Where the registry stops being a theorem and becomes a conjecture with certified evidence. Every entry below is a concrete lead for a human proof: a recognised candidate constant, the extremal that attains it, and exactly what is missing to close it. The certification levels are explained in the FAQ; the engine itself in Methods.
The branch-and-bound rigorously proves max |L| ≤ V + ε and an explicit member attains V, so the sharp constant is trapped in a thin bracket and the candidate below is matched to the extremal. What is open: a local second-variation / KKT (or sum-of-squares) certificate that closes ε → 0, turning the candidate into a theorem.
| Functional | Class | Candidate constant | Certified bracket | To close |
|---|---|---|---|---|
| \(|a_2a_3-a_4|\)
Gen. Zalcman (2,3) |
Starlike S*
✓ Generalized Zalcman, Ravichandran–Verma 2016 (|aₙaₘ−a_{n+m−1}| ≤ (n−1)(m−1) = 2)
|
\(2\) | [2.00000, 2.00000] | prove the candidate is attained sharply (ε → 0) |
| \(|a_2a_4-a_3^{2}|\)
Hankel H₂(2) |
Bean S*_𝔅
✓ Kumar & Verma 2025 (Filomat)
|
\(\tfrac{1}{16}\) | [0.06250, 0.07250] | prove the candidate is attained sharply (ε → 0) |
| \(|a_2a_4-a_3^{2}|\)
Hankel H₂(2) |
Cosh-sqrt | \(\tfrac{1}{16}\) | [0.06250, 0.07250] | prove the candidate is attained sharply (ε → 0) |
| \(|a_2a_4-a_3^{2}|\)
Hankel H₂(2) |
Epicycloid S*_{3ℒ} | \(\tfrac{9}{64}\) | [0.14062, 0.16625] | prove the candidate is attained sharply (ε → 0) |
| \(|a_2a_4-a_3^{2}|\)
Hankel H₂(2) |
Four-leaf S*_{4L} | \(\tfrac{25}{144}\) | [0.17361, 0.20750] | prove the candidate is attained sharply (ε → 0) |
| \(|a_2a_4-a_3^{2}|\)
Hankel H₂(2) |
Lemniscate S*_L
✓ Lee, Ravichandran & Supramaniam 2013 (#455)
|
\(\tfrac{1}{16}\) | [0.06250, 0.07250] | prove the candidate is attained sharply (ε → 0) |
| \(|a_2a_4-a_3^{2}|\)
Hankel H₂(2) |
Rational S*_R | \(3/4 - sqrt(2)/2\) | [0.04289, 0.05289] | prove the candidate is attained sharply (ε → 0) |
| \(|a_2a_4-a_3^{2}|\)
Hankel H₂(2) |
Sigmoid S*_{SG}
✓ Riaz, Raza & Thomas 2022 (Forum Math.)
|
\(\tfrac{1}{16}\) | [0.06250, 0.07250] | prove the candidate is attained sharply (ε → 0) |
| \(|a_2a_4-a_3^{2}|\)
Hankel H₂(2) |
Three-leaf | \(\tfrac{4}{25}\) | [0.16000, 0.18667] | prove the candidate is attained sharply (ε → 0) |
| \(|a_2a_4-a_3^{2}|\)
Hankel H₂(2) |
limacon_0.3 | \(\tfrac{9}{100}\) | [0.09000, 0.11375] | prove the candidate is attained sharply (ε → 0) |
| \(|a_2a_4-a_3^{2}|\)
Hankel H₂(2) |
order_0.75 | \(\tfrac{1}{16}\) | [0.06250, 0.08250] | prove the candidate is attained sharply (ε → 0) |
| \(|a_2a_4-a_3^{2}|\)
Hankel H₂(2) |
strongly_0.25 | \(\tfrac{1}{16}\) | [0.06250, 0.07250] | prove the candidate is attained sharply (ε → 0) |
For these the maximum sits on a flat / degenerate face where interval arithmetic stays loose, so the branch-and-bound does not close at any useful slack (the same wall that the single-harmonic reduction sidesteps for Fekete–Szegő, but which has no analogue for the multi-angle Hankel / Zalcman functionals). There is no certificate yet - only a differential-evolution maximum and a recognised candidate, with the explicit extremal. What is open: a method that certifies the bound at all (sum-of-squares / moment relaxation, or a closed-form reduction). See the roadmap in Methods.
| Functional | Class | Conjectured constant | Numerical max | Extremal |
|---|---|---|---|---|
| \(|a_2a_4-a_3^{2}|\)
Hankel H₂(2) |
Bell S*_B | \(\tfrac{1}{4}\) | 0.250000 | \(\omega(z) = z^2\) |
| \(|a_2a_4-a_3^{2}|\)
Hankel H₂(2) |
Cardioid S*_C | \(\tfrac{4}{9}\) | 0.444444 | \(\omega(z) = z^2\) |
| \(|a_2a_4-a_3^{2}|\)
Hankel H₂(2) |
Cardioid S*_℘ | \(\tfrac{1}{4}\) | 0.250000 | \(\omega(z) = z^2\) |
| \(|a_2a_4-a_3^{2}|\)
Hankel H₂(2) |
Cissoid S*_{cs} | \(\tfrac{1}{4}\) | 0.250000 | \(\omega(z) = z^2\) |
| \(|a_2a_4-a_3^{2}|\)
Hankel H₂(2) |
Crescent | \(\tfrac{1}{4}\) | 0.250000 | \(\omega(z) = z^2\) |
| \(|a_2a_4-a_3^{2}|\)
Hankel H₂(2) |
Epicycloid S*_{6ℒ} | \(\tfrac{9}{49}\) | 0.183673 | \(\omega(z) = z^2\) |
| \(|a_2a_4-a_3^{2}|\)
Hankel H₂(2) |
Exponential S*_e | \(\tfrac{1}{4}\) | 0.250000 | \(\omega(z) = z^2\) |
| \(|a_2a_4-a_3^{2}|\)
Hankel H₂(2) |
Nephroid S*_{Ne} | \(\tfrac{1}{4}\) | 0.250000 | \(\omega(z) = z^2\) |
| \(|a_2a_4-a_3^{2}|\)
Hankel H₂(2) |
Non-convex S*_{nc} | \(\tfrac{1}{4}\) | 0.250000 | \(\omega(z) = z^2\) |
| \(|a_2a_4-a_3^{2}|\)
Hankel H₂(2) |
Parabolic S_p | no simple form | 0.164256 | \(\omega(z) = z^2\) |
| \(|a_2a_4-a_3^{2}|\)
Hankel H₂(2) |
Petal S*_ρ | \(\tfrac{1}{4}\) | 0.250000 | \(\omega(z) = z^2\) |
| \(|a_2a_4-a_3^{2}|\)
Hankel H₂(2) |
Sine S*_{\sin} | \(\tfrac{1}{4}\) | 0.250000 | \(\omega(z) = z^2\) |
| \(|a_2a_4-a_3^{2}|\)
Hankel H₂(2) |
Starlike S* | \(1\) | 1.000000 | \(\omega(z) = z\) |
| \(|a_2a_4-a_3^{2}|\)
Hankel H₂(2) |
Strip S*_τ | \(\tfrac{1}{4}\) | 0.250000 | \(\omega(z) = z^2\) |
| \(|a_2a_4-a_3^{2}|\)
Hankel H₂(2) |
Tanh S*_{\tanh} | \(\tfrac{1}{4}\) | 0.250000 | \(\omega(z) = z^2\) |
| \(|a_2a_4-a_3^{2}|\)
Hankel H₂(2) |
janowski_A0.5_B-0.5 | \(\tfrac{1}{4}\) | 0.250000 | \(\omega(z) = z^2\) |
| \(|a_2a_4-a_3^{2}|\)
Hankel H₂(2) |
janowski_A0.75_B-0.25 | \(\tfrac{1}{4}\) | 0.250000 | \(\omega(z) = z^2\) |
| \(|a_2a_4-a_3^{2}|\)
Hankel H₂(2) |
janowski_A1_B-0.5 | \(\tfrac{9}{16}\) | 0.562500 | \(\omega(z) = z^2\) |
| \(|a_2a_4-a_3^{2}|\)
Hankel H₂(2) |
janowski_A1_B0 | \(\tfrac{1}{4}\) | 0.250000 | \(\omega(z) = z^2\) |
| \(|a_2a_4-a_3^{2}|\)
Hankel H₂(2) |
limacon_0.5 | \(\tfrac{1}{4}\) | 0.250000 | \(\omega(z) = z^2\) |
| \(|a_2a_4-a_3^{2}|\)
Hankel H₂(2) |
limacon_0.707 | \(\tfrac{1}{2}\) | 0.500000 | \(\omega(z) = z^2\) |
| \(|a_2a_4-a_3^{2}|\)
Hankel H₂(2) |
order_0.25 | \(\tfrac{9}{16}\) | 0.562500 | \(\omega(z) = z^2\) |
| \(|a_2a_4-a_3^{2}|\)
Hankel H₂(2) |
order_0.5 | \(\tfrac{1}{4}\) | 0.250000 | \(\omega(z) = z^2\) |
| \(|a_2a_4-a_3^{2}|\)
Hankel H₂(2) |
strongly_0.5 | \(\tfrac{1}{4}\) | 0.250000 | \(\omega(z) = z^2\) |
| \(|a_2a_4-a_3^{2}|\)
Hankel H₂(2) |
strongly_0.75 | \(\tfrac{9}{16}\) | 0.562500 | \(\omega(z) = z^2\) |
| \(|H_{3}(1)|\)
Hankel H₃(1) |
Bean S*_𝔅 | \(\tfrac{1}{36}\) | 0.027778 | \(\omega(z) = z^3\) |
| \(|H_{3}(1)|\)
Hankel H₃(1) |
Bell S*_B | \(\tfrac{1}{9}\) | 0.111111 | \(\omega(z) = z^3\) |
| \(|H_{3}(1)|\)
Hankel H₃(1) |
Cardioid S*_C | no simple form | 0.197531 | \(\omega(z) = z^3\) |
| \(|H_{3}(1)|\)
Hankel H₃(1) |
Cardioid S*_℘ | \(\tfrac{1}{9}\) | 0.111111 | \(\omega(z) = z^3\) |
| \(|H_{3}(1)|\)
Hankel H₃(1) |
Cissoid S*_{cs} | \(\tfrac{1}{9}\) | 0.111111 | \(\omega(z) = z^3\) |
| \(|H_{3}(1)|\)
Hankel H₃(1) |
Cosh-sqrt | \(\tfrac{1}{36}\) | 0.027778 | \(\omega(z) = z^3\) |
| \(|H_{3}(1)|\)
Hankel H₃(1) |
Crescent | \(\tfrac{1}{9}\) | 0.111111 | \(\omega(z) = z^3\) |
| \(|H_{3}(1)|\)
Hankel H₃(1) |
Epicycloid S*_{3ℒ} | \(\tfrac{1}{16}\) | 0.062500 | \(\omega(z) = z^3\) |
| \(|H_{3}(1)|\)
Hankel H₃(1) |
Epicycloid S*_{6ℒ} | \(\tfrac{4}{49}\) | 0.081633 | \(\omega(z) = z^3\) |
| \(|H_{3}(1)|\)
Hankel H₃(1) |
Exponential S*_e | \(\tfrac{1}{9}\) | 0.111111 | \(\omega(z) = z^3\) |
| \(|H_{3}(1)|\)
Hankel H₃(1) |
Four-leaf S*_{4L} | no simple form | 0.077160 | \(\omega(z) = z^3\) |
| \(|H_{3}(1)|\)
Hankel H₃(1) |
Lemniscate S*_L
✓ Banga & Sivaprasad Kumar 2020 (Math. Slovaca; #294)
|
\(\tfrac{1}{36}\) | 0.027778 | \(\omega(z) = z^3\) |
| \(|H_{3}(1)|\)
Hankel H₃(1) |
Nephroid S*_{Ne} | \(\tfrac{1}{9}\) | 0.111111 | \(\omega(z) = z^3\) |
| \(|H_{3}(1)|\)
Hankel H₃(1) |
Non-convex S*_{nc} | \(\tfrac{1}{9}\) | 0.111111 | \(\omega(z) = z^3\) |
| \(|H_{3}(1)|\)
Hankel H₃(1) |
Parabolic S_p | no simple form | 0.073003 | \(\omega(z) = z^3\) |
| \(|H_{3}(1)|\)
Hankel H₃(1) |
Petal S*_ρ | \(\tfrac{1}{9}\) | 0.111111 | \(\omega(z) = z^3\) |
| \(|H_{3}(1)|\)
Hankel H₃(1) |
Rational S*_R | no simple form | 0.019064 | \(\omega(z) = z^3\) |
| \(|H_{3}(1)|\)
Hankel H₃(1) |
Sigmoid S*_{SG} | \(\tfrac{1}{36}\) | 0.027778 | \(\omega(z) = z^3\) |
| \(|H_{3}(1)|\)
Hankel H₃(1) |
Sine S*_{\sin} | \(\tfrac{1}{9}\) | 0.111111 | \(\omega(z) = z^3\) |
| \(|H_{3}(1)|\)
Hankel H₃(1) |
Starlike S*
✓ Kowalczyk–Lecko–Thomas 2022
|
\(\tfrac{4}{9}\) | 0.444444 | \(\omega(z) = z^3\) |
| \(|H_{3}(1)|\)
Hankel H₃(1) |
Strip S*_τ | \(\tfrac{1}{9}\) | 0.111111 | \(\omega(z) = z^3\) |
| \(|H_{3}(1)|\)
Hankel H₃(1) |
Tanh S*_{\tanh} | \(\tfrac{1}{9}\) | 0.111111 | \(\omega(z) = z^3\) |
| \(|H_{3}(1)|\)
Hankel H₃(1) |
Three-leaf | no simple form | 0.071111 | \(\omega(z) = z^3\) |
| \(|H_{3}(1)|\)
Hankel H₃(1) |
janowski_A0.5_B-0.5 | \(\tfrac{1}{9}\) | 0.111111 | \(\omega(z) = z^3\) |
| \(|H_{3}(1)|\)
Hankel H₃(1) |
janowski_A0.75_B-0.25 | \(\tfrac{1}{9}\) | 0.111111 | \(\omega(z) = z^3\) |
| \(|H_{3}(1)|\)
Hankel H₃(1) |
janowski_A1_B-0.5 | \(\tfrac{1}{4}\) | 0.250000 | \(\omega(z) = z^3\) |
| \(|H_{3}(1)|\)
Hankel H₃(1) |
janowski_A1_B0 | \(\tfrac{1}{9}\) | 0.111111 | \(\omega(z) = z^3\) |
| \(|H_{3}(1)|\)
Hankel H₃(1) |
limacon_0.3 | \(\tfrac{1}{25}\) | 0.040000 | \(\omega(z) = z^3\) |
| \(|H_{3}(1)|\)
Hankel H₃(1) |
limacon_0.5 | \(\tfrac{1}{9}\) | 0.111111 | \(\omega(z) = z^3\) |
| \(|H_{3}(1)|\)
Hankel H₃(1) |
limacon_0.707 | \(\tfrac{2}{9}\) | 0.222222 | \(\omega(z) = z^3\) |
| \(|H_{3}(1)|\)
Hankel H₃(1) |
order_0.25 | \(\tfrac{1}{4}\) | 0.250000 | \(\omega(z) = z^3\) |
| \(|H_{3}(1)|\)
Hankel H₃(1) |
order_0.5 | \(\tfrac{1}{9}\) | 0.111111 | \(\omega(z) = z^3\) |
| \(|H_{3}(1)|\)
Hankel H₃(1) |
order_0.75 | \(\tfrac{1}{36}\) | 0.027778 | \(\omega(z) = z^3\) |
| \(|H_{3}(1)|\)
Hankel H₃(1) |
strongly_0.25 | \(\tfrac{1}{36}\) | 0.027778 | \(\omega(z) = z^3\) |
| \(|H_{3}(1)|\)
Hankel H₃(1) |
strongly_0.5 | \(\tfrac{1}{9}\) | 0.111111 | \(\omega(z) = z^3\) |
| \(|H_{3}(1)|\)
Hankel H₃(1) |
strongly_0.75 | \(\tfrac{1}{4}\) | 0.250000 | \(\omega(z) = z^3\) |
| \(|a_3^{2}-a_5|\)
Zalcman (n=3) |
Bean S*_𝔅 | \(\tfrac{1}{8}\) | 0.125000 | \(\omega(z) = z^4\) |
| \(|a_3^{2}-a_5|\)
Zalcman (n=3) |
Bell S*_B | no simple form | 0.484991 | interior |
| \(|a_3^{2}-a_5|\)
Zalcman (n=3) |
Cardioid S*_C | no simple form | 1.010288 | \(\omega(z) = z\) |
| \(|a_3^{2}-a_5|\)
Zalcman (n=3) |
Cardioid S*_℘ | no simple form | 0.514205 | interior |
| \(|a_3^{2}-a_5|\)
Zalcman (n=3) |
Cissoid S*_{cs} | no simple form | 0.368177 | interior |
| \(|a_3^{2}-a_5|\)
Zalcman (n=3) |
Cosh-sqrt | \(\tfrac{1}{8}\) | 0.125000 | \(\omega(z) = z^4\) |
| \(|a_3^{2}-a_5|\)
Zalcman (n=3) |
Crescent | \(\tfrac{19}{48}\) | 0.395833 | \(\omega(z) = z\) |
| \(|a_3^{2}-a_5|\)
Zalcman (n=3) |
Epicycloid S*_{3ℒ} | \(\tfrac{3}{16}\) | 0.187500 | \(\omega(z) = z^4\) |
| \(|a_3^{2}-a_5|\)
Zalcman (n=3) |
Epicycloid S*_{6ℒ} | \(\tfrac{3}{14}\) | 0.214286 | \(\omega(z) = z^4\) |
| \(|a_3^{2}-a_5|\)
Zalcman (n=3) |
Exponential S*_e | no simple form | 0.339888 | interior |
| \(|a_3^{2}-a_5|\)
Zalcman (n=3) |
Four-leaf S*_{4L} | \(\tfrac{5}{24}\) | 0.208333 | \(\omega(z) = z^4\) |
| \(|a_3^{2}-a_5|\)
Zalcman (n=3) |
Lemniscate S*_L
✓ Banga & Sivaprasad Kumar 2020 (Math. Slovaca; #294)
|
\(\tfrac{1}{8}\) | 0.125000 | \(\omega(z) = z^4\) |
| \(|a_3^{2}-a_5|\)
Zalcman (n=3) |
Nephroid S*_{Ne} | no simple form | 0.319444 | \(\omega(z) = z\) |
| \(|a_3^{2}-a_5|\)
Zalcman (n=3) |
Non-convex S*_{nc} | \(\tfrac{1}{3}\) | 0.333333 | interior |
| \(|a_3^{2}-a_5|\)
Zalcman (n=3) |
Parabolic S_p | no simple form | 0.255494 | interior |
| \(|a_3^{2}-a_5|\)
Zalcman (n=3) |
Petal S*_ρ | no simple form | 0.263889 | \(\omega(z) = z\) |
| \(|a_3^{2}-a_5|\)
Zalcman (n=3) |
Rational S*_R | no simple form | 0.111859 | interior |
| \(|a_3^{2}-a_5|\)
Zalcman (n=3) |
Sigmoid S*_{SG} | \(\tfrac{1}{8}\) | 0.125000 | \(\omega(z) = z^4\) |
| \(|a_3^{2}-a_5|\)
Zalcman (n=3) |
Sine S*_{\sin} | no simple form | 0.263889 | \(\omega(z) = z\) |
| \(|a_3^{2}-a_5|\)
Zalcman (n=3) |
Starlike S*
✓ Krushkal (Zalcman conjecture, n=3)
|
\(4\) | 4.000000 | \(\omega(z) = z\) |
| \(|a_3^{2}-a_5|\)
Zalcman (n=3) |
Strip S*_τ | no simple form | 0.319444 | \(\omega(z) = z\) |
| \(|a_3^{2}-a_5|\)
Zalcman (n=3) |
Tanh S*_{\tanh} | no simple form | 0.319444 | \(\omega(z) = z\) |
| \(|a_3^{2}-a_5|\)
Zalcman (n=3) |
Three-leaf | \(\tfrac{1}{5}\) | 0.200000 | \(\omega(z) = z^4\) |
| \(|a_3^{2}-a_5|\)
Zalcman (n=3) |
janowski_A0.5_B-0.5 | no simple form | 0.328720 | interior |
| \(|a_3^{2}-a_5|\)
Zalcman (n=3) |
janowski_A0.75_B-0.25 | no simple form | 0.262708 | interior |
| \(|a_3^{2}-a_5|\)
Zalcman (n=3) |
janowski_A1_B-0.5 | \(\tfrac{21}{16}\) | 1.312500 | \(\omega(z) = z\) |
| \(|a_3^{2}-a_5|\)
Zalcman (n=3) |
janowski_A1_B0 | \(\tfrac{1}{4}\) | 0.250000 | \(\omega(z) = z^4\) |
| \(|a_3^{2}-a_5|\)
Zalcman (n=3) |
limacon_0.3 | \(\tfrac{3}{20}\) | 0.150000 | \(\omega(z) = z^4\) |
| \(|a_3^{2}-a_5|\)
Zalcman (n=3) |
limacon_0.5 | no simple form | 0.278676 | interior |
| \(|a_3^{2}-a_5|\)
Zalcman (n=3) |
limacon_0.707 | no simple form | 1.114583 | \(\omega(z) = z\) |
| \(|a_3^{2}-a_5|\)
Zalcman (n=3) |
order_0.25 | no simple form | 1.168269 | interior |
| \(|a_3^{2}-a_5|\)
Zalcman (n=3) |
order_0.5 | \(\tfrac{8}{17}\) | 0.470588 | interior |
| \(|a_3^{2}-a_5|\)
Zalcman (n=3) |
order_0.75 | no simple form | 0.170259 | interior |
| \(|a_3^{2}-a_5|\)
Zalcman (n=3) |
strongly_0.25 | \(\tfrac{1}{8}\) | 0.125000 | \(\omega(z) = z^4\) |
| \(|a_3^{2}-a_5|\)
Zalcman (n=3) |
strongly_0.5 | no simple form | 0.320261 | interior |
| \(|a_3^{2}-a_5|\)
Zalcman (n=3) |
strongly_0.75 | no simple form | 1.081731 | interior |
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