| import Mathlib |
|
|
| namespace OAI |
|
|
| noncomputable section |
| open Set MeasureTheory Filter |
| open scoped Topology |
| 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 integralMean (f : ℂ → ℂ) (t r : ℝ) : ℝ := |
| (2 * Real.pi)⁻¹ * ∫ θ in -Real.pi..Real.pi, |
| ‖deriv f ((r : ℂ) * Complex.exp ((θ : ℂ) * Complex.I))‖ ^ t |
|
|
| def beta (f : ℂ → ℂ) (t : ℝ) : EReal := |
| Filter.limsup (fun r : ℝ => |
| ((Real.log (integralMean f t r) / Real.log (1 / (1 - r)) : ℝ) : EReal)) |
| (𝓝[<] (1 : ℝ)) |
|
|
| def spectrum (t : ℝ) : EReal := |
| sSup {b | ∃ f : ℂ → ℂ, Schlicht f ∧ beta f t = b} |
|
|
| def SphericalBoundaryNontrivial (W : Set ℂ) : Prop := |
| Set.Nontrivial (frontier (((↑) : ℂ → OnePoint ℂ) '' W)) |
|
|
| def MainStatement : Prop := |
| (∀ ε : ℝ, 0 < ε → ∃ C : ℝ, ∀ f : ℂ → ℂ, Schlicht f → |
| ∀ r : ℝ, 1 / 2 ≤ r → r < 1 → |
| integralMean f (-2) r ≤ C * (1 - r) ^ (-1 - ε)) ∧ |
| spectrum (-2) = 1 ∧ |
| (∀ W : Set ℂ, IsOpen W → IsConnected W → SimplyConnectedSpace W → |
| SphericalBoundaryNontrivial W → ∀ φ : ℂ → ℂ, |
| DifferentiableOn ℂ φ W → Set.BijOn φ W disk → |
| ∀ s : ℝ, 4 / 3 < s → s < 4 → |
| IntegrableOn (fun z => ‖deriv φ z‖ ^ s) W volume) ∧ |
| (∀ f : ℂ → ℂ, UnivalentOn f disk → ∀ t : ℝ, |
| -2 < t → t < 2 / 3 → |
| IntegrableOn (fun z => ‖deriv f z‖ ^ t) disk volume) |
|
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| theorem main_theorem : MainStatement := by |
| sorry |
|
|
| end Brennan |
| end |
|
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| end OAI |
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|
