| import Mathlib | |
| namespace OAI | |
| noncomputable section | |
| open Set MeasureTheory Filter | |
| open scoped Topology ENNReal | |
| namespace Brennan | |
| def disk : Set ℂ := Metric.ball 0 1 | |
| def UnivalentOn (f : ℂ → ℂ) (U : Set ℂ) : Prop := | |
| DifferentiableOn ℂ f U ∧ Set.InjOn f U | |
| def Schlicht (f : ℂ → ℂ) : Prop := | |
| UnivalentOn f disk ∧ f 0 = 0 ∧ deriv f 0 = 1 | |
| def koebeMap (z : ℂ) : ℂ := z/(1-z)^2 | |
| namespace Sharp | |
| def koebeDomain : Set ℂ := koebeMap '' disk | |
| def koebeInverse : ℂ → ℂ := Function.invFunOn koebeMap disk | |
| def areaMoment (f : ℂ → ℂ) (U : Set ℂ) (t : ℝ) : ℝ≥0∞ := | |
| ∫⁻ z in U, ENNReal.ofReal (‖deriv f z‖ ^ t) ∂volume | |
| def SharpEndpointStatement : Prop := | |
| Schlicht koebeMap ∧ | |
| areaMoment koebeMap disk (-2) = ⊤ ∧ | |
| areaMoment koebeMap disk (2 / 3) = ⊤ ∧ | |
| ¬ IntegrableOn (fun z => ‖deriv koebeInverse z‖ ^ (4 : ℝ)) koebeDomain volume ∧ | |
| ¬ IntegrableOn (fun z => ‖deriv koebeInverse z‖ ^ (4 / 3 : ℝ)) koebeDomain volume | |
| theorem sharp_endpoints : SharpEndpointStatement := by | |
| sorry | |
| end Sharp | |
| end Brennan | |
| end | |
| end OAI | |