-0.1.0-reaudit-20260610
pointwise_starlike_iv starlike disproves pass 0.990 [0.9605927690132365, -0.2395026766436712] gft-0.1.0-reaudit-20260610
pointwise_starlike_iv starlike disproves pass 0.990 [-0.49500000000000044, -0.857365149746594] gft-0.1.0-reaudit-20260610
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 proves pass 0.999 - boundary_v3.0
boundary_scan_polynomial starlike proves pass 0.950 - boundary_v3.0
boundary_scan_polynomial starlike disproves pass 0.950 0.99*exp(i*6.024038) boundary_v3.0
boundary_scan_polynomial starlike proves pass 0.900 - boundary_v3.0

Parameter Sweep (64 verified instances, starlikeness × disk radius vs parameter)

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

Instances (25)

{"alpha": 0.1} {"alpha": 0.25} {"alpha": 0.5} {"alpha": 0.75} {"alpha": 0.9} {"alpha": 0.0} {"alpha": 0.0521} {"alpha": 0.1042} {"alpha": 0.1563} {"alpha": 0.2084} {"alpha": 0.2605} {"alpha": 0.3126} {"alpha": 0.3647} {"alpha": 0.4168} {"alpha": 0.4689} {"alpha": 0.5211} {"alpha": 0.5732} {"alpha": 0.6253} {"alpha": 0.6774} {"alpha": 0.7295} {"alpha": 0.7816} {"alpha": 0.8337} {"alpha": 0.8858} {"alpha": 0.9379} {"alpha": 0.99}
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