The containment poset of the function classes: an arrow A → B means A ⊂ B (every A is a B). Broader classes sit at the top. Click any class to highlight its super-classes (above, in blue) and sub-classes (below, in green).
What the registry has established for each class × coefficient functional. proven exact = sharp theorem (bracket closed to a point); bracket = certified enclosure, candidate not yet proven sharp; open = numerically conjectured, not yet certified (open problems →); - = none. A ✦ marks a result already in the literature.
36 proven-exact · 11 brackets · 97 open · across 39 classes.
| Class | Fekete–Szegő | H₂(2) | H₃(1) | Zalcman |
|---|---|---|---|---|
| Bean S*_𝔅 | ✓ | ≈✦ | ? | ? |
| Bell S*_B | ✓ | ? | ? | ? |
| booth_0.3 | - | - | - | - |
| booth_0.7 | - | - | - | - |
| Cardioid S*_C | ✓ | ? | ? | ? |
| Cardioid S*_℘ | ✓ | ? | ? | ? |
| Cissoid S*_{cs} | ✓ | ? | ? | ? |
| Cosh-sqrt | ✓ | ≈ | ? | ? |
| Crescent | ✓ | ? | ? | ? |
| Epicycloid S*_{3ℒ} | ✓ | ≈ | ? | ? |
| Epicycloid S*_{6ℒ} | ✓ | ? | ? | ? |
| Exponential S*_e | ✓ | ? | ? | ? |
| Four-leaf S*_{4L} | ✓ | ≈ | ? | ? |
| janowski_A0.5_B-0.5 | ✓ | ? | ? | ? |
| janowski_A0.75_B-0.25 | ✓ | ? | ? | ? |
| janowski_A0_B-1 | - | - | - | - |
| janowski_A1_B-0.5 | ✓ | ? | ? | ? |
| janowski_A1_B0 | ✓ | ? | ? | ? |
| Lemniscate S*_L | ✓ | ≈✦ | ?✦ | ?✦ |
| limacon_0.3 | ✓ | ≈ | ? | ? |
| limacon_0.5 | ✓ | ? | ? | ? |
| limacon_0.707 | ✓ | ? | ? | ? |
| Nephroid S*_{Ne} | ✓ | ? | ? | ? |
| Non-convex S*_{nc} | ✓ | ? | ? | ? |
| order_0.25 | ✓ | ? | ? | ? |
| order_0.5 | ✓ | ? | ? | ? |
| order_0.75 | ✓ | ≈ | ? | ? |
| Parabolic S_p | ✓ | ? | ? | ? |
| Petal S*_ρ | ✓ | ? | ? | ? |
| Rational S*_R | ✓ | ≈ | ? | ? |
| Sigmoid S*_{SG} | ✓ | ≈✦ | ? | ? |
| Sine S*_{\sin} | ✓ | ? | ? | ? |
| Starlike S* | ✓✦ | ? | ?✦ | ?✦ |
| Strip S*_τ | ✓ | ? | ? | ? |
| strongly_0.25 | ✓ | ≈ | ? | ? |
| strongly_0.5 | ✓ | ? | ? | ? |
| strongly_0.75 | ✓ | ? | ? | ? |
| Tanh S*_{\tanh} | ✓ | ? | ? | ? |
| Three-leaf | ✓ | ≈ | ? | ? |