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Exponential starlike — φ(z) = e^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_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.

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
 →
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 {} ✓ 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 (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.

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”
Stud. Univ. Babe¸s-Bolyai Math. 70(2025), No. 1, 15–32 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”
EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS 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 · 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 | Boletín de la Sociedad Matemática Mexicana | Springer Nature Link high
Unknown authors · 2024
“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)”
JAMSI, 18 (2022), No. 2 high
Gagandeep Singh, Gagandeep Singh · 2023
“If f ∈Rα(ez), then |a2| ≤ 1 1+α , (2) |a3| ≤ 1 1+2α , (3) |a”
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”
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
“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)”
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”
Mathematics & Statistics high
Nisha Bohra, Sushil Kumar, V. Ravichandran · 2019
“Let p(z) + βzp′(z) ≺ez, β > 0. Then the following are true: (a) 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”
Third Hankel Determinant for A Class of Functions with Respect to high
K. Ganesh, R. B. Sharma, K. Rajya Laxmi · 2019
“Theorem 6 If f ∈S∗ s(ez) then |a2| ≤1 2, |a3| ≤1 2, |a4| ≤19 48, |a5| ≤13 24.”
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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