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465b10d | 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 | 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
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