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Ruscheweyh family R(λ)

Ruscheweyh · parametric, arity 1

Canonical Key
ruscheweyh
Parameter Space
lambda ∈ (0, inf) (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.

lambda=0.5
lambda=0.5 lambda=4.0
Taylor series

Unit disk explorer - hover to evaluate

Hover over the disk

Related

0
Counterexamples
 →
35
Facts
 →
50
Verification runs
 →
23
Instances
 →
3
Cited in papers

Facts (35)

Property Kind Params Holds Value / r Status Confidence
K membership {"lambda": 0.5} ✗ -71.1186 unscreened -
K membership {"lambda": 1.0} ✗ -17.5859 unscreened -
K membership {"lambda": 1.5} ✗ -90.4840 unscreened -
K membership {"lambda": 2.0} ✗ -161.9735 unscreened -
r* radius {"lambda": 0.5} - r ≤ 0.906 unscreened -
r* radius {"lambda": 1.0} - r ≤ 0.838 unscreened -
r* radius {"lambda": 1.5} - r ≤ 0.778 unscreened -
r* radius {"lambda": 2.0} - r ≤ 0.920 unscreened -
S* membership {"lambda": 4.0} - - retracted -
S* membership {"lambda": 3.5} - - retracted -
S* membership {"lambda": 3.0} - - retracted -
S* membership {"lambda": 2.75} - - retracted -
S* membership {"lambda": 2.5} - - retracted -
S* membership {"lambda": 2.25} - - retracted -
S* membership {"lambda": 1.9} - - retracted -
S* membership {"lambda": 1.75} - - retracted -
S* membership {"lambda": 1.6} - - retracted -
S* membership {"lambda": 1.4} - - retracted -
S* membership {"lambda": 1.25} - - retracted -
S* membership {"lambda": 1.1} - - retracted -
S* membership {"lambda": 0.9} - - retracted -
S* membership {"lambda": 0.8} - - retracted -
S* membership {"lambda": 0.75} - - retracted -
S* membership {"lambda": 0.6} - - retracted -
S* membership {"lambda": 0.4} - - retracted -
S* membership {"lambda": 0.25} - - retracted -
S* membership {"lambda": 0.1} - - retracted -
S* membership {"lambda": 0.5} - -0.1613 retracted -
S* membership {"lambda": 1.0} - -3.8277 retracted -
S* membership {"lambda": 1.5} - -41.3492 retracted -
S* membership {"lambda": 2.0} - -197.5407 retracted -
S membership {"lambda": 0.5} ✗ - unscreened -
S membership {"lambda": 1.0} ✗ - unscreened -
S membership {"lambda": 1.5} ✗ - unscreened -
S membership {"lambda": 2.0} ✗ - unscreened -

Verification Runs (50)

Verifier Property Direction Outcome Domain r Witness Engine
boundary_scan_centered_v31 starlike disproves pass - - boundary_v3.1
boundary_scan_centered_v31 starlike disproves pass - - boundary_v3.1
boundary_scan_centered_v31 starlike disproves pass - - boundary_v3.1
boundary_scan_centered_v31 starlike disproves pass - - boundary_v3.1
boundary_scan_centered_v31 starlike disproves pass - - boundary_v3.1
boundary_scan_centered_v31 starlike disproves pass - - boundary_v3.1
boundary_scan_centered_v31 starlike disproves pass - - boundary_v3.1
boundary_scan_centered_v31 starlike disproves pass - - boundary_v3.1
boundary_scan_centered_v31 starlike disproves pass - - boundary_v3.1
boundary_scan_centered_v31 starlike disproves pass - - boundary_v3.1
boundary_scan_centered_v31 starlike disproves pass - - boundary_v3.1
boundary_scan_centered_v31 starlike disproves pass - - boundary_v3.1
boundary_scan_centered_v31 starlike disproves pass - - boundary_v3.1
boundary_scan_centered_v31 starlike disproves pass - - boundary_v3.1
boundary_scan_centered_v31 starlike disproves pass - - boundary_v3.1
boundary_scan_centered_v31 starlike disproves pass - - boundary_v3.1
boundary_scan_centered_v31 starlike disproves pass 0.900 0.95*exp(i*0.581379) boundary_v3.1
boundary_scan_centered_v31 starlike disproves pass 0.900 0.95*exp(i*5.707415) 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 - - boundary_v3.1
boundary_scan_centered_v31 starlike disproves pass - - boundary_v3.1
boundary_scan_centered_v31 starlike disproves pass - - boundary_v3.1
boundary_scan_centered_v31 starlike disproves pass 0.900 0.95*exp(i*6.004001) boundary_v3.1
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, 0.2728809822588292] 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 [0.8207471968294914, -0.5536009744360391] gft-0.1.0-reaudit-20260610
pointwise_starlike_iv starlike disproves pass 0.990 [0.8207471968294912, 0.5536009744360394] gft-0.1.0-reaudit-20260610
pointwise_starlike_iv starlike disproves pass 0.982 [-0.49105042016806766, -0.8505242768091371] gft-0.1.0-reaudit-20260610
pointwise_starlike_iv starlike disproves pass 0.966 [0.06740585643560469, -0.9639486566334218] gft-0.1.0-reaudit-20260610
pointwise_starlike_iv starlike disproves pass 0.951 [0.5851888191099597, 0.7490075322727422] gft-0.1.0-reaudit-20260610
pointwise_starlike_iv starlike disproves pass 0.919 [0.06409975129344615, -0.9166691503858133] gft-0.1.0-reaudit-20260610
pointwise_starlike_iv starlike disproves pass 0.903 [0.06299771624605997, -0.9009093149699438] gft-0.1.0-reaudit-20260610
pointwise_starlike_iv starlike disproves pass 0.903 [0.7306307259283663, 0.5308342946543873] gft-0.1.0-reaudit-20260610
pointwise_starlike_iv starlike disproves pass 0.856 [0.29267152264582247, -0.8041083997867986] gft-0.1.0-reaudit-20260610
pointwise_starlike_iv starlike disproves pass 0.832 [0.2845665074846594, 0.7818400536471662] gft-0.1.0-reaudit-20260610
pointwise_starlike_iv starlike disproves pass 0.808 [-0.19555014401455498, 0.7843087891637655] gft-0.1.0-reaudit-20260610
pointwise_starlike_iv starlike disproves pass 0.800 [0.4704751704273787, 0.6475535186068382] gft-0.1.0-reaudit-20260610
pointwise_starlike_iv starlike disproves pass 0.753 [-0.18217328625786747, 0.7306571431226222] gft-0.1.0-reaudit-20260610
pointwise_starlike_iv starlike disproves pass 0.714 [-0.53025569017887, 0.47744436794899714] 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 [-0.49500000000000044, -0.857365149746594] gft-0.1.0-reaudit-20260610
pointwise_starlike_iv starlike disproves pass 0.935 [0.06520178634083233, -0.9324289858016828] gft-0.1.0-reaudit-20260610
pointwise_starlike_iv starlike disproves pass 0.895 [0.7242401715190179, -0.5261912850984637] gft-0.1.0-reaudit-20260610
boundary_scan_polynomial starlike proves pass 0.300 - boundary_v3.0
boundary_scan_polynomial starlike proves pass 0.300 - boundary_v3.0
boundary_scan_polynomial starlike proves pass 0.300 - boundary_v3.0
boundary_scan_polynomial starlike proves pass 0.300 - boundary_v3.0

Parameter Sweep (69 recorded instances, starlikeness evidence × disk radius vs parameter)

Legacy “proven” label (scope not recorded)   ·   Recorded finite-radius evidence   ·   Disproven (witness)

Instances (23)

{"lambda": 0.5} {"lambda": 1.0} {"lambda": 1.5} {"lambda": 2.0} {"lambda": 0.1} {"lambda": 0.25} {"lambda": 0.4} {"lambda": 0.6} {"lambda": 0.75} {"lambda": 0.8} {"lambda": 0.9} {"lambda": 1.1} {"lambda": 1.25} {"lambda": 1.4} {"lambda": 1.6} {"lambda": 1.75} {"lambda": 1.9} {"lambda": 2.25} {"lambda": 2.5} {"lambda": 2.75} {"lambda": 3.0} {"lambda": 3.5} {"lambda": 4.0}

Cited in Papers (3)

Universal convexity and range problems of shifted hypergeometric functions
Toshiyuki Sugawa, Li-Mei Wang, Chengfa Wu · 2023
Sufficiency for Nephroid Starlikeness using Hypergeometric Functions
A. Swaminathan, Lateef Ahmad Wani · 2021
Univalence criterion for meromorphic functions and Loewner chains
Erhan Deníz, Halit Orhan · 2011
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