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This function is certified starlike - the sharp, proven-exact coefficient bounds for the starlike class apply to it. See the certified proofs →
Canonical Key
ma_minda_lemniscate

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* ≈ 0.999
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

Facts (4)

Property Kind Params Holds Value / r Status Confidence
K membership {} 0.0312 🔬 screened numerical_strong
r* radius {} - r ≤ 0.700 🔬 screened numerical_strong
S* membership {} r ≤ 0.999 ✓ proven rigorous_verified
S membership {} - 🔬 screened numerical_strong

Transitively Implied Properties

Because starlikeness is proven, these properties follow by classical implications (unconditional edges in the lattice):

S C

Verification Runs (2)

Verifier Property Direction Outcome Domain r Witness Engine
boundary_scan_centered_v31 starlike proves pass 0.999 - boundary_v3.1
boundary_scan_polynomial starlike proves pass 0.999 - boundary_v3.0

Instances (1)

{}

Papers in this φ-class (30)

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.

Vasudevarao Allu, Raju Biswas, Rajib Mandal · 2025
“Theorem 3.2. Let f ∈S∗ L. Then the pre-Schwarzian norm satisfies the following inequality”
Baskar Babujee Janani, V. Ravichandran, Nisha Bohra · 2025
“The radius of lemniscate convexity for the functions whose derivative belongs to K(α, β)”
Md Firoz Ali, Md Nurezzaman, Sanjit Pal · 2024
“the class S*(q1) =: S*( √ 1 + z).”
V. Ravichandran et al. · 2024
“the class S∗(q1) =: S∗( √ 1 + z)”
S. Sivaprasad Kumar, Mridula Mundalia · 2023
“f ∈S∗ L in |z|< tanh2”
Asha Sebastian, V. Ravichandran · 2023 · Studia Univ. Babeș-Bolyai Math. 68(1), 161-170
“S*_L = S*(√(1+z)) (lemniscate of Bernoulli)”
Priyanka Goel, S. Sivaprasad Kumar · 2022
“S∗ L := S∗(√1 + z) and S∗ e := S∗(ez)”
Surya Giri, S. S. Kumar · 2022
“Class S*_L: zf'/f subordinate to sqrt(1+z)”
Asha Sebastian, V. Ravichandran · 2021 · Mathematica Slovaca 71(1), 83-104
“S*_L = S*(√(1+z)) (lemniscate of Bernoulli)”
Khalil Ullah, Jihad Younis, Khurshid Ahmad, A. Manickam, Bilal Khan, Mirajul Haq · 2021
“Corollary 3. Let f 2 S ... These inequalities are best possible and for these see”
Somya Malik, V. Ravichandran · 2020
“The Lemniscate starlike radius, RS∗ L = 2 √ 3− √ 6 4”
Kanika Khatter, See Keong Lee, V. Ravichandran · 2020
“(2) The S∗ L-radius is RS∗ L = ( √ 5 −2)/( √ 2 + 1)”
R. Kanaga, V. Ravichandran · 2020
“(ii) The radius RS∗ L (≈0.1645) is the smallest positive root of the polynomial”
Adam Lecko, V. Ravichandran, Asha Sebastian · 2020
“S*_L = S*(√(1+z))”
Ahmad Sulaiman Ahmad El-Faqeer, Maisarah Haji Mohd, V. Ravichandran, Shamani Supramaniam · 2020
“(2) The S∗ L radius is at least RS∗ L”
Nazar Khan, Muhammad Shafiq, Maslina Darus, Bilal Khan and Qazi Zahoor Ahmad · 2020
“subclass of q-Starlike functions associated with the lemniscate of Bernoulli”
Adiba Naz, Sushil Kumar, V. Ravichandran · 2019
“A_2*A_4 - A_3^2 (inverse H_2(2) for S*_L) ≤ 19/280 for class S*_L”
Vibha Madaan, Ajay Kumar, V. Ravichandran · 2019
“p(z) ≺√1 + z”
İbrahim Aktaş · 2019
“q(z) = √1 + z”
Nisha Bohra, Sushil Kumar, V. Ravichandran · 2019
“If β ≥βL ≃2.35, then p(z) ≺√1 + z where βL is the unique root”
Shagun Banga, S. Sivaprasad Kumar · 2019
“Class SL*: f in A with zf'(z)/f(z) subordinate to sqrt(1+z)”
Vibha Madaan, Ajay Kumar, V. Ravichandran · 2018
“Starlikeness associated with lemniscate of Bernoulli”
Om P. Ahuja, Sushil Kumar, V. Ravichandran · 2018
“1+βzp′(z) ≺√1 + z. Then the following subordination results hold”
Lee See Keong, V. Ravichandran, Shamani Supramaniam · 2013
“Class S*_L: S*(sqrt(1+z))”
Rashidah Omar, Suzeini Abdul Halim · 2013
“then p(z) ≺ √ 1 + z”
S. Sivaprasad Kumar, Virendra Kumar, V. Ravichandran, Nak Eun Cho · 2013
“p(z) + βzp′(z) ≺ √1 + z”
Suzeini Abdul Halim, Rashidah Omar · 2012
“then p(z) ≺ √1 + z .”
Yong Sun, Zhi-Gang Wang, Antti Rasila, Janusz Sokol · 2020
“Theorem 8. Let π/2 ≤α < π. Then MS(α) ⊂SL (|z| ≤r0)”
Vibha Madaan, Ajay Kumar, V. Ravichandran · 2019
“lemniscate starlike radii”
R. Kargar, L. Trojnar-Spelina · 2018
“then f ∈SL∗.”
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