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Mocanu α-convex

Mocanu · parametric, arity 1

Canonical Key
mocanu
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=0.25
alpha=0.25 alpha=1.0
Taylor series

Unit disk explorer - hover to evaluate

Hover over the disk

Related

0
Counterexamples
 →
30
Facts
 →
49
Verification runs
 →
21
Instances
 →
9
Cited in papers

Facts (30)

Property Kind Params Holds Value / r Status Confidence
K membership {"alpha": 0.25} ✗ -8.1287 unscreened -
K membership {"alpha": 0.5} ✗ -1.2411 unscreened -
K membership {"alpha": 0.75} ✓ 0.0364 unscreened -
r* radius {"alpha": 0.25} - r ≤ 0.920 unscreened -
r* radius {"alpha": 0.5} - r ≤ 0.920 unscreened -
r* radius {"alpha": 0.75} - r ≤ 0.920 unscreened -
S* membership {"alpha": 0.35} - - retracted -
S* membership {"alpha": 0.4} - - retracted -
S* membership {"alpha": 0.45} - - retracted -
S* membership {"alpha": 0.55} - - retracted -
S* membership {"alpha": 0.6} - - retracted -
S* membership {"alpha": 0.65} - - retracted -
S* membership {"alpha": 0.7} - - retracted -
S* membership {"alpha": 0.8} - - retracted -
S* membership {"alpha": 0.85} - - retracted -
S* membership {"alpha": 0.9} - - retracted -
S* membership {"alpha": 0.95} - - retracted -
S* membership {"alpha": 1.0} - - retracted -
S* membership {"alpha": 0.1} - - retracted -
S* membership {"alpha": 0.15} - - retracted -
S* membership {"alpha": 0.2} - - retracted -
S* membership {"alpha": 0.3} - - retracted -
S* membership {"alpha": 0.0} - - retracted -
S* membership {"alpha": 0.05} - - retracted -
S* membership {"alpha": 0.25} - 0.0214 retracted -
S* membership {"alpha": 0.5} - 0.3995 retracted -
S* membership {"alpha": 0.75} - 0.6216 retracted -
S membership {"alpha": 0.25} ✗ - unscreened -
S membership {"alpha": 0.5} ✗ - unscreened -
S membership {"alpha": 0.75} ✗ - unscreened -

Verification Runs (49)

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 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*2.368466) boundary_v3.1
boundary_scan_centered_v31 starlike disproves pass 0.800 0.9*exp(i*2.396078) boundary_v3.1
boundary_scan_centered_v31 starlike disproves pass 0.950 0.99*exp(i*3.884039) 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*2.399146) 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*2.374602) boundary_v3.1
boundary_scan_centered_v31 starlike disproves pass 0.800 0.9*exp(i*3.951535) boundary_v3.1
boundary_scan_centered_v31 starlike disproves pass 0.800 0.9*exp(i*3.951535) boundary_v3.1
pointwise_starlike_iv starlike disproves pass 0.990 [-0.7121464023352646, -0.6877117867544074] gft-0.1.0-reaudit-20260610
pointwise_starlike_iv starlike disproves pass 0.990 [-0.7121464023352646, -0.6877117867544074] gft-0.1.0-reaudit-20260610
pointwise_starlike_iv starlike disproves pass 0.990 [-0.7121464023352646, -0.6877117867544074] gft-0.1.0-reaudit-20260610
pointwise_starlike_iv starlike disproves pass 0.990 [-0.7121464023352646, -0.6877117867544074] gft-0.1.0-reaudit-20260610
pointwise_starlike_iv starlike disproves pass 0.990 [-0.7121464023352646, -0.6877117867544074] gft-0.1.0-reaudit-20260610
pointwise_starlike_iv starlike disproves pass 0.990 [-0.7357133772226201, 0.6624393002952698] gft-0.1.0-reaudit-20260610
pointwise_starlike_iv starlike disproves pass 0.990 [-0.7357133772226203, -0.6624393002952697] 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 0.999 - 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 0.800 0.9*exp(i*2.368466) boundary_v3.0
boundary_scan_polynomial starlike disproves pass 0.800 0.9*exp(i*2.386874) boundary_v3.0
boundary_scan_polynomial starlike disproves pass 0.800 0.9*exp(i*2.40068) boundary_v3.0
boundary_scan_polynomial starlike disproves pass 0.950 0.99*exp(i*2.40835) boundary_v3.0
boundary_scan_polynomial starlike disproves pass 0.600 0.8*exp(i*2.374602) boundary_v3.0
boundary_scan_polynomial starlike disproves pass 0.800 0.9*exp(i*2.380738) boundary_v3.0
boundary_scan_polynomial starlike disproves pass 0.900 0.95*exp(i*2.402214) 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 (49 recorded instances, starlikeness evidence × disk radius vs parameter)

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

Instances (21)

{"alpha": 0.25} {"alpha": 0.5} {"alpha": 0.75} {"alpha": 0.0} {"alpha": 0.05} {"alpha": 0.1} {"alpha": 0.15} {"alpha": 0.2} {"alpha": 0.3} {"alpha": 0.35} {"alpha": 0.4} {"alpha": 0.45} {"alpha": 0.55} {"alpha": 0.6} {"alpha": 0.65} {"alpha": 0.7} {"alpha": 0.8} {"alpha": 0.85} {"alpha": 0.9} {"alpha": 0.95} {"alpha": 1.0}

Cited in Papers (9)

Criteria for Starlikeness Using Schwarzian Derivatives
Asha Sebastian, V. Ravichandran · 2024 · J. Analysis 32(4), 1973-1990
On Starlike Functions Associated with a Bean Shaped Domain
S. Sivaprasad Kumar, Pooja Yadav · 2024
On the difference of the moduli of the two initial logarithmic coefficients
Milutin Obradović, Nikola Tuneski · 2024
Starlikeness of analytic functions using special functions and subordination
Meghna Sharma; Naveen Kumar Jain; Sushil Kumar · 2024 · Boletín de la Sociedad Matemática Mexicana
Starlikeness Using Special Functions and Subordination
Meghna Sharma, Naveen Kumar Jain, Sushil Kumar · 2023
New subclass of the class of close-to-convex harmonic mappings defined by a third-order differential inequality
Serkan Çakmak, Elif Yaşar, Sibel Yalçın · 2021
Sufficiency for Nephroid Starlikeness using Hypergeometric Functions
A. Swaminathan, Lateef Ahmad Wani · 2021
Differential Subordinations for Starlike Functions Associated With A Nephroid Domain
Lateef Ahmad Wani, A. Swaminathan · 2019
A generalization of starlike functions of order alpha
Sarita Agrawal, Swadesh K. Sahoo · 2014
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