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import FormalConjecturesUtil
namespace BrennanConjectureopen Complex Filter MeasureTheory Set Topologydef 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 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 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 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 brennan_universalSpectrumBounded :
universalSpectrumBounded (-2) = 1 := ⊢ universalSpectrumBounded (-2) = 1
All goals completed! 🐙Brennan's conjecture, part 3: $B(-2) = B_b(-2)$.
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