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