| import Mathlib |
|
|
| namespace OAI |
|
|
| open Set Filter MeasureTheory |
| open scoped Topology |
|
|
| namespace StrictInverseFirstPower |
|
|
| def disk : Set ℂ := Metric.ball 0 1 |
|
|
| def NormalizedUnivalent (f : ℂ → ℂ) : Prop := |
| DifferentiableOn ℂ f disk ∧ Set.InjOn f disk ∧ f 0 = 0 ∧ deriv f 0 = 1 |
|
|
| def BoundedNormalizedUnivalent (f : ℂ → ℂ) : Prop := |
| NormalizedUnivalent f ∧ ∃ R : ℝ, ∀ z ∈ disk, ‖f z‖ ≤ R |
|
|
| noncomputable def integralMean (p : ℝ) (f : ℂ → ℂ) (r : ℝ) : ℝ := |
| (2 * Real.pi)⁻¹ * ∫ θ in -Real.pi..Real.pi, |
| ‖deriv f ((r : ℂ) * Complex.exp ((θ : ℂ) * Complex.I))‖ ^ p |
|
|
| noncomputable def growthExponent (p : ℝ) (f : ℂ → ℂ) : EReal := |
| Filter.limsup (fun r : ℝ => |
| ((Real.log (integralMean p f r) / Real.log (1 / (1 - r)) : ℝ) : EReal)) |
| (𝓝[<] (1 : ℝ)) |
|
|
| noncomputable def boundedSpectrum (p : ℝ) : EReal := |
| ⨆ (f : ℂ → ℂ) (_ : BoundedNormalizedUnivalent f), growthExponent p f |
|
|
| noncomputable def kraetzerPrediction (p : ℝ) : EReal := |
| if |p| ≤ 2 then ((p ^ (2 : ℕ) / 4 : ℝ) : EReal) |
| else ((|p| - 1 : ℝ) : EReal) |
|
|
| theorem main : |
| (∃ ε C : ℝ, 0 < ε ∧ ε < 1 / 4 ∧ |
| ∀ (f : ℂ → ℂ), NormalizedUnivalent f → |
| ∀ r : ℝ, 1 / 2 ≤ r → r < 1 → |
| integralMean (-1) f r ≤ C * (1 - r) ^ (-(1 / 4 : ℝ) + ε)) ∧ |
| boundedSpectrum (-1) < (1 / 4 : EReal) ∧ |
| (¬ ∀ p : ℝ, boundedSpectrum p = kraetzerPrediction p) := by |
| sorry |
|
|
| end StrictInverseFirstPower |
|
|
| end OAI |
|
|
