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₂F₁(a, b; c; z) — Gauss hypergeometric

Hypergeometric · parametric, arity 3

This function is certified starlike - the sharp, proven-exact coefficient bounds for the starlike class apply to it. See the certified proofs →
Canonical Key
hyp2f1
Parameter Space
a ∈ R (real) ; b ∈ R (real) ; c ∈ R\Z_neg (real)

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.

a=1.0, b=1.0, c=2.0
a=1.0, b=1.0, c=2.0 a=0.25, b=0.75, c=3.0
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

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9
Counterexamples
 →
22
Facts
 →
30
Verification runs
 →
13
Instances
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16
Cited in papers

Facts (22)

Property Kind Params Holds Value / r Status Confidence
K membership {"a": 5.0, "b": 5.0, "c": 15.0} ✓ 0.3870 🔬 screened numerical_strong
K membership {"a": 1.0, "b": 1.0, "c": 2.0} ✗ -13.2440 unscreened -
K membership {"a": 1.0, "b": 2.0, "c": 3.0} ✗ -155.3993 unscreened -
r* radius {"a": 5.0, "b": 5.0, "c": 15.0} - r ≤ 0.960 🔬 screened numerical_strong
r* radius {"a": 1.0, "b": 1.0, "c": 2.0} - r ≤ 0.960 🔬 screened numerical_strong
r* radius {"a": 1.0, "b": 2.0, "c": 3.0} - r ≤ 0.960 🔬 screened numerical_strong
S* membership {"a": 0.25, "b": 0.75, "c": 2.0} ✓ r ≤ 0.999 ✓ proven rigorous_verified
S* membership {"a": 0.25, "b": 0.75, "c": 3.0} ✓ r ≤ 0.999 ✓ proven rigorous_verified
S* membership {"a": 0.5, "b": 0.5, "c": 1.5} ✓ r ≤ 0.999 ✓ proven rigorous_verified
S* membership {"a": 0.5, "b": 0.5, "c": 2.5} ✓ r ≤ 0.999 ✓ proven rigorous_verified
S* membership {"a": 0.5, "b": 0.5, "c": 3.5} ✓ r ≤ 0.999 ✓ proven rigorous_verified
S* membership {"a": 1.0, "b": 1.0, "c": 4.5} ✓ r ≤ 0.999 ✓ proven rigorous_verified
S* membership {"a": 1.0, "b": 1.0, "c": 3.5} ✓ r ≤ 0.999 ✓ proven rigorous_verified
S* membership {"a": 1.0, "b": 1.0, "c": 3.0} ✓ r ≤ 0.999 ✓ proven rigorous_verified
S* membership {"a": 1.0, "b": 1.0, "c": 2.5} ✓ r ≤ 0.999 ✓ proven rigorous_verified
S* membership {"a": 1.0, "b": 1.0, "c": 1.5} ✗ r ≤ 0.900 ✗ disproven proven
S* membership {"a": 5.0, "b": 5.0, "c": 15.0} ✗ r ≤ 0.800 ✗ disproven proven
S* membership {"a": 1.0, "b": 1.0, "c": 2.0} ✗ r ≤ 0.950 ✗ disproven proven
S* membership {"a": 1.0, "b": 2.0, "c": 3.0} ✗ r ≤ 0.950 ✗ disproven proven
S membership {"a": 5.0, "b": 5.0, "c": 15.0} ✗ - unscreened -
S membership {"a": 1.0, "b": 1.0, "c": 2.0} ✗ - unscreened -
S membership {"a": 1.0, "b": 2.0, "c": 3.0} ✗ - unscreened -

Transitively Implied Properties

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

S C

Verification Runs (30)

Verifier Property Direction Outcome Domain r Witness Engine
boundary_scan_centered_v31 starlike proves pass 0.999 - boundary_v3.1
boundary_scan_centered_v31 starlike proves pass 0.999 - boundary_v3.1
boundary_scan_centered_v31 starlike proves pass 0.999 - boundary_v3.1
boundary_scan_centered_v31 starlike proves pass 0.999 - boundary_v3.1
boundary_scan_centered_v31 starlike proves pass 0.999 - boundary_v3.1
boundary_scan_centered_v31 starlike proves pass 0.999 - boundary_v3.1
boundary_scan_centered_v31 starlike proves pass 0.999 - boundary_v3.1
boundary_scan_centered_v31 starlike proves pass 0.999 - boundary_v3.1
boundary_scan_centered_v31 starlike proves pass 0.999 - boundary_v3.1
boundary_scan_centered_v31 starlike disproves pass 0.900 0.95*exp(i*6.004001) boundary_v3.1
boundary_scan_centered_v31 starlike disproves pass 0.950 0.99*exp(i*0.266913) boundary_v3.1
boundary_scan_centered_v31 starlike disproves pass 0.800 0.9*exp(i*2.988195) boundary_v3.1
boundary_scan_centered_v31 starlike disproves pass 0.950 0.99*exp(i*6.023943) boundary_v3.1
pointwise_starlike_iv starlike disproves pass 0.990 [0.9516490789789357, 0.2728809822588292] gft-0.1.0-reaudit-20260610
pointwise_starlike_iv starlike disproves pass 0.990 [-0.9803653880541546, 0.13778136995046467] gft-0.1.0-reaudit-20260610
pointwise_starlike_iv starlike disproves pass 0.990 [0.9605927690132365, -0.2395026766436712] gft-0.1.0-reaudit-20260610
pointwise_starlike_iv starlike disproves pass 0.990 [0.9516490789789357, 0.2728809822588292] gft-0.1.0-reaudit-20260610
boundary_scan_polynomial starlike proves pass 0.999 - boundary_v3.0
boundary_scan_polynomial starlike proves pass 0.999 - boundary_v3.0
boundary_scan_polynomial starlike proves pass 0.999 - boundary_v3.0
boundary_scan_polynomial starlike proves pass 0.999 - boundary_v3.0
boundary_scan_polynomial starlike proves pass 0.999 - boundary_v3.0
boundary_scan_polynomial starlike proves pass 0.999 - boundary_v3.0
boundary_scan_polynomial starlike proves pass 0.999 - boundary_v3.0
boundary_scan_polynomial starlike proves pass 0.999 - boundary_v3.0
boundary_scan_polynomial starlike proves pass 0.999 - boundary_v3.0
boundary_scan_polynomial starlike proves pass 0.800 - boundary_v3.0
boundary_scan_polynomial starlike proves pass 0.600 - boundary_v3.0
boundary_scan_polynomial starlike disproves pass 0.950 0.99*exp(i*0.261927) boundary_v3.0
boundary_scan_polynomial starlike proves pass 0.900 - boundary_v3.0

Parameter Sweep (30 verified instances, starlikeness × disk radius vs parameter)

Proven on full disk (r ≥ 0.999)   ·   Proven on partial disk   ·   Disproven (witness)

Instances (13)

{"a": 1.0, "b": 1.0, "c": 2.0} {"a": 5.0, "b": 5.0, "c": 15.0} {"a": 1.0, "b": 2.0, "c": 3.0} {"a": 1.0, "b": 1.0, "c": 1.5} {"a": 1.0, "b": 1.0, "c": 2.5} {"a": 1.0, "b": 1.0, "c": 3.0} {"a": 1.0, "b": 1.0, "c": 3.5} {"a": 1.0, "b": 1.0, "c": 4.5} {"a": 0.5, "b": 0.5, "c": 1.5} {"a": 0.5, "b": 0.5, "c": 2.5} {"a": 0.5, "b": 0.5, "c": 3.5} {"a": 0.25, "b": 0.75, "c": 2.0} {"a": 0.25, "b": 0.75, "c": 3.0}

Cited in Papers (16)

Hadamard product of convex functions and Jackson operator
K. Piejko, J. Sokół, K. Trcabka-Wiȩc\a{l}aw · 2026
Exponential radii of starlikeness and convexity of some special functions
Unknown authors · 2024
Starlikeness of analytic functions using special functions and subordination
Unknown authors · 2024
Starlikeness Using Special Functions and Subordination
Meghna Sharma, Naveen Kumar Jain, Sushil Kumar · 2023
Universal convexity and range problems of shifted hypergeometric functions
Toshiyuki Sugawa, Li-Mei Wang, Chengfa Wu · 2023
New subclass of the class of close-to-convex harmonic mappings defined by a third-order differential inequality
Serkan Çakmak, Elif Yaşar, Sibel Yalçın · 2021
Sufficiency for Nephroid Starlikeness using Hypergeometric Functions
A. Swaminathan, Lateef Ahmad Wani · 2021
Nehari's univalence criteria, pre-Schwarzian derivative and applications
Sarita Agrawal, Swadesh Kumar Sahoo · 2020
Differential Subordinations for Starlike Functions Associated With A Nephroid Domain
Lateef Ahmad Wani, A. Swaminathan · 2019
Inclusion properties for bi-univalent functions of complex order defined by combining of Faber polynomial expansions and Fibonacci numbers
Sahsene Altinkaya, Samaneh G. Hamidi, Jay M. Jahangiri, Sibel Yalcin · 2019
Radii of starlikeness and convexity of generalized $k-$Bessel functions
Evrim Toklu · 2019
Asymptotics for Hankel Determinants Associated to a Hermite Weight with a Varying Discontinuity
Christophe Charlier, Alfredo Deaño · 2017
Asymptotics of Hankel determinants with a one-cut regular potential and Fisher-Hartwig singularities
Christophe Charlier · 2017
Certain results on $q$-starlike and $q$-convex error functions
S. Kanas, C. Ramachandran, L. Vanitha · 2016
Logarithmic Coefficients and Generalized Multifractality of Whole-Plane SLE
Bertrand Duplantier, Xuan Hieu Ho, Thanh Binh Le, Michel Zinsmeister · 2015
A generalization of starlike functions of order alpha
Sarita Agrawal, Swadesh K. Sahoo · 2014
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