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Canonical Key
ma_minda_nephroid

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.

Taylor series
r* ≈ 0.950
Radius of starlikeness r* ≈ 0.9500
disproven f is starlike on |z| < r*; the colored disk shows the certified region.

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Related

Facts (4)

Property Kind Params Holds Value / r Status Confidence
K membership {} -0.9510 unscreened -
r* radius {} - r ≤ 0.700 🔬 screened numerical_strong
S* membership {} r ≤ 0.950 ✗ disproven proven
S membership {} - unscreened -

Verification Runs (3)

Verifier Property Direction Outcome Domain r Witness Engine
boundary_scan_centered_v31 starlike disproves pass 0.950 0.99*exp(i*4.092661) boundary_v3.1
pointwise_starlike_iv starlike disproves pass 0.990 [-0.5819073997695485, -0.8009268244311979] gft-0.1.0-reaudit-20260610
boundary_scan_polynomial starlike disproves pass 0.950 0.99*exp(i*4.089593) boundary_v3.0

Instances (1)

{}

Papers in this φ-class (14)

Papers that explicitly study S*(φ) for this φ - conservatively tagged from the corpus (explicit φ formula, class symbol, or unambiguous name). Quote = the supporting passage.

Sushil Kumar KUMAR, Asena Çetınkaya · 2022
“is in the class S∗ Ne if and only if 1 z [ f(z) ∗z −Lz2”
Neha Verma, S. Sivaprasad Kumar · 2022
“If f ∈ S∗ Ne. Then |H3(1)| ≤ 32/81”
Priyanka Goel, S. Sivaprasad Kumar · 2022
“RS∗ SG(S∗ Ne) = rNe ≈ 0.43473”
Somya Malik, V. Ravichandran · 2022
“S∗ Ne = S∗(ϕNe) with ϕNe(z) = 1 + z −z3/3”
Muhammad Ghaffar Khan, Bakhtiar Ahmad, Wali Khan Mashwani, Timilehin Gideon Shaba, Muhammad Arif · 2021
“Theorem 1. Let f ∈S∗ Ne of the form (1.1). Then |a2| ≤ 1, |a3| ≤ 1 2, |a4| ≤ 7 18”
Shalu Yadav, Kanika Sharma, V. Ravichandran · 2021
“the class S∗ Ne. (i) RS∗ Ne(T1) = 1/4 = 0.25”
Rosihan M. Ali, Kanika Sharma, V. Ravichandran · 2021
“The following are the S∗ Ne-radius for the classes T1 and T2”
A. Swaminathan, Lateef Ahmad Wani · 2021
“Class S*_Ne: f in A with zf'(z)/f(z) subordinate to 1 + z - z^3/3”
Somya Malik, V. Ravichandran · 2020
“(3) The nephroid radius, RS∗”
Adam Lecko, V. Ravichandran, Asha Sebastian · 2020
“S*_Ne = S*(1 + z − z^3/3) (nephroid)”
Lateef Ahmad Wani, A. Swaminathan · 2019
“ϕNe(z) := 1 + z −z3/3”
Lateef Ahmad Wani, A. Swaminathan · 2019
“ΩNe := ϕNe(D), the region bounded by the nephroid (1.4)”
Lateef Ahmad Wani, A. Swaminathan · 2019
“The function ϕNe(z) = 1 + z −z3/3 maps D onto the region bounded by the nephroid”
Baskar Babujee Janani, V. Ravichandran, Nisha Bohra · 2025
“The sharp CVNe radius for functions whose derivative belongs to K(α, β)”
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