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Brennan's Conjecture

Reference:

namespace BrennanConjecture open Complex Filter MeasureTheory Set Topology def unitDisk : Set ℂ := {z | ‖z‖ < 1}

The standard class $\mathcal{S}$ of normalised univalent functions on $\mathbb{D}$.

structure IsUnivalentNormalized (f : ℂ → ℂ) : Prop where analyticOn : AnalyticOn ℂ f unitDisk injOn : InjOn f unitDisk map_zero : f 0 = 0 deriv_zero : deriv f 0 = 1

$\beta_f(\tau) := \limsup_{r \to 1^-} \frac{\log \int_{-\pi}^{\pi} |f'(re^{i\theta})|^\tau , d\theta}{|\log(1-r)|}$

noncomputable def integralMeansSpectrum (f : ℂ → ℂ) (τ : ℝ) : ℝ := limsup (fun r => Real.log (∫ θ in Ioc (-Real.pi) Real.pi, ‖deriv f (r • exp (Complex.I * θ))‖ ^ τ) / |Real.log (1 - r)|) (𝓝[Iio 1] (1 : ℝ)) noncomputable def universalSpectrum (τ : ℝ) : ℝ := sSup {β | ∃ f : ℂ → ℂ, IsUnivalentNormalized f ∧ β = integralMeansSpectrum f τ} noncomputable def universalSpectrumBounded (τ : ℝ) : ℝ := sSup {β | ∃ f : ℂ → ℂ, IsUnivalentNormalized f ∧ Bornology.IsBounded (f '' unitDisk) ∧ β = integralMeansSpectrum f τ} @[category API, AMS 30] theorem declaration uses 'sorry'universalSpectrumBounded_le (τ : ℝ) : universalSpectrumBounded τ ≤ universalSpectrum τ := τ:ℝ⊢ universalSpectrumBounded τ ≤ universalSpectrum τ τ:ℝ⊢ BddAbove {β | ∃ f, IsUnivalentNormalized f ∧ β = integralMeansSpectrum f τ}τ:ℝ⊢ {β | ∃ f, IsUnivalentNormalized f ∧ Bornology.IsBounded (f '' unitDisk) ∧ β = integralMeansSpectrum f τ}.Nonemptyτ:ℝ⊢ {β | ∃ f, IsUnivalentNormalized f ∧ Bornology.IsBounded (f '' unitDisk) ∧ β = integralMeansSpectrum f τ} ⊆ {β | ∃ f, IsUnivalentNormalized f ∧ β = integralMeansSpectrum f τ} τ:ℝ⊢ BddAbove {β | ∃ f, IsUnivalentNormalized f ∧ β = integralMeansSpectrum f τ} All goals completed! 🐙 τ:ℝ⊢ {β | ∃ f, IsUnivalentNormalized f ∧ Bornology.IsBounded (f '' unitDisk) ∧ β = integralMeansSpectrum f τ}.Nonempty All goals completed! 🐙 τ:ℝ⊢ {β | ∃ f, IsUnivalentNormalized f ∧ Bornology.IsBounded (f '' unitDisk) ∧ β = integralMeansSpectrum f τ} ⊆ {β | ∃ f, IsUnivalentNormalized f ∧ β = integralMeansSpectrum f τ} τ:ℝf:ℂ → ℂhf:IsUnivalentNormalized fleft✝:Bornology.IsBounded (f '' unitDisk)⊢ integralMeansSpectrum f τ ∈ {β | ∃ f, IsUnivalentNormalized f ∧ β = integralMeansSpectrum f τ}; All goals completed! 🐙 @[category test, AMS 30] theorem declaration uses 'sorry'integralMeansSpectrum_id (τ : ℝ) : integralMeansSpectrum id τ = 0 := τ:ℝ⊢ integralMeansSpectrum id τ = 0 All goals completed! 🐙

Brennan's conjecture, part 1: $B(-2) = 1$.

@[category research open, AMS 30] theorem declaration uses 'sorry'brennan_universalSpectrum : universalSpectrum (-2) = 1 := ⊢ universalSpectrum (-2) = 1 All goals completed! 🐙

Brennan's conjecture, part 2: $B_b(-2) = 1$.

@[category research open, AMS 30] theorem declaration uses 'sorry'brennan_universalSpectrumBounded : universalSpectrumBounded (-2) = 1 := ⊢ universalSpectrumBounded (-2) = 1 All goals completed! 🐙

Brennan's conjecture, part 3: $B(-2) = B_b(-2)$.

@[category API, AMS 30] theorem brennan_spectra_eq : universalSpectrum (-2) = universalSpectrumBounded (-2) := ⊢ universalSpectrum (-2) = universalSpectrumBounded (-2) All goals completed! 🐙

Brennan's conjecture: $B(-2) = B_b(-2) = 1$.

@[category API, AMS 30] theorem brennan : universalSpectrum (-2) = 1 ∧ universalSpectrumBounded (-2) = 1 := ⟨brennan_universalSpectrum, brennan_universalSpectrumBounded⟩ end BrennanConjecture