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Wright φ(α,β; z)

Wright · parametric, arity 2

This function is certified starlike - the sharp, proven-exact coefficient bounds for the starlike class apply to it. See the certified proofs →
Canonical Key
wright_normalized
Parameter Space
alpha ∈ (0, inf) (real) ; beta ∈ R (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=5.0, beta=1.0
alpha=5.0, beta=1.0 alpha=2.0, beta=2.0
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

→
81
Counterexamples
 →
60
Facts
 →
50
Verification runs
 →
45
Instances
 →
3
Cited in papers

Facts (60)

Property Kind Params Holds Value / r Status Confidence
K membership {"alpha": 5.0, "beta": 1.0} ✗ -909.1245 unscreened -
K membership {"alpha": 0.5, "beta": 0.5} ✗ -3.0094 unscreened -
K membership {"alpha": 0.5, "beta": 1.5} ✗ -517.2295 unscreened -
K membership {"alpha": 1.0, "beta": 5.0} ✗ -179.3449 unscreened -
K membership {"alpha": 1.0, "beta": 1.0} ✗ -151.2139 unscreened -
r* radius {"alpha": 5.0, "beta": 1.0} - r ≤ 0.707 🔬 screened numerical_strong
r* radius {"alpha": 0.5, "beta": 0.5} - r ≤ 0.960 🔬 screened numerical_strong
r* radius {"alpha": 0.5, "beta": 1.5} - r ≤ 0.764 🔬 screened numerical_strong
r* radius {"alpha": 1.0, "beta": 5.0} - r ≤ 0.607 🔬 screened numerical_strong
r* radius {"alpha": 1.0, "beta": 1.0} - r ≤ 0.635 🔬 screened numerical_strong
S* membership {"alpha": 0.75, "beta": 2.0} ✓ r ≤ 0.999 ✓ proven rigorous_verified
S* membership {"alpha": 1.25, "beta": 0.5} ✗ r ≤ 0.300 ✗ disproven proven
S* membership {"alpha": 1.25, "beta": 0.75} ✗ r ≤ 0.300 ✗ disproven proven
S* membership {"alpha": 1.25, "beta": 1.0} ✗ r ≤ 0.600 ✗ disproven proven
S* membership {"alpha": 1.25, "beta": 1.25} ✗ r ≤ 0.600 ✗ disproven proven
S* membership {"alpha": 1.25, "beta": 1.5} ✗ r ≤ 0.950 ✗ disproven proven
S* membership {"alpha": 1.25, "beta": 2.0} ✓ r ≤ 0.999 ✓ proven rigorous_verified
S* membership {"alpha": 1.5, "beta": 0.5} ✗ r ≤ 0.300 ✗ disproven proven
S* membership {"alpha": 1.5, "beta": 0.75} ✗ r ≤ 0.600 ✗ disproven proven
S* membership {"alpha": 1.5, "beta": 1.0} ✗ r ≤ 0.600 ✗ disproven proven
S* membership {"alpha": 1.5, "beta": 1.25} ✗ r ≤ 0.900 ✗ disproven proven
S* membership {"alpha": 1.5, "beta": 1.5} ✓ r ≤ 0.999 ✓ proven rigorous_verified
S* membership {"alpha": 1.5, "beta": 2.0} ✓ r ≤ 0.999 ✓ proven rigorous_verified
S* membership {"alpha": 2.0, "beta": 0.5} ✗ r ≤ 0.600 ✗ disproven proven
S* membership {"alpha": 2.0, "beta": 0.75} ✗ r ≤ 0.800 ✗ disproven proven
S* membership {"alpha": 2.0, "beta": 1.0} ✓ r ≤ 0.999 ✓ proven rigorous_verified
S* membership {"alpha": 2.0, "beta": 1.25} ✓ r ≤ 0.999 ✓ proven rigorous_verified
S* membership {"alpha": 2.0, "beta": 1.5} ✓ r ≤ 0.999 ✓ proven rigorous_verified
S* membership {"alpha": 2.0, "beta": 2.0} ✓ r ≤ 0.999 ✓ proven rigorous_verified
S* membership {"alpha": 0.5, "beta": 2.0} ✓ r ≤ 0.999 ✓ proven rigorous_verified
S* membership {"alpha": 0.75, "beta": 0.5} ✗ r ≤ 0.800 ✗ disproven proven
S* membership {"alpha": 0.75, "beta": 0.75} ✗ r ≤ 0.600 ✗ disproven proven
S* membership {"alpha": 0.75, "beta": 1.0} ✗ r ≤ 0.600 ✗ disproven proven
S* membership {"alpha": 0.75, "beta": 1.25} ✗ r ≤ 0.600 ✗ disproven proven
S* membership {"alpha": 0.75, "beta": 1.5} ✗ r ≤ 0.600 ✗ disproven proven
S* membership {"alpha": 0.25, "beta": 2.0} ✓ r ≤ 0.999 ✓ proven rigorous_verified
S* membership {"alpha": 0.5, "beta": 0.75} ✗ r ≤ 0.800 ✗ disproven proven
S* membership {"alpha": 0.5, "beta": 1.0} ✗ r ≤ 0.600 ✗ disproven proven
S* membership {"alpha": 0.5, "beta": 1.25} ✗ r ≤ 0.600 ✗ disproven proven
S* membership {"alpha": 0.25, "beta": 1.0} ✗ r ≤ 0.800 ✗ disproven proven
S* membership {"alpha": 0.25, "beta": 1.25} ✗ r ≤ 0.600 ✗ disproven proven
S* membership {"alpha": 0.25, "beta": 1.5} ✗ r ≤ 0.600 ✗ disproven proven
S* membership {"alpha": 0.25, "beta": 0.75} ✓ r ≤ 0.999 ✓ proven rigorous_verified
S* membership {"alpha": 0.1, "beta": 2.0} ✓ r ≤ 0.999 ✓ proven rigorous_verified
S* membership {"alpha": 0.25, "beta": 0.5} ✓ r ≤ 0.999 ✓ proven rigorous_verified
S* membership {"alpha": 0.1, "beta": 1.25} ✗ r ≤ 0.600 ✗ disproven proven
S* membership {"alpha": 0.1, "beta": 1.5} ✗ r ≤ 0.600 ✗ disproven proven
S* membership {"alpha": 0.1, "beta": 0.75} ✓ r ≤ 0.999 ✓ proven rigorous_verified
S* membership {"alpha": 0.1, "beta": 1.0} ✗ r ≤ 0.900 ✗ disproven proven
S* membership {"alpha": 0.1, "beta": 0.5} ✓ r ≤ 0.999 ✓ proven rigorous_verified
S* membership {"alpha": 5.0, "beta": 1.0} ✓ r ≤ 0.999 ✓ proven rigorous_verified
S* membership {"alpha": 0.5, "beta": 0.5} ✓ r ≤ 0.999 ✓ proven rigorous_verified
S* membership {"alpha": 0.5, "beta": 1.5} ✗ r ≤ 0.600 ✗ disproven proven
S* membership {"alpha": 1.0, "beta": 5.0} ✓ r ≤ 0.999 ✓ proven rigorous_verified
S* membership {"alpha": 1.0, "beta": 1.0} ✗ r ≤ 0.600 ✗ disproven proven
S membership {"alpha": 5.0, "beta": 1.0} ✗ - unscreened -
S membership {"alpha": 0.5, "beta": 0.5} ✗ - unscreened -
S membership {"alpha": 0.5, "beta": 1.5} ✗ - unscreened -
S membership {"alpha": 1.0, "beta": 5.0} ✗ - unscreened -
S membership {"alpha": 1.0, "beta": 1.0} ✗ - 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 disproves pass 0.600 0.8*exp(i*3.092505) boundary_v3.1
boundary_scan_centered_v31 starlike disproves pass 0.600 0.8*exp(i*3.166136) 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 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 0.600 0.8*exp(i*3.19068) boundary_v3.1
boundary_scan_centered_v31 starlike disproves pass 0.800 0.9*exp(i*3.166136) boundary_v3.1
boundary_scan_centered_v31 starlike disproves pass 0.600 0.8*exp(i*3.141784) boundary_v3.1
boundary_scan_centered_v31 starlike disproves pass 0.950 0.99*exp(i*3.144661) boundary_v3.1
boundary_scan_centered_v31 starlike proves pass 0.999 - boundary_v3.1
boundary_scan_centered_v31 starlike disproves pass 0.300 0.6*exp(i*3.166136) boundary_v3.1
boundary_scan_centered_v31 starlike disproves pass 0.300 0.6*exp(i*3.141784) boundary_v3.1
boundary_scan_centered_v31 starlike disproves pass 0.600 0.8*exp(i*3.092505) boundary_v3.1
boundary_scan_centered_v31 starlike disproves pass 0.600 0.8*exp(i*3.144661) boundary_v3.1
boundary_scan_centered_v31 starlike proves pass 0.999 - boundary_v3.1
boundary_scan_centered_v31 starlike disproves pass 0.300 0.6*exp(i*3.166136) boundary_v3.1
boundary_scan_centered_v31 starlike disproves pass 0.600 0.8*exp(i*2.99433) boundary_v3.1
boundary_scan_centered_v31 starlike disproves pass 0.600 0.8*exp(i*3.19068) boundary_v3.1
boundary_scan_centered_v31 starlike disproves pass 0.600 0.8*exp(i*3.19068) boundary_v3.1
boundary_scan_centered_v31 starlike proves pass 0.999 - boundary_v3.1
boundary_scan_centered_v31 starlike disproves pass 0.800 0.9*exp(i*3.411573) boundary_v3.1
boundary_scan_centered_v31 starlike disproves pass 0.600 0.8*exp(i*3.092505) boundary_v3.1
boundary_scan_centered_v31 starlike disproves pass 0.600 0.8*exp(i*3.092505) boundary_v3.1
boundary_scan_centered_v31 starlike proves pass 0.999 - boundary_v3.1
boundary_scan_centered_v31 starlike disproves pass 0.800 0.9*exp(i*3.067962) boundary_v3.1
boundary_scan_centered_v31 starlike disproves pass 0.600 0.8*exp(i*3.067962) boundary_v3.1
boundary_scan_centered_v31 starlike disproves pass 0.600 0.8*exp(i*3.215224) boundary_v3.1
boundary_scan_centered_v31 starlike proves pass 0.999 - boundary_v3.1
boundary_scan_centered_v31 starlike disproves pass 0.800 0.9*exp(i*3.018874) 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 0.600 0.8*exp(i*3.178408) boundary_v3.1
boundary_scan_centered_v31 starlike disproves pass 0.600 0.8*exp(i*3.067962) boundary_v3.1
boundary_scan_centered_v31 starlike disproves pass 0.900 0.95*exp(i*3.172272) boundary_v3.1
boundary_scan_centered_v31 starlike proves pass 0.999 - boundary_v3.1
boundary_scan_centered_v31 starlike disproves pass 0.600 0.8*exp(i*3.166136) 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 0.600 0.8*exp(i*3.19068) 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 [-0.99, 1.2124003311558797e-16] gft-0.1.0-reaudit-20260610

Parameter Sweep (117 verified instances, starlikeness × disk radius vs parameter)

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

Instances (45)

{"alpha": 5.0, "beta": 1.0} {"alpha": 0.5, "beta": 0.5} {"alpha": 1.0, "beta": 1.0} {"alpha": 0.5, "beta": 1.5} {"alpha": 1.0, "beta": 5.0} {"alpha": 0.1, "beta": 0.5} {"alpha": 0.1, "beta": 0.75} {"alpha": 0.1, "beta": 1.0} {"alpha": 0.1, "beta": 1.25} {"alpha": 0.1, "beta": 1.5} {"alpha": 0.1, "beta": 2.0} {"alpha": 0.25, "beta": 0.5} {"alpha": 0.25, "beta": 0.75} {"alpha": 0.25, "beta": 1.0} {"alpha": 0.25, "beta": 1.25} {"alpha": 0.25, "beta": 1.5} {"alpha": 0.25, "beta": 2.0} {"alpha": 0.5, "beta": 0.75} {"alpha": 0.5, "beta": 1.0} {"alpha": 0.5, "beta": 1.25} {"alpha": 0.5, "beta": 2.0} {"alpha": 0.75, "beta": 0.5} {"alpha": 0.75, "beta": 0.75} {"alpha": 0.75, "beta": 1.0} {"alpha": 0.75, "beta": 1.25} {"alpha": 0.75, "beta": 1.5} {"alpha": 0.75, "beta": 2.0} {"alpha": 1.25, "beta": 0.5} {"alpha": 1.25, "beta": 0.75} {"alpha": 1.25, "beta": 1.0} {"alpha": 1.25, "beta": 1.25} {"alpha": 1.25, "beta": 1.5} {"alpha": 1.25, "beta": 2.0} {"alpha": 1.5, "beta": 0.5} {"alpha": 1.5, "beta": 0.75} {"alpha": 1.5, "beta": 1.0} {"alpha": 1.5, "beta": 1.25} {"alpha": 1.5, "beta": 1.5} {"alpha": 1.5, "beta": 2.0} {"alpha": 2.0, "beta": 0.5} {"alpha": 2.0, "beta": 0.75} {"alpha": 2.0, "beta": 1.0} {"alpha": 2.0, "beta": 1.25} {"alpha": 2.0, "beta": 1.5} {"alpha": 2.0, "beta": 2.0}

Cited in Papers (3)

Exponential radii of starlikeness and convexity of some special functions
Unknown authors · 2024
Radii of starlikeness and convexity of generalized $k-$Bessel functions
Evrim Toklu · 2019
Radii of starlikeness and convexity of generalized Mittag-Leffler functions
Árpád Baricz, Anuja Prajapati · 2019
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