144 lines
4.0 KiB
Plaintext
144 lines
4.0 KiB
Plaintext
import Nevanlinna.harmonicAt_examples
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import Nevanlinna.harmonicAt_meanValue
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theorem jensen_case_R_eq_one
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(f : ℂ → ℂ)
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(h₁f : Differentiable ℂ f)
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(h₂f : f 0 ≠ 0)
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(S : Finset ℕ)
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(a : S → ℂ)
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(ha : ∀ s, a s ∈ Metric.ball 0 1)
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(F : ℂ → ℂ)
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(h₁F : Differentiable ℂ F)
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(h₂F : ∀ z, F z ≠ 0)
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(h₃F : f = fun z ↦ (F z) * ∏ s : S, (z - a s))
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:
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Real.log ‖f 0‖ = -∑ s, Real.log (‖a s‖⁻¹) + (2 * Real.pi)⁻¹ * ∫ (x : ℝ) in (0)..2 * Real.pi, Real.log ‖f (circleMap 0 1 x)‖ := by
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let logAbsF := fun w ↦ Real.log ‖F w‖
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have t₀ : ∀ z, HarmonicAt logAbsF z := by
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intro z
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apply logabs_of_holomorphicAt_is_harmonic
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apply h₁F.holomorphicAt
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exact h₂F z
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have t₁ : (∫ (x : ℝ) in (0)..2 * Real.pi, logAbsF (circleMap 0 1 x)) = 2 * Real.pi * logAbsF 0 := by
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apply harmonic_meanValue t₀ 1
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exact Real.zero_lt_one
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have t₂ : ∀ s, f (a s) = 0 := by
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intro s
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rw [h₃F]
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simp
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right
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apply Finset.prod_eq_zero_iff.2
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use s
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simp
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let logAbsf := fun w ↦ Real.log ‖f w‖
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have s₀ : ∀ z, f z ≠ 0 → logAbsf z = logAbsF z + ∑ s, Real.log ‖z - a s‖ := by
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intro z hz
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dsimp [logAbsf]
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rw [h₃F]
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simp_rw [Complex.abs.map_mul]
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rw [Complex.abs_prod]
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rw [Real.log_mul]
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rw [Real.log_prod]
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rfl
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intro s hs
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simp
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by_contra ha'
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rw [ha'] at hz
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exact hz (t₂ s)
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-- Complex.abs (F z) ≠ 0
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simp
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exact h₂F z
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-- ∏ I : { x // x ∈ S }, Complex.abs (z - a I) ≠ 0
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by_contra h'
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obtain ⟨s, h's, h''⟩ := Finset.prod_eq_zero_iff.1 h'
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simp at h''
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rw [h''] at hz
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let A := t₂ s
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exact hz A
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have s₁ : ∀ z, f z ≠ 0 → logAbsF z = logAbsf z - ∑ s, Real.log ‖z - a s‖ := by
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sorry
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rw [s₁ 0 h₂f] at t₁
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have {x : ℝ} : f (circleMap 0 1 x) ≠ 0 := by
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sorry
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simp_rw [s₁ (circleMap 0 1 _) this] at t₁
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rw [intervalIntegral.integral_sub] at t₁
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rw [intervalIntegral.integral_finset_sum] at t₁
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have {i : S} : ∫ (x : ℝ) in (0)..2 * Real.pi, Real.log ‖circleMap 0 1 x - a i‖ = 0 := by
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sorry
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simp_rw [this] at t₁
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simp at t₁
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rw [t₁]
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simp
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have {w : ℝ} : Real.pi⁻¹ * 2⁻¹ * (2 * Real.pi * w) = w := by
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ring_nf
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simp [mul_inv_cancel Real.pi_ne_zero]
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rw [this]
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simp
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rfl
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-- ∀ i ∈ Finset.univ, IntervalIntegrable (fun x => Real.log ‖circleMap 0 1 x - a i‖) MeasureTheory.volume 0 (2 * Real.pi)
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intro i hi
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apply Continuous.intervalIntegrable
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apply continuous_iff_continuousAt.2
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intro x
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have : (fun x => Real.log ‖circleMap 0 1 x - a i‖) = Real.log ∘ Complex.abs ∘ (fun x ↦ circleMap 0 1 x - a i) :=
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rfl
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rw [this]
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apply ContinuousAt.comp
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apply Real.continuousAt_log
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simp
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by_contra ha'
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let A := ha i
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rw [← ha'] at A
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simp at A
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apply ContinuousAt.comp
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apply Complex.continuous_abs.continuousAt
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fun_prop
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-- IntervalIntegrable (fun x => logAbsf (circleMap 0 1 x)) MeasureTheory.volume 0 (2 * Real.pi)
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apply Continuous.intervalIntegrable
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apply continuous_iff_continuousAt.2
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intro x
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have : (fun x => logAbsf (circleMap 0 1 x)) = Real.log ∘ Complex.abs ∘ f ∘ (fun x ↦ circleMap 0 1 x) :=
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rfl
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rw [this]
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apply ContinuousAt.comp
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simp
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sorry
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apply ContinuousAt.comp
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apply Complex.continuous_abs.continuousAt
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apply ContinuousAt.comp
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apply h₁f.continuous.continuousAt
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let A := continuous_circleMap 0 1
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apply A.continuousAt
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-- IntervalIntegrable (fun x => ∑ s : { x // x ∈ S }, Real.log ‖circleMap 0 1 x - a s‖) MeasureTheory.volume 0 (2 * Real.pi)
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apply Continuous.intervalIntegrable
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apply continuous_finset_sum
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intro i hi
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apply continuous_iff_continuousAt.2
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intro x
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have : (fun x => Real.log ‖circleMap 0 1 x - a i‖) = Real.log ∘ Complex.abs ∘ (fun x ↦ circleMap 0 1 x - a i) :=
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rfl
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rw [this]
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apply ContinuousAt.comp
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apply Real.continuousAt_log
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simp
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by_contra ha'
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let A := ha i
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rw [← ha'] at A
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simp at A
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apply ContinuousAt.comp
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apply Complex.continuous_abs.continuousAt
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fun_prop
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