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Exponential starlike — φ(z) = e^z

Ma-Minda

Canonical Key
ma_minda_exp

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.

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Related

0
Counterexamples
 →
4
Facts
 →
2
Verification runs
 →
1
Instances
0
Cited in papers
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32
In this φ-class

Facts (4)

Property Kind Params Holds Value / r Status Confidence
K membership {} ✗ -1.2578 unscreened -
r* radius {} - r ≤ 0.850 🔬 screened numerical_strong
S* membership {} - 0.4144 retracted -
S membership {} ✗ - unscreened -

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 (32)

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.

Ma-Minda starlikeness of certain analytic functions high
Baskar Babujee Janani, V. Ravichandran · 2025
“The ST e radius is given by RST e = ( e−1 eα+1 if α ⩾1 e−1 e+α if α ⩽1”
On Coefficient problems for classes S_e and C_e high
Sujoy Majumder, Nabadwip Sarkar and Molla Basir Ahamed · 2025
“Class S*_e: f in S with z*f'(z)/f(z) subordinate to e^z”
Hankel and Toeplitz Determinants of Logarithmic Coefficients of Inverse Functions for the Subclass of Starlike Functions with Respect to Symmetric Conjugate Points high
Nur Hazwani Aqilah Abdul Wahid, Adawiyah Tumiran, Timilehin Gideon Shaba · 2024
“ϕ (z) = ez, is an analytic univalent function”
Inverse logarithmic coefficient bounds for starlike functions subordinated to the exponential functions high
Lei Shi; Muhammad Abbas; Mohsan Raza; Muhammad Arif; Poom Kumam · 2024
“Theorem 3.1 Let f ∈S∗ e”
Pre-Schwarzian norm estimate for certain Ma-Minda Class of functions high
Md Firoz Ali, Md Nurezzaman, Sanjit Pal · 2024
“the sharp estimate of the pre-Schwarzian norm for functions in S*(ez).”
Radius Problems for Certain Classes of Analytic Functions high
V. Ravichandran et al. · 2024
“For the particular value λ = 1, we get the sharp estimate of the pre-Schwarzian norm for functions in S∗(ez).”
Third Hankel determinant for starlike and convex functions associated with the exponential function high
Moshan Raza; Amina Riaz; Derek K. Thomas; Paweł Zaprawa · 2024 · Boletín de la Sociedad Matemática Mexicana
“Theorem 2 Let \(f\in {\mathcal {S}}^{*}_{e}\) and be given by (1).”
Differential Subordination of Certain Class of Starlike Functions high
Neha Verma, S. Sivaprasad Kumar · 2023
“f(z) ∈ S* e if Yf(z) ≺ (1+Cz)/(1+Dz)”
On a Class of Non-Univalent functions Associated high
S. Sivaprasad Kumar, Mridula Mundalia · 2023
“The S∗e −radius is r1 = tanh2(λπ), where λ = (1/2) p (e −1)/”
Radius of Starlikeness Through Subordination high
Asha Sebastian, V. Ravichandran · 2023 · Studia Univ. Babeș-Bolyai Math. 68(1), 161-170
“S*_e = S*(e^z) (exponential)”
Starlikeness Using Special Functions and Subordination high
Meghna Sharma, Naveen Kumar Jain, Sushil Kumar · 2023
“f ∈ A to be exponential starlike in D”
Coefficient inequalities for a subclass of analytic functions associated with exponential function high
Gagandeep Singh, Gagandeep Singh · 2022
“If f ∈Rα(ez), then |a2| ≤ 1 1+α , (2) |a3| ≤ 1 1+2α , (3) |a”
On coefficient problems for functions starlike with respect to symmetric points high
Paweł Zaprawa · 2022
“Theorem 1 If f 2 S SðezÞ is of the form (1.1), then ja4j  1 4 and ja5j  1 4 : The bound”
Radius Constants of Sigmoid Starlike Functions high
Priyanka Goel, S. Sivaprasad Kumar · 2022
“S∗ e := S∗(ez)”
Starlikeness of a product of starlike functions with non-vanishing polynomials high
Somya Malik, V. Ravichandran · 2022
“radius of starlikeness asscociated with the exponential function for the function F”
Third Hankel Determinant For A Class Of Starlike Functions Associated With Exponential Function high
Kunal Joshi, S. Sivaprasad Kumar · 2022
“Class S*_e: f in S such that zf'(z)/f(z) is subordinate to e^z”
Radius of Starlikeness of Certain Analytic Functions high
Asha Sebastian, V. Ravichandran · 2021 · Mathematica Slovaca 71(1), 83-104
“S*_e = S*(e^z) (exponential)”
Radius of starlikeness for some classes containing non-univalent functions high
Shalu Yadav, Kanika Sharma, V. Ravichandran · 2021
“the class S∗ e. (i) RS∗e (T1)”
Starlikeness of Analytic Functions with Subordinate Ratios high
Rosihan M. Ali, Kanika Sharma, V. Ravichandran · 2021
“S∗ exp := S∗(ez) the class associated with ϕ(z) := ez”
Initial successive coefficients for certain classes of univalent functions involving the exponential function high
Lei Shi; Zhi-Gang Wang; Ren-Li Su; Muhammad Arif · 2020 · Journal of Mathematical Inequalities
“Let 0 ≤p ≤2 and f(z) ∈S∗ e(p). Then |a3 −a2| ≤1/16(−5p2 + 8p + 8)”
Radius of Starlikeness for Bloch Functions high
Somya Malik, V. Ravichandran · 2020
“The S∗ e-radius of the class B of Bloch functions is RS∗e (B)”
Radius of Starlikeness for Classes of Analytic Functions high
Kanika Khatter, See Keong Lee, V. Ravichandran · 2020
“(4) The S∗ e-radius is the smallest positive real root”
Starlikeness for Certain Close-to-Star Functions high
R. Kanaga, V. Ravichandran · 2020
“(iii) The radius RS∗e (≈0.16628) is the same as RS∗(1/e)”
Starlikeness of Certain Non-Univalent Functions high
Adam Lecko, V. Ravichandran, Asha Sebastian · 2020
“S*_e = S*(e^z)”
Third Hankel Determinant for A Class of Functions with Respect to Symmetric Points Associated with Exponential Function high
K. Ganesh, R. B. Sharma, K. Rajya Laxmi · 2020
“Theorem 6 If f ∈S∗ s(ez) then |a2| ≤1 2, |a3| ≤1 2, |a4| ≤19 48, |a5| ≤13 24.”
Coefficients of the Inverse Functions and Radius Estimates of Certain Starlike Functions high
Adiba Naz, Sushil Kumar, V. Ravichandran · 2019
“Class S*_e: f in A: zf'(z)/f(z) subordinate to e^z”
Exponential starlikeness and convexity of confluent hypergeometric, Lommel and Struve functions high
Adiba Naz, Sumit Nagpal, V. Ravichandran · 2019
“p(z) ≺ez”
Lemniscate and Exponential Starlikeness of Regular Coulomb Wave Functions high
İbrahim Aktaş · 2019
“p(z) ≺ez”
New upper bounds of the third Hankel determinant for some classes of univalent functions high
Milutin Obradovic, Nikola Tuneski · 2019
“Class S*_e: f in A : z*f'(z)/f(z) subordinate to e^z”
Starlikeness Associated With The Exponential Function high
Adiba Naz, Sumit Nagpal, V. Ravichandran · 2019
“p(z) ≺ez”
Some Special Differential Subordinations high
Nisha Bohra, Sushil Kumar, V. Ravichandran · 2018
“Let p(z) + βzp′(z) ≺ez, β > 0. Then the following are true: (a) p(z) ≺ez.”
Bohr radius for certain classes of starlike and convex univalent functions medium
Vasudevarao Allu, Himadri Halder · 2020
“for α = 0 in Corollary 2.4, we obtain the sharp Bohr radius for the class S∗ e”
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