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Sine starlike — φ(z) = 1+sin(z)

Ma-Minda

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_sine

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

0
Counterexamples
 →
4
Facts
 →
2
Verification runs
 →
1
Instances
0
Cited in papers
 →
18
In this φ-class

Facts (4)

Property Kind Params Holds Value / r Status Confidence
K membership {} ✗ -0.8694 unscreened -
r* radius {} - r ≤ 0.700 🔬 screened numerical_strong
S* membership {} ✓ r ≤ 0.999 ✓ proven rigorous_verified
S membership {} ✗ - unscreened -

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 (18)

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.

Higher order differential subordinations for certain starlike functions high
Neha Verma, S. Sivaprasad Kumar · 2024
“q(z) := 1 + sin z and define the admissibility class”
EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS high
Daud Mohamad, Nur Hazwani Aqilah Abdul Wahid, Nurul Natasya Hasni · 2023
“belongs to S∗ SC (sin z) , then |a2| ⩽1 2, |a3| ⩽1 2, |a4| ⩽”
Hermitian-Toeplitz determinants for certain univalent functions high
Surya Giri, S. Sivaprasad Kumar · 2022
“If f ∈ S∗ sin, then 0 ≤ det T2,1(f) ≤ 1”
Radius Constants of Sigmoid Starlike Functions high
Priyanka Goel, S. Sivaprasad Kumar · 2022
“Let f ∈ S∗ S, then zf'(z)/f(z) ≺ 1 + sin z”
Sharp Estimate of Fifth Coefficient for Ma Minda Starlike and Convex Functions high
Surya Giri, S. S. Kumar · 2022
“Class S*_sin: zf'/f subordinate to 1+sin(z)”
J. Math. Computer Sci., 25 (2022), 29–36 high
Muhammad Ghaffar Khan, Bakhtiar Ahmad, G. Murugusundaramoorthy, Wali Khan Mashwani, Sibel Yalçın, Timilehin Gideon Shaba et al. · 2021
“estimates for the class S∗ s (sin) . Further we evaluate Fekete-”
Radius of Starlikeness of Certain Analytic Functions high
Asha Sebastian, V. Ravichandran · 2021 · Mathematica Slovaca 71(1), 83-104
“S*_sin = S*(1 + sin z) (sine)”
Radius of starlikeness for some classes containing non-univalent functions high
Shalu Yadav, Kanika Sharma, V. Ravichandran · 2021
“the class S∗ sin. (i) RS∗ sin(T1)”
Starlikeness of Analytic Functions with Subordinate Ratios high
Rosihan M. Ali, Kanika Sharma, V. Ravichandran · 2021
“The following are the S∗ sin-radius for each class T1 and T2”
Miskolc Mathematical Notes high
Muhammad Arif, Mohsan Raza, Miraj Ul Haq, Gautam Srivastava · 2020
“BERNARDI’S INTEGRAL OPERATOR ASSOCIATED WITH SINE FUNCTION”
Radius of Starlikeness for Bloch Functions high
Somya Malik, V. Ravichandran · 2020
“associated with the sine function, RS∗ sin = √ 3 sin 1 2+2 sin 1”
Radius of Starlikeness for Classes of Analytic Functions high
Kanika Khatter, See Keong Lee, V. Ravichandran · 2020
“Ωsin is the image of the unit disk D under the function 1+sin z”
Starlikeness for Certain Close-to-Star Functions high
R. Kanaga, V. Ravichandran · 2020
“(iv) The radius RS∗ sin (≈0.2142) is the same as RS∗(1−sin 1)”
Starlikeness of Certain Non-Univalent Functions high
Adam Lecko, V. Ravichandran, Asha Sebastian · 2020
“S*_sin = S*(1 + sin z)”
Starlikeness of certain analytic functions high
Ahmad Sulaiman Ahmad El-Faqeer, Maisarah Haji Mohd, V. Ravichandran, Shamani Supramaniam · 2020
“ϕs(z) = 1 + sin z. This proves that the S∗ sin radius is at least RS∗ sin”
Mathematics & Statistics high
Nisha Bohra, Sushil Kumar, V. Ravichandran · 2019
“then p(z) ≺φs(z) = 1 + sin z. (c) If β ≥2(1 + √ 2)”
Mathematics & Statistics medium
Baskar Babujee Janani, V. Ravichandran, Nisha Bohra · 2025
“The sharp CVsin radius for functions f such that f′ ∈ K(α, β)”
Stud. Univ. Babe¸s-Bolyai Math. 69(2024), No. 4, 789–800 medium
Gagandeep Singh, Gurcharanjit Singh · 2024
“If f ∈Rα sin, then |a2| ≤ 1 1 + α”
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