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dae740b | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 | 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
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