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Janowski S*[A,B] extremal

Janowski · 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
janowski_starlike_extremal
Parameter Space
A ∈ [-1,1] (real) ; B ∈ [-1,1] (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.

A=1.0, B=-1.0
A=1.0, B=-1.0 A=1.0, B=0.5
Taylor series
r* ≈ 0.600
Radius of starlikeness r* ≈ 0.6000
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

→
28
Counterexamples
 →
38
Facts
 →
48
Verification runs
 →
20
Instances
 →
6
Cited in papers

Facts (38)

Property Kind Params Holds Value / r Status Confidence
K membership {"A": 0.5, "B": 0.0} ✓ 0.5373 unscreened -
K membership {"A": 0.5, "B": -0.5} ✗ -18.9751 unscreened -
K membership {"A": 0.0, "B": -1.0} ✗ -17.4745 unscreened -
K membership {"A": 1.0, "B": 0.0} ✗ -0.7037 unscreened -
K membership {"A": 1.0, "B": -0.5} ✗ -8.0233 unscreened -
K membership {"A": 1.0, "B": -1.0} ✗ -9.9165 unscreened -
r* radius {"A": 0.5, "B": 0.0} - r ≤ 0.920 unscreened -
r* radius {"A": 0.5, "B": -0.5} - r ≤ 0.888 unscreened -
r* radius {"A": 0.0, "B": -1.0} - r ≤ 0.664 unscreened -
r* radius {"A": 1.0, "B": 0.0} - r ≤ 0.920 unscreened -
r* radius {"A": 1.0, "B": -0.5} - r ≤ 0.760 unscreened -
r* radius {"A": 1.0, "B": -1.0} - r ≤ 0.920 unscreened -
S* membership {"A": 0.75, "B": -0.75} ✗ r ≤ 0.600 ✗ disproven proven
S* membership {"A": 0.75, "B": -0.25} ✓ r ≤ 0.999 ✓ proven rigorous_verified
S* membership {"A": 0.75, "B": 0.0} ✓ r ≤ 0.999 ✓ proven rigorous_verified
S* membership {"A": 0.75, "B": 0.5} ✓ r ≤ 0.999 ✓ proven rigorous_verified
S* membership {"A": 1.0, "B": -0.75} ✗ r ≤ 0.300 ✗ disproven proven
S* membership {"A": 1.0, "B": -0.25} ✓ r ≤ 0.999 ✓ proven rigorous_verified
S* membership {"A": 1.0, "B": 0.5} ✓ r ≤ 0.999 ✓ proven rigorous_verified
S* membership {"A": -0.5, "B": -1.0} ✗ r ≤ 0.900 ✗ disproven proven
S* membership {"A": -0.5, "B": -0.75} ✓ r ≤ 0.999 ✓ proven rigorous_verified
S* membership {"A": 0.25, "B": -1.0} ✗ r ≤ 0.600 ✗ disproven proven
S* membership {"A": 0.25, "B": -0.75} ✗ r ≤ 0.600 ✗ disproven proven
S* membership {"A": 0.25, "B": -0.25} ✓ r ≤ 0.999 ✓ proven rigorous_verified
S* membership {"A": 0.25, "B": 0.0} ✓ r ≤ 0.999 ✓ proven rigorous_verified
S* membership {"A": 0.75, "B": -1.0} ✗ r ≤ 0.000 ✗ disproven proven
S* membership {"A": 0.5, "B": 0.0} ✓ 0.8179 🔬 screened numerical_weak
S* membership {"A": 0.5, "B": -0.5} ✗ r ≤ 0.800 ✗ disproven proven
S* membership {"A": 0.0, "B": -1.0} ✗ r ≤ 0.600 ✗ disproven proven
S* membership {"A": 1.0, "B": 0.0} ✓ 0.4776 🔬 screened numerical_weak
S* membership {"A": 1.0, "B": -0.5} ✗ r ≤ 0.600 ✗ disproven proven
S* membership {"A": 1.0, "B": -1.0} ✗ r ≤ 0.000 ✗ disproven proven
S membership {"A": 0.5, "B": 0.0} ✗ - unscreened -
S membership {"A": 0.5, "B": -0.5} ✗ - unscreened -
S membership {"A": 0.0, "B": -1.0} ✗ - unscreened -
S membership {"A": 1.0, "B": 0.0} ✗ - unscreened -
S membership {"A": 1.0, "B": -0.5} ✗ - unscreened -
S membership {"A": 1.0, "B": -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 (48)

Verifier Property Direction Outcome Domain r Witness Engine
boundary_scan_centered_v31 starlike disproves pass 0.600 0.8*exp(i*2.012583) 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*4.232253) 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*1.969631) boundary_v3.1
boundary_scan_centered_v31 starlike disproves pass 0.600 0.8*exp(i*4.466952) boundary_v3.1
boundary_scan_centered_v31 starlike disproves pass 0.600 0.8*exp(i*2.086214) boundary_v3.1
boundary_scan_centered_v31 starlike disproves pass 0.900 0.95*exp(i*4.491496) 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*1.914408) boundary_v3.1
boundary_scan_centered_v31 starlike disproves pass 0.600 0.8*exp(i*1.889864) 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 disproves pass 0.000 - boundary_v3.1
pointwise_starlike_iv starlike disproves pass 0.990 [-0.20583257390958173, 0.9683661247264677] gft-0.1.0-reaudit-20260610
pointwise_starlike_iv starlike disproves pass 0.990 [-0.33859994189241205, 0.9302956945780493] gft-0.1.0-reaudit-20260610
pointwise_starlike_iv starlike disproves pass 0.990 [-0.33859994189241205, 0.9302956945780493] gft-0.1.0-reaudit-20260610
pointwise_starlike_iv starlike disproves pass 0.903 [-0.36732762211223724, -0.8250313473348766] gft-0.1.0-reaudit-20260610
pointwise_starlike_iv starlike disproves pass 0.990 [-0.40266927664504226, 0.9044100030661748] gft-0.1.0-reaudit-20260610
pointwise_starlike_iv starlike disproves pass 0.990 [-0.46477684715803186, -0.8741181169303377] gft-0.1.0-reaudit-20260610
pointwise_starlike_iv starlike disproves pass 0.990 [-0.3708605274817532, -0.9179120160211194] gft-0.1.0-reaudit-20260610
pointwise_starlike_iv starlike disproves pass 0.990 [-0.2728809822588289, -0.9516490789789358] 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 0.856 [-0.3751204527523677, 0.7691109053331442] gft-0.1.0-reaudit-20260610
boundary_scan_polynomial starlike disproves pass 0.600 0.8*exp(i*2.012583) 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*2.054) 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 0.900 0.95*exp(i*4.491496) boundary_v3.0
boundary_scan_polynomial starlike proves pass 0.999 - boundary_v3.0
boundary_scan_polynomial starlike disproves pass 0.600 0.8*exp(i*4.417865) boundary_v3.0
boundary_scan_polynomial starlike disproves pass 0.600 0.8*exp(i*4.393321) 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*2.012583) boundary_v3.0
boundary_scan_none starlike proves not_applicable - - boundary_v3.0
boundary_scan_polynomial starlike disproves pass 0.800 0.9*exp(i*1.969631) boundary_v3.0
boundary_scan_polynomial starlike disproves pass 0.600 0.8*exp(i*4.417865) boundary_v3.0
boundary_scan_none starlike proves not_applicable - - boundary_v3.0
boundary_scan_polynomial starlike disproves pass 0.600 0.8*exp(i*2.098486) boundary_v3.0
boundary_scan_polynomial starlike disproves pass 0.300 0.6*exp(i*2.012583) boundary_v3.0

Parameter Sweep (48 verified instances, starlikeness × disk radius vs parameter)

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

Instances (20)

{"A": 1.0, "B": -1.0} {"A": 1.0, "B": 0.0} {"A": 0.5, "B": -0.5} {"A": 1.0, "B": -0.5} {"A": 0.5, "B": 0.0} {"A": 0.0, "B": -1.0} {"A": -0.5, "B": -1.0} {"A": -0.5, "B": -0.75} {"A": 0.25, "B": -1.0} {"A": 0.25, "B": -0.75} {"A": 0.25, "B": -0.25} {"A": 0.25, "B": 0.0} {"A": 0.75, "B": -1.0} {"A": 0.75, "B": -0.75} {"A": 0.75, "B": -0.25} {"A": 0.75, "B": 0.0} {"A": 0.75, "B": 0.5} {"A": 1.0, "B": -0.75} {"A": 1.0, "B": -0.25} {"A": 1.0, "B": 0.5}

Cited in Papers (6)

Exponential radii of starlikeness and convexity of some special functions
Unknown authors · 2024
Starlikeness of analytic functions using special functions and subordination
Unknown authors · 2024
Starlikeness Using Special Functions and Subordination
Meghna Sharma, Naveen Kumar Jain, Sushil Kumar · 2023
On a subfamily of starlike functions related to Hyperbolic Cosine function
Mridula Mundalia, S. Sivaprasad Kumar · 2022
Differential Subordinations for Starlike Functions Associated With A Nephroid Domain
Lateef Ahmad Wani, A. Swaminathan · 2019
Certain results on $q$-starlike and $q$-convex error functions
S. Kanas, C. Ramachandran, L. Vanitha · 2016
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