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Two-parameter Mittag-Leffler E_{a,b}

Mittag-Leffler · parametric, arity 0 · decay: gamma_ratio

This function is certified starlike - the sharp, proven-exact coefficient bounds for the starlike class apply to it. See the certified proofs →
Canonical Key
mittag_leffler_2p
Generating Definition
f(z) = z + sum_{n>=2} [Gamma(b)/Gamma(a*(n-1)+b)] z^n
Parameter Space
{"alpha": "(0,inf)", "beta": "(0,inf)"}

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=0.25, beta=0.5
alpha=0.25, beta=0.5 alpha=2.0, beta=3.0
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

→
14
Counterexamples
 →
36
Facts
 →
50
Verification runs
 →
36
Instances
0
Cited in papers

Facts (36)

Property Kind Params Holds Value / r Status Confidence
S* membership {"alpha": 1.0, "beta": 1.0} ✓ r ≤ 0.999 ✓ proven rigorous_verified
S* membership {"alpha": 1.0, "beta": 1.5} ✓ r ≤ 0.999 ✓ proven rigorous_verified
S* membership {"alpha": 1.0, "beta": 2.0} ✓ r ≤ 0.999 ✓ proven rigorous_verified
S* membership {"alpha": 1.0, "beta": 2.5} ✓ r ≤ 0.999 ✓ proven rigorous_verified
S* membership {"alpha": 1.0, "beta": 3.0} ✓ r ≤ 0.999 ✓ proven rigorous_verified
S* membership {"alpha": 1.5, "beta": 0.5} ✗ r ≤ 0.000 ✗ disproven proven
S* membership {"alpha": 1.5, "beta": 1.0} ✗ r ≤ 0.800 ✗ 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": 1.5, "beta": 2.5} ✓ r ≤ 0.999 ✓ proven rigorous_verified
S* membership {"alpha": 1.5, "beta": 3.0} ✓ r ≤ 0.999 ✓ proven rigorous_verified
S* membership {"alpha": 2.0, "beta": 0.5} ✗ r ≤ 0.000 ✗ disproven proven
S* membership {"alpha": 2.0, "beta": 1.0} ✓ 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": 2.0, "beta": 2.5} ✓ r ≤ 0.999 ✓ proven rigorous_verified
S* membership {"alpha": 2.0, "beta": 3.0} ✓ r ≤ 0.999 ✓ proven rigorous_verified
S* membership {"alpha": 0.75, "beta": 2.5} ✓ r ≤ 0.999 ✓ proven rigorous_verified
S* membership {"alpha": 0.75, "beta": 3.0} ✓ r ≤ 0.999 ✓ proven rigorous_verified
S* membership {"alpha": 1.0, "beta": 0.5} ✗ r ≤ 0.000 ✗ disproven proven
S* membership {"alpha": 0.75, "beta": 1.5} ✓ r ≤ 0.999 ✓ proven rigorous_verified
S* membership {"alpha": 0.75, "beta": 2.0} ✓ r ≤ 0.999 ✓ proven rigorous_verified
S* membership {"alpha": 0.75, "beta": 1.0} ✓ r ≤ 0.999 ✓ proven rigorous_verified
S* membership {"alpha": 0.5, "beta": 3.0} ✓ r ≤ 0.999 ✓ proven rigorous_verified
S* membership {"alpha": 0.75, "beta": 0.5} ✗ r ≤ 0.300 ✗ disproven proven
S* membership {"alpha": 0.5, "beta": 2.0} ✓ r ≤ 0.999 ✓ proven rigorous_verified
S* membership {"alpha": 0.5, "beta": 2.5} ✓ r ≤ 0.999 ✓ proven rigorous_verified
S* membership {"alpha": 0.5, "beta": 1.5} ✓ r ≤ 0.999 ✓ proven rigorous_verified
S* membership {"alpha": 0.5, "beta": 1.0} ✓ r ≤ 0.999 ✓ proven rigorous_verified
S* membership {"alpha": 0.5, "beta": 0.5} ✗ r ≤ 0.800 ✗ disproven proven
S* membership {"alpha": 0.25, "beta": 3.0} ✓ r ≤ 0.999 ✓ proven rigorous_verified
S* membership {"alpha": 0.25, "beta": 2.5} ✓ r ≤ 0.999 ✓ proven rigorous_verified
S* membership {"alpha": 0.25, "beta": 2.0} ✓ r ≤ 0.999 ✓ proven rigorous_verified
S* membership {"alpha": 0.25, "beta": 1.5} ✓ r ≤ 0.999 ✓ proven rigorous_verified
S* membership {"alpha": 0.25, "beta": 1.0} ✓ r ≤ 0.999 ✓ proven rigorous_verified
S* membership {"alpha": 0.25, "beta": 0.5} ✗ r ≤ 0.900 ✗ disproven proven

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*3.141976) 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 0.000 - 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 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 0.000 - boundary_v3.1
boundary_scan_centered_v31 starlike proves pass 0.999 - 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 proves pass 0.999 - boundary_v3.1
boundary_scan_centered_v31 starlike disproves pass 0.300 0.6*exp(i*2.99433) 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 0.800 0.9*exp(i*2.853204) 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 0.900 0.95*exp(i*1.23102) 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 0.848 [-0.8478151260504201, 1.0382740803863567e-16] gft-0.1.0-reaudit-20260610
pointwise_starlike_iv starlike disproves pass 0.690 [-0.6898319327731092, 8.448004684202203e-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
pointwise_starlike_iv starlike disproves pass 0.824 [-0.8241176470588235, 1.0092530385914363e-16] 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 disproves pass 0.300 0.6*exp(i*3.19068) boundary_v3.0
boundary_scan_polynomial starlike disproves pass 0.800 0.9*exp(i*3.150797) 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

Parameter Sweep (77 verified instances, starlikeness × disk radius vs parameter)

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

Instances (36)

{"alpha": 0.25, "beta": 0.5} {"alpha": 0.25, "beta": 1.0} {"alpha": 0.25, "beta": 1.5} {"alpha": 0.25, "beta": 2.0} {"alpha": 0.25, "beta": 2.5} {"alpha": 0.25, "beta": 3.0} {"alpha": 0.5, "beta": 0.5} {"alpha": 0.5, "beta": 1.0} {"alpha": 0.5, "beta": 1.5} {"alpha": 0.5, "beta": 2.0} {"alpha": 0.5, "beta": 2.5} {"alpha": 0.5, "beta": 3.0} {"alpha": 0.75, "beta": 0.5} {"alpha": 0.75, "beta": 1.0} {"alpha": 0.75, "beta": 1.5} {"alpha": 0.75, "beta": 2.0} {"alpha": 0.75, "beta": 2.5} {"alpha": 0.75, "beta": 3.0} {"alpha": 1.0, "beta": 0.5} {"alpha": 1.0, "beta": 1.0} {"alpha": 1.0, "beta": 1.5} {"alpha": 1.0, "beta": 2.0} {"alpha": 1.0, "beta": 2.5} {"alpha": 1.0, "beta": 3.0} {"alpha": 1.5, "beta": 0.5} {"alpha": 1.5, "beta": 1.0} {"alpha": 1.5, "beta": 1.5} {"alpha": 1.5, "beta": 2.0} {"alpha": 1.5, "beta": 2.5} {"alpha": 1.5, "beta": 3.0} {"alpha": 2.0, "beta": 0.5} {"alpha": 2.0, "beta": 1.0} {"alpha": 2.0, "beta": 1.5} {"alpha": 2.0, "beta": 2.0} {"alpha": 2.0, "beta": 2.5} {"alpha": 2.0, "beta": 3.0}
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