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.
None right now - every certified enclosure has been closed to a proven-exact constant.
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) |
Janowski S*[1,-1/2] | \(\tfrac{9}{16}\) | 0.562500 | \(\omega(z) = z^2\) |
| \(|a_2a_4-a_3^{2}|\)
Hankel H₂(2) |
Janowski S*[1,0] | \(\tfrac{1}{4}\) | 0.250000 | \(\omega(z) = z^2\) |
| \(|a_2a_4-a_3^{2}|\)
Hankel H₂(2) |
Janowski S*[1/2,-1/2] | \(\tfrac{1}{4}\) | 0.250000 | \(\omega(z) = z^2\) |
| \(|a_2a_4-a_3^{2}|\)
Hankel H₂(2) |
Janowski S*[3/4,-1/4] | \(\tfrac{1}{4}\) | 0.250000 | \(\omega(z) = z^2\) |
| \(|a_2a_4-a_3^{2}|\)
Hankel H₂(2) |
Limaçon S*(s=1/2) | \(\tfrac{1}{4}\) | 0.250000 | \(\omega(z) = z^2\) |
| \(|a_2a_4-a_3^{2}|\)
Hankel H₂(2) |
Limaçon S*(s=√2/2) | \(\tfrac{1}{2}\) | 0.500000 | \(\omega(z) = z^2\) |
| \(|a_2a_4-a_3^{2}|\)
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) |
Starlike of order 1/2 | \(\tfrac{1}{4}\) | 0.250000 | \(\omega(z) = z^2\) |
| \(|a_2a_4-a_3^{2}|\)
Hankel H₂(2) |
Starlike of order 1/4 | \(\tfrac{9}{16}\) | 0.562500 | \(\omega(z) = z^2\) |
| \(|a_2a_4-a_3^{2}|\)
Hankel H₂(2) |
Strip S*_τ | \(\tfrac{1}{4}\) | 0.250000 | \(\omega(z) = z^2\) |
| \(|a_2a_4-a_3^{2}|\)
Hankel H₂(2) |
Strongly starlike SS*(1/2) | \(\tfrac{1}{4}\) | 0.250000 | \(\omega(z) = z^2\) |
| \(|a_2a_4-a_3^{2}|\)
Hankel H₂(2) |
Strongly starlike SS*(3/4) | \(\tfrac{9}{16}\) | 0.562500 | \(\omega(z) = z^2\) |
| \(|a_2a_4-a_3^{2}|\)
Hankel H₂(2) |
Tanh S*_{\tanh} | \(\tfrac{1}{4}\) | 0.250000 | \(\omega(z) = z^2\) |
| \(|H_{3}(1)|\)
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) |
Janowski S*[1,-1/2] | \(\tfrac{1}{4}\) | 0.250000 | \(\omega(z) = z^3\) |
| \(|H_{3}(1)|\)
Hankel H₃(1) |
Janowski S*[1,0] | \(\tfrac{1}{9}\) | 0.111111 | \(\omega(z) = z^3\) |
| \(|H_{3}(1)|\)
Hankel H₃(1) |
Janowski S*[1/2,-1/2] | \(\tfrac{1}{9}\) | 0.111111 | \(\omega(z) = z^3\) |
| \(|H_{3}(1)|\)
Hankel H₃(1) |
Janowski S*[3/4,-1/4] | \(\tfrac{1}{9}\) | 0.111111 | \(\omega(z) = z^3\) |
| \(|H_{3}(1)|\)
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) |
Limaçon S*(s=1/2) | \(\tfrac{1}{9}\) | 0.111111 | \(\omega(z) = z^3\) |
| \(|H_{3}(1)|\)
Hankel H₃(1) |
Limaçon S*(s=3/10) | \(\tfrac{1}{25}\) | 0.040000 | \(\omega(z) = z^3\) |
| \(|H_{3}(1)|\)
Hankel H₃(1) |
Limaçon S*(s=√2/2) | \(\tfrac{2}{9}\) | 0.222222 | \(\omega(z) = z^3\) |
| \(|H_{3}(1)|\)
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) |
Starlike of order 1/2 | \(\tfrac{1}{9}\) | 0.111111 | \(\omega(z) = z^3\) |
| \(|H_{3}(1)|\)
Hankel H₃(1) |
Starlike of order 1/4 | \(\tfrac{1}{4}\) | 0.250000 | \(\omega(z) = z^3\) |
| \(|H_{3}(1)|\)
Hankel H₃(1) |
Starlike of order 3/4 | \(\tfrac{1}{36}\) | 0.027778 | \(\omega(z) = z^3\) |
| \(|H_{3}(1)|\)
Hankel H₃(1) |
Strip S*_τ | \(\tfrac{1}{9}\) | 0.111111 | \(\omega(z) = z^3\) |
| \(|H_{3}(1)|\)
Hankel H₃(1) |
Strongly starlike SS*(1/2) | \(\tfrac{1}{9}\) | 0.111111 | \(\omega(z) = z^3\) |
| \(|H_{3}(1)|\)
Hankel H₃(1) |
Strongly starlike SS*(1/4) | \(\tfrac{1}{36}\) | 0.027778 | \(\omega(z) = z^3\) |
| \(|H_{3}(1)|\)
Hankel H₃(1) |
Strongly starlike SS*(3/4) | \(\tfrac{1}{4}\) | 0.250000 | \(\omega(z) = z^3\) |
| \(|H_{3}(1)|\)
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 | \(\omega(z) = z^3\) |
| \(|a_3^{2}-a_5|\)
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) |
Janowski S*[1,-1/2] | \(\tfrac{21}{16}\) | 1.312500 | \(\omega(z) = z\) |
| \(|a_3^{2}-a_5|\)
Zalcman (n=3) |
Janowski S*[1,0] | \(\tfrac{1}{4}\) | 0.250000 | \(\omega(z) = z^4\) |
| \(|a_3^{2}-a_5|\)
Zalcman (n=3) |
Janowski S*[1/2,-1/2] | no simple form | 0.328720 | interior |
| \(|a_3^{2}-a_5|\)
Zalcman (n=3) |
Janowski S*[3/4,-1/4] | no simple form | 0.262708 | interior |
| \(|a_3^{2}-a_5|\)
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) |
Limaçon S*(s=1/2) | no simple form | 0.278676 | interior |
| \(|a_3^{2}-a_5|\)
Zalcman (n=3) |
Limaçon S*(s=3/10) | \(\tfrac{3}{20}\) | 0.150000 | \(\omega(z) = z^4\) |
| \(|a_3^{2}-a_5|\)
Zalcman (n=3) |
Limaçon S*(s=√2/2) | no simple form | 1.114583 | \(\omega(z) = z\) |
| \(|a_3^{2}-a_5|\)
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) |
Starlike of order 1/2 | \(\tfrac{8}{17}\) | 0.470588 | interior |
| \(|a_3^{2}-a_5|\)
Zalcman (n=3) |
Starlike of order 1/4 | no simple form | 1.168269 | interior |
| \(|a_3^{2}-a_5|\)
Zalcman (n=3) |
Starlike of order 3/4 | no simple form | 0.170259 | interior |
| \(|a_3^{2}-a_5|\)
Zalcman (n=3) |
Strip S*_τ | no simple form | 0.319444 | \(\omega(z) = z\) |
| \(|a_3^{2}-a_5|\)
Zalcman (n=3) |
Strongly starlike SS*(1/2) | no simple form | 0.320261 | interior |
| \(|a_3^{2}-a_5|\)
Zalcman (n=3) |
Strongly starlike SS*(1/4) | \(\tfrac{1}{8}\) | 0.125000 | \(\omega(z) = z^4\) |
| \(|a_3^{2}-a_5|\)
Zalcman (n=3) |
Strongly starlike SS*(3/4) | no simple form | 1.081731 | interior |
| \(|a_3^{2}-a_5|\)
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 | \(\omega(z) = z^4\) |
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