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9eeb9c7 | 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 46 47 48 49 50 51 | 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)
end Brennan
end
end OAI
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