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dba4e2d9c4
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712be956d0
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@ -1,449 +1,172 @@
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import Mathlib.Analysis.Complex.CauchyIntegral
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import Mathlib.Analysis.Analytic.IsolatedZeros
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import Nevanlinna.analyticOn_zeroSet
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import Nevanlinna.harmonicAt_examples
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import Nevanlinna.harmonicAt_meanValue
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import Nevanlinna.specialFunctions_CircleIntegral_affine
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open Real
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theorem jensen_case_R_eq_one
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(f : ℂ → ℂ)
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(h₁f : AnalyticOn ℂ f (Metric.closedBall 0 1))
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(h₂f : f 0 ≠ 0) :
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log ‖f 0‖ = -∑ᶠ s, (h₁f.order s).toNat * log (‖s.1‖⁻¹) + (2 * π)⁻¹ * ∫ (x : ℝ) in (0)..(2 * π), log ‖f (circleMap 0 1 x)‖ := by
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(h₁f : ∀ z ∈ Metric.closedBall (0 : ℂ) 1, HolomorphicAt f z)
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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 : ∀ z ∈ Metric.closedBall (0 : ℂ) 1, HolomorphicAt F z)
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(h₂F : ∀ z ∈ Metric.closedBall (0 : ℂ) 1, F z ≠ 0)
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(h₃F : f = fun z ↦ (F z) * ∏ s : S, (z - a s)) :
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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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have h₁U : IsPreconnected (Metric.closedBall (0 : ℂ) 1) :=
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(convex_closedBall (0 : ℂ) 1).isPreconnected
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have h₁U : IsPreconnected (Metric.closedBall (0 : ℂ) 1) := by sorry
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have h₂U : IsCompact (Metric.closedBall (0 : ℂ) 1) := by sorry
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have h₁f : AnalyticOn ℂ f (Metric.closedBall (0 : ℂ) 1) := by sorry
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have h₂f : ∃ u ∈ (Metric.closedBall (0 : ℂ) 1), f u ≠ 0 := by sorry
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have h₂U : IsCompact (Metric.closedBall (0 : ℂ) 1) :=
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isCompact_closedBall 0 1
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let α := AnalyticOnCompact.eliminateZeros h₁U h₂U h₁f h₂f
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obtain ⟨g, A, h'₁g, h₂g, h₃g⟩ := α
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have h₁g : ∀ z ∈ Metric.closedBall 0 1, HolomorphicAt F z := by sorry
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have h'₂f : ∃ u ∈ (Metric.closedBall (0 : ℂ) 1), f u ≠ 0 := by
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use 0; simp; exact h₂f
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obtain ⟨F, h₁F, h₂F, h₃F⟩ := AnalyticOnCompact.eliminateZeros₂ h₁U h₂U h₁f h'₂f
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have h'₁F : ∀ z ∈ Metric.closedBall (0 : ℂ) 1, HolomorphicAt F z := by
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intro z h₁z
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apply AnalyticAt.holomorphicAt
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exact h₁F z h₁z
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let logAbsF := fun w ↦ Real.log ‖F w‖
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let G := fun z ↦ log ‖F z‖ + ∑ s ∈ (finiteZeros h₁U h₂U h₁f h'₂f).toFinset, (h₁f.order s).toNat * log ‖z - s‖
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have t₀ : ∀ z ∈ Metric.closedBall 0 1, HarmonicAt logAbsF z := by
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intro z hz
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apply logabs_of_holomorphicAt_is_harmonic
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apply h₁F z hz
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exact h₂F z hz
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have decompose_f : ∀ z ∈ Metric.closedBall (0 : ℂ) 1, f z ≠ 0 → log ‖f z‖ = G z := by
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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_meanValue₁ 1 Real.zero_lt_one t₀
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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 ∈ Metric.closedBall (0 : ℂ) 1, f z ≠ 0 → logAbsf z = logAbsF z + ∑ s, Real.log ‖z - a s‖ := by
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intro z h₁z h₂z
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conv =>
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left
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arg 1
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rw [h₃F]
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rw [smul_eq_mul]
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rw [norm_mul]
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rw [norm_prod]
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left
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arg 2
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intro b
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rw [norm_pow]
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simp only [Complex.norm_eq_abs, Finset.univ_eq_attach]
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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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conv =>
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left
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left
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arg 2
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intro s
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rw [Real.log_pow]
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dsimp [G]
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abel
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-- ∀ x ∈ ⋯.toFinset, Complex.abs (z - ↑x) ^ (h'₁f.order x).toNat ≠ 0
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have : ∀ x ∈ (finiteZeros h₁U h₂U h₁f h'₂f).toFinset, Complex.abs (z - ↑x) ^ (h₁f.order x).toNat ≠ 0 := by
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intro s hs
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simp at hs
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simp
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intro h₂s
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rw [h₂s] at h₂z
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tauto
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exact this
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-- ∏ x ∈ ⋯.toFinset, Complex.abs (z - ↑x) ^ (h'₁f.order x).toNat ≠ 0
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have : ∀ x ∈ (finiteZeros h₁U h₂U h₁f h'₂f).toFinset, Complex.abs (z - ↑x) ^ (h₁f.order x).toNat ≠ 0 := by
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intro s hs
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simp at hs
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simp
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intro h₂s
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rw [h₂s] at h₂z
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tauto
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rw [Finset.prod_ne_zero_iff]
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exact this
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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 h₂z
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exact h₂z (t₂ s)
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-- Complex.abs (F z) ≠ 0
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simp
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exact h₂F z h₁z
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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 h₂z
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let A := t₂ s
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exact h₂z A
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have int_logAbs_f_eq_int_G : ∫ (x : ℝ) in (0)..2 * π, log ‖f (circleMap 0 1 x)‖ = ∫ (x : ℝ) in (0)..2 * π, G (circleMap 0 1 x) := by
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rw [intervalIntegral.integral_congr_ae]
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rw [MeasureTheory.ae_iff]
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apply Set.Countable.measure_zero
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have s₁ : ∀ z ∈ Metric.closedBall (0 : ℂ) 1, f z ≠ 0 → logAbsF z = logAbsf z - ∑ s, Real.log ‖z - a s‖ := by
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intro z h₁z h₂z
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rw [s₀ z h₁z]
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simp
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assumption
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have t₀ : {a | a ∈ Ι 0 (2 * π) ∧ ¬log ‖f (circleMap 0 1 a)‖ = G (circleMap 0 1 a)}
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⊆ (circleMap 0 1)⁻¹' (Metric.closedBall 0 1 ∩ f⁻¹' {0}) := by
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intro a ha
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simp at ha
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simp
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by_contra C
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have : (circleMap 0 1 a) ∈ Metric.closedBall 0 1 :=
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circleMap_mem_closedBall 0 (zero_le_one' ℝ) a
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exact ha.2 (decompose_f (circleMap 0 1 a) this C)
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have : 0 ∈ Metric.closedBall (0 : ℂ) 1 := by simp
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rw [s₁ 0 this h₂f] at t₁
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apply Set.Countable.mono t₀
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apply Set.Countable.preimage_circleMap
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apply Set.Finite.countable
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let A := finiteZeros h₁U h₂U h₁f h'₂f
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have : (Metric.closedBall 0 1 ∩ f ⁻¹' {0}) = (Metric.closedBall 0 1).restrict f ⁻¹' {0} := by
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ext z
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simp
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tauto
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rw [this]
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exact Set.Finite.image Subtype.val A
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exact Ne.symm (zero_ne_one' ℝ)
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have decompose_int_G : ∫ (x : ℝ) in (0)..2 * π, G (circleMap 0 1 x)
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= (∫ (x : ℝ) in (0)..2 * π, log (Complex.abs (F (circleMap 0 1 x))))
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+ ∑ x ∈ (finiteZeros h₁U h₂U h₁f h'₂f).toFinset, (h₁f.order x).toNat * ∫ (x_1 : ℝ) in (0)..2 * π, log (Complex.abs (circleMap 0 1 x_1 - ↑x)) := by
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dsimp [G]
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rw [intervalIntegral.integral_add]
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rw [intervalIntegral.integral_finset_sum]
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simp_rw [intervalIntegral.integral_const_mul]
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-- ∀ i ∈ (finiteZeros h₁U h₂U h'₁f h'₂f).toFinset,
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-- IntervalIntegrable (fun x => (h'₁f.order i).toNat *
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-- log (Complex.abs (circleMap 0 1 x - ↑i))) MeasureTheory.volume 0 (2 * π)
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intro i _
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apply IntervalIntegrable.const_mul
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--simp at this
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by_cases h₂i : ‖i.1‖ = 1
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-- case pos
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exact int'₂ h₂i
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-- case neg
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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 => log (Complex.abs (circleMap 0 1 x - ↑i))) = log ∘ Complex.abs ∘ (fun x ↦ circleMap 0 1 x - ↑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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have h₀ {x : ℝ} : f (circleMap 0 1 x) ≠ 0 := by
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rw [h₃F]
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simp
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constructor
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· have : (circleMap 0 1 x) ∈ Metric.closedBall (0 : ℂ) 1 := by simp
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exact h₂F (circleMap 0 1 x) this
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· by_contra h'
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obtain ⟨s, _, h₂s⟩ := Finset.prod_eq_zero_iff.1 h'
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have : circleMap 0 1 x = a s := by
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rw [← sub_zero (circleMap 0 1 x)]
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nth_rw 2 [← h₂s]
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simp
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let A := ha s
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rw [← this] at A
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simp at A
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by_contra ha'
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conv at h₂i =>
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arg 1
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rw [← ha']
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rw [Complex.norm_eq_abs]
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rw [abs_circleMap_zero 1 x]
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simp
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tauto
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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 => log (Complex.abs (F (circleMap 0 1 x)))) MeasureTheory.volume 0 (2 * π)
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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 => log (Complex.abs (F (circleMap 0 1 x)))) = 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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apply Real.continuousAt_log
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simp [h₂F]
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-- ContinuousAt (⇑Complex.abs ∘ F ∘ fun x => circleMap 0 1 x) x
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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 DifferentiableAt.continuousAt (𝕜 := ℂ )
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apply HolomorphicAt.differentiableAt
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simp [h'₁F]
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-- ContinuousAt (fun x => circleMap 0 1 x) x
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apply Continuous.continuousAt
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apply continuous_circleMap
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have {θ} : (circleMap 0 1 θ) ∈ Metric.closedBall (0 : ℂ) 1 := by simp
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simp_rw [s₁ (circleMap 0 1 _) this h₀] 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 : (fun x => ∑ s ∈ (finiteZeros h₁U h₂U h₁f h'₂f).toFinset, (h₁f.order s).toNat * log (Complex.abs (circleMap 0 1 x - ↑s)))
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= ∑ s ∈ (finiteZeros h₁U h₂U h₁f h'₂f).toFinset, (fun x => (h₁f.order s).toNat * log (Complex.abs (circleMap 0 1 x - ↑s))) := by
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funext x
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simp
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rw [this]
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apply IntervalIntegrable.sum
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intro i _
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apply IntervalIntegrable.const_mul
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--have : i.1 ∈ Metric.closedBall (0 : ℂ) 1 := i.2
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--simp at this
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by_cases h₂i : ‖i.1‖ = 1
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-- case pos
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exact int'₂ h₂i
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-- case neg
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--have : i.1 ∈ Metric.ball (0 : ℂ) 1 := by sorry
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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 => log (Complex.abs (circleMap 0 1 x - ↑i))) = log ∘ Complex.abs ∘ (fun x ↦ circleMap 0 1 x - ↑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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conv at h₂i =>
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arg 1
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rw [← ha']
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rw [Complex.norm_eq_abs]
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rw [abs_circleMap_zero 1 x]
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simp
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tauto
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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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have t₁ : (∫ (x : ℝ) in (0)..2 * Real.pi, log ‖F (circleMap 0 1 x)‖) = 2 * Real.pi * log ‖F 0‖ := by
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let logAbsF := fun w ↦ Real.log ‖F w‖
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have t₀ : ∀ z ∈ Metric.closedBall 0 1, HarmonicAt logAbsF z := by
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intro z hz
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apply logabs_of_holomorphicAt_is_harmonic
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apply h'₁F z hz
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exact h₂F z hz
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apply harmonic_meanValue₁ 1 Real.zero_lt_one t₀
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simp_rw [← Complex.norm_eq_abs] at decompose_int_G
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rw [t₁] at decompose_int_G
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conv at decompose_int_G =>
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right
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right
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arg 2
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intro x
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right
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rw [int₃ x.2]
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simp at decompose_int_G
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rw [int_logAbs_f_eq_int_G]
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rw [decompose_int_G]
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rw [h₃F]
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simp_rw [int₀ (ha _)] 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 {l : ℝ} : π⁻¹ * 2⁻¹ * (2 * π * l) = l := by
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calc π⁻¹ * 2⁻¹ * (2 * π * l)
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_ = π⁻¹ * (2⁻¹ * 2) * π * l := by ring
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_ = π⁻¹ * π * l := by ring
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_ = (π⁻¹ * π) * l := by ring
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_ = 1 * l := by
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rw [inv_mul_cancel₀]
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exact pi_ne_zero
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_ = l := by 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_cancel₀ Real.pi_ne_zero]
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rw [this]
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rw [log_mul]
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rw [log_prod]
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simp
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rw [finsum_eq_sum_of_support_subset _ (s := (finiteZeros h₁U h₂U h₁f h'₂f).toFinset)]
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simp
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simp
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intro x ⟨h₁x, _⟩
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simp
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dsimp [AnalyticOn.order] at h₁x
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simp only [Function.mem_support, ne_eq, Nat.cast_eq_zero, not_or] at h₁x
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exact AnalyticAt.supp_order_toNat (AnalyticOn.order.proof_1 h₁f x) h₁x
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--
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intro x hx
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simp at hx
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simp
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intro h₁x
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nth_rw 1 [← h₁x] at h₂f
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tauto
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--
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rw [Finset.prod_ne_zero_iff]
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intro x hx
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simp at hx
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simp
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intro h₁x
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nth_rw 1 [← h₁x] at h₂f
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tauto
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--
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simp
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apply h₂F
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simp
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lemma const_mul_circleMap_zero
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{R θ : ℝ} :
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circleMap 0 R θ = R * circleMap 0 1 θ := by
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rw [circleMap_zero, circleMap_zero]
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simp
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theorem jensen
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{R : ℝ}
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(hR : 0 < R)
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(f : ℂ → ℂ)
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(h₁f : AnalyticOn ℂ f (Metric.closedBall 0 R))
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(h₂f : f 0 ≠ 0) :
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log ‖f 0‖ = -∑ᶠ s, (h₁f.order s).toNat * log (R * ‖s.1‖⁻¹) + (2 * π)⁻¹ * ∫ (x : ℝ) in (0)..(2 * π), log ‖f (circleMap 0 R x)‖ := by
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let ℓ : ℂ ≃L[ℂ] ℂ :=
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{
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toFun := fun x ↦ R * x
|
||||
map_add' := fun x y => DistribSMul.smul_add R x y
|
||||
map_smul' := fun m x => mul_smul_comm m (↑R) x
|
||||
invFun := fun x ↦ R⁻¹ * x
|
||||
left_inv := by
|
||||
intro x
|
||||
simp
|
||||
rw [← mul_assoc, mul_comm, inv_mul_cancel₀, mul_one]
|
||||
simp
|
||||
exact ne_of_gt hR
|
||||
right_inv := by
|
||||
intro x
|
||||
simp
|
||||
rw [← mul_assoc, mul_inv_cancel₀, one_mul]
|
||||
simp
|
||||
exact ne_of_gt hR
|
||||
continuous_toFun := continuous_const_smul R
|
||||
continuous_invFun := continuous_const_smul R⁻¹
|
||||
}
|
||||
|
||||
|
||||
let F := f ∘ ℓ
|
||||
|
||||
have h₁F : AnalyticOn ℂ F (Metric.closedBall 0 1) := by
|
||||
apply AnalyticOn.comp (t := Metric.closedBall 0 R)
|
||||
exact h₁f
|
||||
intro x _
|
||||
apply ℓ.toContinuousLinearMap.analyticAt x
|
||||
|
||||
intro x hx
|
||||
have : ℓ x = R * x := by rfl
|
||||
rw [this]
|
||||
simp
|
||||
simp at hx
|
||||
rw [abs_of_pos hR]
|
||||
calc R * Complex.abs x
|
||||
_ ≤ R * 1 := by exact (mul_le_mul_iff_of_pos_left hR).mpr hx
|
||||
_ = R := by simp
|
||||
|
||||
have h₂F : F 0 ≠ 0 := by
|
||||
dsimp [F]
|
||||
have : ℓ 0 = R * 0 := by rfl
|
||||
rw [this]
|
||||
simpa
|
||||
|
||||
let A := jensen_case_R_eq_one F h₁F h₂F
|
||||
|
||||
dsimp [F] at A
|
||||
have {x : ℂ} : ℓ x = R * x := by rfl
|
||||
repeat
|
||||
simp_rw [this] at A
|
||||
simp at A
|
||||
simp
|
||||
rw [A]
|
||||
simp_rw [← const_mul_circleMap_zero]
|
||||
simp
|
||||
|
||||
let e : (Metric.closedBall (0 : ℂ) 1) → (Metric.closedBall (0 : ℂ) R) := by
|
||||
intro ⟨x, hx⟩
|
||||
have hy : R • x ∈ Metric.closedBall (0 : ℂ) R := by
|
||||
simp
|
||||
simp at hx
|
||||
have : R = |R| := by exact Eq.symm (abs_of_pos hR)
|
||||
rw [← this]
|
||||
norm_num
|
||||
calc R * Complex.abs x
|
||||
_ ≤ R * 1 := by exact (mul_le_mul_iff_of_pos_left hR).mpr hx
|
||||
_ = R := by simp
|
||||
exact ⟨R • x, hy⟩
|
||||
|
||||
let e' : (Metric.closedBall (0 : ℂ) R) → (Metric.closedBall (0 : ℂ) 1) := by
|
||||
intro ⟨x, hx⟩
|
||||
have hy : R⁻¹ • x ∈ Metric.closedBall (0 : ℂ) 1 := by
|
||||
simp
|
||||
simp at hx
|
||||
have : R = |R| := by exact Eq.symm (abs_of_pos hR)
|
||||
rw [← this]
|
||||
norm_num
|
||||
calc R⁻¹ * Complex.abs x
|
||||
_ ≤ R⁻¹ * R := by
|
||||
apply mul_le_mul_of_nonneg_left hx
|
||||
apply inv_nonneg.mpr
|
||||
exact abs_eq_self.mp (id (Eq.symm this))
|
||||
_ = 1 := by
|
||||
apply inv_mul_cancel₀
|
||||
exact Ne.symm (ne_of_lt hR)
|
||||
exact ⟨R⁻¹ • x, hy⟩
|
||||
|
||||
apply finsum_eq_of_bijective e
|
||||
|
||||
|
||||
apply Function.bijective_iff_has_inverse.mpr
|
||||
use e'
|
||||
constructor
|
||||
· apply Function.leftInverse_iff_comp.mpr
|
||||
funext x
|
||||
dsimp only [e, e', id_eq, eq_mp_eq_cast, Function.comp_apply]
|
||||
conv =>
|
||||
left
|
||||
arg 1
|
||||
rw [← smul_assoc, smul_eq_mul]
|
||||
rw [inv_mul_cancel₀ (Ne.symm (ne_of_lt hR))]
|
||||
rw [one_smul]
|
||||
· apply Function.rightInverse_iff_comp.mpr
|
||||
funext x
|
||||
dsimp only [e, e', id_eq, eq_mp_eq_cast, Function.comp_apply]
|
||||
conv =>
|
||||
left
|
||||
arg 1
|
||||
rw [← smul_assoc, smul_eq_mul]
|
||||
rw [mul_inv_cancel₀ (Ne.symm (ne_of_lt hR))]
|
||||
rw [one_smul]
|
||||
|
||||
rfl
|
||||
-- ∀ i ∈ Finset.univ, IntervalIntegrable (fun x => Real.log ‖circleMap 0 1 x - a i‖) MeasureTheory.volume 0 (2 * Real.pi)
|
||||
intro i _
|
||||
apply Continuous.intervalIntegrable
|
||||
apply continuous_iff_continuousAt.2
|
||||
intro x
|
||||
have : (fun x => Real.log ‖circleMap 0 1 x - a i‖) = Real.log ∘ Complex.abs ∘ (fun x ↦ circleMap 0 1 x - a i) :=
|
||||
rfl
|
||||
rw [this]
|
||||
apply ContinuousAt.comp
|
||||
apply Real.continuousAt_log
|
||||
simp
|
||||
by_cases hx : x = (0 : ℂ)
|
||||
rw [hx]
|
||||
by_contra ha'
|
||||
let A := ha i
|
||||
rw [← ha'] at A
|
||||
simp at A
|
||||
apply ContinuousAt.comp
|
||||
apply Complex.continuous_abs.continuousAt
|
||||
fun_prop
|
||||
-- IntervalIntegrable (fun x => logAbsf (circleMap 0 1 x)) MeasureTheory.volume 0 (2 * Real.pi)
|
||||
apply Continuous.intervalIntegrable
|
||||
apply continuous_iff_continuousAt.2
|
||||
intro x
|
||||
have : (fun x => logAbsf (circleMap 0 1 x)) = Real.log ∘ Complex.abs ∘ f ∘ (fun x ↦ circleMap 0 1 x) :=
|
||||
rfl
|
||||
rw [this]
|
||||
apply ContinuousAt.comp
|
||||
simp
|
||||
|
||||
rw [log_mul, log_mul, log_inv, log_inv]
|
||||
have : R = |R| := by exact Eq.symm (abs_of_pos hR)
|
||||
rw [← this]
|
||||
exact h₀
|
||||
apply ContinuousAt.comp
|
||||
apply Complex.continuous_abs.continuousAt
|
||||
apply ContinuousAt.comp
|
||||
apply ContDiffAt.continuousAt (f := f) (𝕜 := ℝ) (n := 1)
|
||||
apply HolomorphicAt.contDiffAt
|
||||
apply h₁f
|
||||
simp
|
||||
left
|
||||
congr 1
|
||||
|
||||
dsimp [AnalyticOn.order]
|
||||
rw [← AnalyticAt.order_comp_CLE ℓ]
|
||||
|
||||
--
|
||||
simpa
|
||||
--
|
||||
have : R = |R| := by exact Eq.symm (abs_of_pos hR)
|
||||
rw [← this]
|
||||
apply inv_ne_zero
|
||||
exact Ne.symm (ne_of_lt hR)
|
||||
--
|
||||
exact Ne.symm (ne_of_lt hR)
|
||||
--
|
||||
let A := continuous_circleMap 0 1
|
||||
apply A.continuousAt
|
||||
-- IntervalIntegrable (fun x => ∑ s : { x // x ∈ S }, Real.log ‖circleMap 0 1 x - a s‖) MeasureTheory.volume 0 (2 * Real.pi)
|
||||
apply Continuous.intervalIntegrable
|
||||
apply continuous_finset_sum
|
||||
intro i _
|
||||
apply continuous_iff_continuousAt.2
|
||||
intro x
|
||||
have : (fun x => Real.log ‖circleMap 0 1 x - a i‖) = Real.log ∘ Complex.abs ∘ (fun x ↦ circleMap 0 1 x - a i) :=
|
||||
rfl
|
||||
rw [this]
|
||||
apply ContinuousAt.comp
|
||||
apply Real.continuousAt_log
|
||||
simp
|
||||
constructor
|
||||
· assumption
|
||||
· exact Ne.symm (ne_of_lt hR)
|
||||
by_contra ha'
|
||||
let A := ha i
|
||||
rw [← ha'] at A
|
||||
simp at A
|
||||
apply ContinuousAt.comp
|
||||
apply Complex.continuous_abs.continuousAt
|
||||
fun_prop
|
||||
|
|
|
|||
|
|
@ -0,0 +1,449 @@
|
|||
import Mathlib.Analysis.Complex.CauchyIntegral
|
||||
import Mathlib.Analysis.Analytic.IsolatedZeros
|
||||
import Nevanlinna.analyticOn_zeroSet
|
||||
import Nevanlinna.harmonicAt_examples
|
||||
import Nevanlinna.harmonicAt_meanValue
|
||||
import Nevanlinna.specialFunctions_CircleIntegral_affine
|
||||
|
||||
open Real
|
||||
|
||||
|
||||
|
||||
theorem jensen_case_R_eq_one
|
||||
(f : ℂ → ℂ)
|
||||
(h₁f : AnalyticOn ℂ f (Metric.closedBall 0 1))
|
||||
(h₂f : f 0 ≠ 0) :
|
||||
log ‖f 0‖ = -∑ᶠ s, (h₁f.order s).toNat * log (‖s.1‖⁻¹) + (2 * π)⁻¹ * ∫ (x : ℝ) in (0)..(2 * π), log ‖f (circleMap 0 1 x)‖ := by
|
||||
|
||||
have h₁U : IsPreconnected (Metric.closedBall (0 : ℂ) 1) :=
|
||||
(convex_closedBall (0 : ℂ) 1).isPreconnected
|
||||
|
||||
have h₂U : IsCompact (Metric.closedBall (0 : ℂ) 1) :=
|
||||
isCompact_closedBall 0 1
|
||||
|
||||
have h'₂f : ∃ u ∈ (Metric.closedBall (0 : ℂ) 1), f u ≠ 0 := by
|
||||
use 0; simp; exact h₂f
|
||||
|
||||
obtain ⟨F, h₁F, h₂F, h₃F⟩ := AnalyticOnCompact.eliminateZeros₂ h₁U h₂U h₁f h'₂f
|
||||
|
||||
have h'₁F : ∀ z ∈ Metric.closedBall (0 : ℂ) 1, HolomorphicAt F z := by
|
||||
intro z h₁z
|
||||
apply AnalyticAt.holomorphicAt
|
||||
exact h₁F z h₁z
|
||||
|
||||
let G := fun z ↦ log ‖F z‖ + ∑ s ∈ (finiteZeros h₁U h₂U h₁f h'₂f).toFinset, (h₁f.order s).toNat * log ‖z - s‖
|
||||
|
||||
have decompose_f : ∀ z ∈ Metric.closedBall (0 : ℂ) 1, f z ≠ 0 → log ‖f z‖ = G z := by
|
||||
intro z h₁z h₂z
|
||||
|
||||
conv =>
|
||||
left
|
||||
arg 1
|
||||
rw [h₃F]
|
||||
rw [smul_eq_mul]
|
||||
rw [norm_mul]
|
||||
rw [norm_prod]
|
||||
left
|
||||
arg 2
|
||||
intro b
|
||||
rw [norm_pow]
|
||||
simp only [Complex.norm_eq_abs, Finset.univ_eq_attach]
|
||||
rw [Real.log_mul]
|
||||
rw [Real.log_prod]
|
||||
conv =>
|
||||
left
|
||||
left
|
||||
arg 2
|
||||
intro s
|
||||
rw [Real.log_pow]
|
||||
dsimp [G]
|
||||
abel
|
||||
|
||||
-- ∀ x ∈ ⋯.toFinset, Complex.abs (z - ↑x) ^ (h'₁f.order x).toNat ≠ 0
|
||||
have : ∀ x ∈ (finiteZeros h₁U h₂U h₁f h'₂f).toFinset, Complex.abs (z - ↑x) ^ (h₁f.order x).toNat ≠ 0 := by
|
||||
intro s hs
|
||||
simp at hs
|
||||
simp
|
||||
intro h₂s
|
||||
rw [h₂s] at h₂z
|
||||
tauto
|
||||
exact this
|
||||
|
||||
-- ∏ x ∈ ⋯.toFinset, Complex.abs (z - ↑x) ^ (h'₁f.order x).toNat ≠ 0
|
||||
have : ∀ x ∈ (finiteZeros h₁U h₂U h₁f h'₂f).toFinset, Complex.abs (z - ↑x) ^ (h₁f.order x).toNat ≠ 0 := by
|
||||
intro s hs
|
||||
simp at hs
|
||||
simp
|
||||
intro h₂s
|
||||
rw [h₂s] at h₂z
|
||||
tauto
|
||||
rw [Finset.prod_ne_zero_iff]
|
||||
exact this
|
||||
|
||||
-- Complex.abs (F z) ≠ 0
|
||||
simp
|
||||
exact h₂F z h₁z
|
||||
|
||||
|
||||
have int_logAbs_f_eq_int_G : ∫ (x : ℝ) in (0)..2 * π, log ‖f (circleMap 0 1 x)‖ = ∫ (x : ℝ) in (0)..2 * π, G (circleMap 0 1 x) := by
|
||||
|
||||
rw [intervalIntegral.integral_congr_ae]
|
||||
rw [MeasureTheory.ae_iff]
|
||||
apply Set.Countable.measure_zero
|
||||
simp
|
||||
|
||||
have t₀ : {a | a ∈ Ι 0 (2 * π) ∧ ¬log ‖f (circleMap 0 1 a)‖ = G (circleMap 0 1 a)}
|
||||
⊆ (circleMap 0 1)⁻¹' (Metric.closedBall 0 1 ∩ f⁻¹' {0}) := by
|
||||
intro a ha
|
||||
simp at ha
|
||||
simp
|
||||
by_contra C
|
||||
have : (circleMap 0 1 a) ∈ Metric.closedBall 0 1 :=
|
||||
circleMap_mem_closedBall 0 (zero_le_one' ℝ) a
|
||||
exact ha.2 (decompose_f (circleMap 0 1 a) this C)
|
||||
|
||||
apply Set.Countable.mono t₀
|
||||
apply Set.Countable.preimage_circleMap
|
||||
apply Set.Finite.countable
|
||||
let A := finiteZeros h₁U h₂U h₁f h'₂f
|
||||
|
||||
have : (Metric.closedBall 0 1 ∩ f ⁻¹' {0}) = (Metric.closedBall 0 1).restrict f ⁻¹' {0} := by
|
||||
ext z
|
||||
simp
|
||||
tauto
|
||||
rw [this]
|
||||
exact Set.Finite.image Subtype.val A
|
||||
exact Ne.symm (zero_ne_one' ℝ)
|
||||
|
||||
|
||||
have decompose_int_G : ∫ (x : ℝ) in (0)..2 * π, G (circleMap 0 1 x)
|
||||
= (∫ (x : ℝ) in (0)..2 * π, log (Complex.abs (F (circleMap 0 1 x))))
|
||||
+ ∑ x ∈ (finiteZeros h₁U h₂U h₁f h'₂f).toFinset, (h₁f.order x).toNat * ∫ (x_1 : ℝ) in (0)..2 * π, log (Complex.abs (circleMap 0 1 x_1 - ↑x)) := by
|
||||
dsimp [G]
|
||||
rw [intervalIntegral.integral_add]
|
||||
rw [intervalIntegral.integral_finset_sum]
|
||||
simp_rw [intervalIntegral.integral_const_mul]
|
||||
|
||||
-- ∀ i ∈ (finiteZeros h₁U h₂U h'₁f h'₂f).toFinset,
|
||||
-- IntervalIntegrable (fun x => (h'₁f.order i).toNat *
|
||||
-- log (Complex.abs (circleMap 0 1 x - ↑i))) MeasureTheory.volume 0 (2 * π)
|
||||
intro i _
|
||||
apply IntervalIntegrable.const_mul
|
||||
--simp at this
|
||||
by_cases h₂i : ‖i.1‖ = 1
|
||||
-- case pos
|
||||
exact int'₂ h₂i
|
||||
-- case neg
|
||||
apply Continuous.intervalIntegrable
|
||||
apply continuous_iff_continuousAt.2
|
||||
intro x
|
||||
have : (fun x => log (Complex.abs (circleMap 0 1 x - ↑i))) = log ∘ Complex.abs ∘ (fun x ↦ circleMap 0 1 x - ↑i) :=
|
||||
rfl
|
||||
rw [this]
|
||||
apply ContinuousAt.comp
|
||||
apply Real.continuousAt_log
|
||||
simp
|
||||
|
||||
by_contra ha'
|
||||
conv at h₂i =>
|
||||
arg 1
|
||||
rw [← ha']
|
||||
rw [Complex.norm_eq_abs]
|
||||
rw [abs_circleMap_zero 1 x]
|
||||
simp
|
||||
tauto
|
||||
apply ContinuousAt.comp
|
||||
apply Complex.continuous_abs.continuousAt
|
||||
fun_prop
|
||||
-- IntervalIntegrable (fun x => log (Complex.abs (F (circleMap 0 1 x)))) MeasureTheory.volume 0 (2 * π)
|
||||
apply Continuous.intervalIntegrable
|
||||
apply continuous_iff_continuousAt.2
|
||||
intro x
|
||||
have : (fun x => log (Complex.abs (F (circleMap 0 1 x)))) = log ∘ Complex.abs ∘ F ∘ (fun x ↦ circleMap 0 1 x) :=
|
||||
rfl
|
||||
rw [this]
|
||||
apply ContinuousAt.comp
|
||||
apply Real.continuousAt_log
|
||||
simp [h₂F]
|
||||
-- ContinuousAt (⇑Complex.abs ∘ F ∘ fun x => circleMap 0 1 x) x
|
||||
apply ContinuousAt.comp
|
||||
apply Complex.continuous_abs.continuousAt
|
||||
apply ContinuousAt.comp
|
||||
apply DifferentiableAt.continuousAt (𝕜 := ℂ )
|
||||
apply HolomorphicAt.differentiableAt
|
||||
simp [h'₁F]
|
||||
-- ContinuousAt (fun x => circleMap 0 1 x) x
|
||||
apply Continuous.continuousAt
|
||||
apply continuous_circleMap
|
||||
|
||||
have : (fun x => ∑ s ∈ (finiteZeros h₁U h₂U h₁f h'₂f).toFinset, (h₁f.order s).toNat * log (Complex.abs (circleMap 0 1 x - ↑s)))
|
||||
= ∑ s ∈ (finiteZeros h₁U h₂U h₁f h'₂f).toFinset, (fun x => (h₁f.order s).toNat * log (Complex.abs (circleMap 0 1 x - ↑s))) := by
|
||||
funext x
|
||||
simp
|
||||
rw [this]
|
||||
apply IntervalIntegrable.sum
|
||||
intro i _
|
||||
apply IntervalIntegrable.const_mul
|
||||
--have : i.1 ∈ Metric.closedBall (0 : ℂ) 1 := i.2
|
||||
--simp at this
|
||||
by_cases h₂i : ‖i.1‖ = 1
|
||||
-- case pos
|
||||
exact int'₂ h₂i
|
||||
-- case neg
|
||||
--have : i.1 ∈ Metric.ball (0 : ℂ) 1 := by sorry
|
||||
apply Continuous.intervalIntegrable
|
||||
apply continuous_iff_continuousAt.2
|
||||
intro x
|
||||
have : (fun x => log (Complex.abs (circleMap 0 1 x - ↑i))) = log ∘ Complex.abs ∘ (fun x ↦ circleMap 0 1 x - ↑i) :=
|
||||
rfl
|
||||
rw [this]
|
||||
apply ContinuousAt.comp
|
||||
apply Real.continuousAt_log
|
||||
simp
|
||||
|
||||
by_contra ha'
|
||||
conv at h₂i =>
|
||||
arg 1
|
||||
rw [← ha']
|
||||
rw [Complex.norm_eq_abs]
|
||||
rw [abs_circleMap_zero 1 x]
|
||||
simp
|
||||
tauto
|
||||
apply ContinuousAt.comp
|
||||
apply Complex.continuous_abs.continuousAt
|
||||
fun_prop
|
||||
|
||||
have t₁ : (∫ (x : ℝ) in (0)..2 * Real.pi, log ‖F (circleMap 0 1 x)‖) = 2 * Real.pi * log ‖F 0‖ := by
|
||||
let logAbsF := fun w ↦ Real.log ‖F w‖
|
||||
have t₀ : ∀ z ∈ Metric.closedBall 0 1, HarmonicAt logAbsF z := by
|
||||
intro z hz
|
||||
apply logabs_of_holomorphicAt_is_harmonic
|
||||
apply h'₁F z hz
|
||||
exact h₂F z hz
|
||||
|
||||
apply harmonic_meanValue₁ 1 Real.zero_lt_one t₀
|
||||
|
||||
simp_rw [← Complex.norm_eq_abs] at decompose_int_G
|
||||
rw [t₁] at decompose_int_G
|
||||
|
||||
conv at decompose_int_G =>
|
||||
right
|
||||
right
|
||||
arg 2
|
||||
intro x
|
||||
right
|
||||
rw [int₃ x.2]
|
||||
simp at decompose_int_G
|
||||
|
||||
rw [int_logAbs_f_eq_int_G]
|
||||
rw [decompose_int_G]
|
||||
rw [h₃F]
|
||||
simp
|
||||
have {l : ℝ} : π⁻¹ * 2⁻¹ * (2 * π * l) = l := by
|
||||
calc π⁻¹ * 2⁻¹ * (2 * π * l)
|
||||
_ = π⁻¹ * (2⁻¹ * 2) * π * l := by ring
|
||||
_ = π⁻¹ * π * l := by ring
|
||||
_ = (π⁻¹ * π) * l := by ring
|
||||
_ = 1 * l := by
|
||||
rw [inv_mul_cancel₀]
|
||||
exact pi_ne_zero
|
||||
_ = l := by simp
|
||||
rw [this]
|
||||
rw [log_mul]
|
||||
rw [log_prod]
|
||||
simp
|
||||
|
||||
rw [finsum_eq_sum_of_support_subset _ (s := (finiteZeros h₁U h₂U h₁f h'₂f).toFinset)]
|
||||
simp
|
||||
simp
|
||||
intro x ⟨h₁x, _⟩
|
||||
simp
|
||||
|
||||
dsimp [AnalyticOn.order] at h₁x
|
||||
simp only [Function.mem_support, ne_eq, Nat.cast_eq_zero, not_or] at h₁x
|
||||
exact AnalyticAt.supp_order_toNat (AnalyticOn.order.proof_1 h₁f x) h₁x
|
||||
|
||||
--
|
||||
intro x hx
|
||||
simp at hx
|
||||
simp
|
||||
intro h₁x
|
||||
nth_rw 1 [← h₁x] at h₂f
|
||||
tauto
|
||||
|
||||
--
|
||||
rw [Finset.prod_ne_zero_iff]
|
||||
intro x hx
|
||||
simp at hx
|
||||
simp
|
||||
intro h₁x
|
||||
nth_rw 1 [← h₁x] at h₂f
|
||||
tauto
|
||||
|
||||
--
|
||||
simp
|
||||
apply h₂F
|
||||
simp
|
||||
|
||||
|
||||
lemma const_mul_circleMap_zero
|
||||
{R θ : ℝ} :
|
||||
circleMap 0 R θ = R * circleMap 0 1 θ := by
|
||||
rw [circleMap_zero, circleMap_zero]
|
||||
simp
|
||||
|
||||
|
||||
theorem jensen
|
||||
{R : ℝ}
|
||||
(hR : 0 < R)
|
||||
(f : ℂ → ℂ)
|
||||
(h₁f : AnalyticOn ℂ f (Metric.closedBall 0 R))
|
||||
(h₂f : f 0 ≠ 0) :
|
||||
log ‖f 0‖ = -∑ᶠ s, (h₁f.order s).toNat * log (R * ‖s.1‖⁻¹) + (2 * π)⁻¹ * ∫ (x : ℝ) in (0)..(2 * π), log ‖f (circleMap 0 R x)‖ := by
|
||||
|
||||
|
||||
let ℓ : ℂ ≃L[ℂ] ℂ :=
|
||||
{
|
||||
toFun := fun x ↦ R * x
|
||||
map_add' := fun x y => DistribSMul.smul_add R x y
|
||||
map_smul' := fun m x => mul_smul_comm m (↑R) x
|
||||
invFun := fun x ↦ R⁻¹ * x
|
||||
left_inv := by
|
||||
intro x
|
||||
simp
|
||||
rw [← mul_assoc, mul_comm, inv_mul_cancel₀, mul_one]
|
||||
simp
|
||||
exact ne_of_gt hR
|
||||
right_inv := by
|
||||
intro x
|
||||
simp
|
||||
rw [← mul_assoc, mul_inv_cancel₀, one_mul]
|
||||
simp
|
||||
exact ne_of_gt hR
|
||||
continuous_toFun := continuous_const_smul R
|
||||
continuous_invFun := continuous_const_smul R⁻¹
|
||||
}
|
||||
|
||||
|
||||
let F := f ∘ ℓ
|
||||
|
||||
have h₁F : AnalyticOn ℂ F (Metric.closedBall 0 1) := by
|
||||
apply AnalyticOn.comp (t := Metric.closedBall 0 R)
|
||||
exact h₁f
|
||||
intro x _
|
||||
apply ℓ.toContinuousLinearMap.analyticAt x
|
||||
|
||||
intro x hx
|
||||
have : ℓ x = R * x := by rfl
|
||||
rw [this]
|
||||
simp
|
||||
simp at hx
|
||||
rw [abs_of_pos hR]
|
||||
calc R * Complex.abs x
|
||||
_ ≤ R * 1 := by exact (mul_le_mul_iff_of_pos_left hR).mpr hx
|
||||
_ = R := by simp
|
||||
|
||||
have h₂F : F 0 ≠ 0 := by
|
||||
dsimp [F]
|
||||
have : ℓ 0 = R * 0 := by rfl
|
||||
rw [this]
|
||||
simpa
|
||||
|
||||
let A := jensen_case_R_eq_one F h₁F h₂F
|
||||
|
||||
dsimp [F] at A
|
||||
have {x : ℂ} : ℓ x = R * x := by rfl
|
||||
repeat
|
||||
simp_rw [this] at A
|
||||
simp at A
|
||||
simp
|
||||
rw [A]
|
||||
simp_rw [← const_mul_circleMap_zero]
|
||||
simp
|
||||
|
||||
let e : (Metric.closedBall (0 : ℂ) 1) → (Metric.closedBall (0 : ℂ) R) := by
|
||||
intro ⟨x, hx⟩
|
||||
have hy : R • x ∈ Metric.closedBall (0 : ℂ) R := by
|
||||
simp
|
||||
simp at hx
|
||||
have : R = |R| := by exact Eq.symm (abs_of_pos hR)
|
||||
rw [← this]
|
||||
norm_num
|
||||
calc R * Complex.abs x
|
||||
_ ≤ R * 1 := by exact (mul_le_mul_iff_of_pos_left hR).mpr hx
|
||||
_ = R := by simp
|
||||
exact ⟨R • x, hy⟩
|
||||
|
||||
let e' : (Metric.closedBall (0 : ℂ) R) → (Metric.closedBall (0 : ℂ) 1) := by
|
||||
intro ⟨x, hx⟩
|
||||
have hy : R⁻¹ • x ∈ Metric.closedBall (0 : ℂ) 1 := by
|
||||
simp
|
||||
simp at hx
|
||||
have : R = |R| := by exact Eq.symm (abs_of_pos hR)
|
||||
rw [← this]
|
||||
norm_num
|
||||
calc R⁻¹ * Complex.abs x
|
||||
_ ≤ R⁻¹ * R := by
|
||||
apply mul_le_mul_of_nonneg_left hx
|
||||
apply inv_nonneg.mpr
|
||||
exact abs_eq_self.mp (id (Eq.symm this))
|
||||
_ = 1 := by
|
||||
apply inv_mul_cancel₀
|
||||
exact Ne.symm (ne_of_lt hR)
|
||||
exact ⟨R⁻¹ • x, hy⟩
|
||||
|
||||
apply finsum_eq_of_bijective e
|
||||
|
||||
|
||||
apply Function.bijective_iff_has_inverse.mpr
|
||||
use e'
|
||||
constructor
|
||||
· apply Function.leftInverse_iff_comp.mpr
|
||||
funext x
|
||||
dsimp only [e, e', id_eq, eq_mp_eq_cast, Function.comp_apply]
|
||||
conv =>
|
||||
left
|
||||
arg 1
|
||||
rw [← smul_assoc, smul_eq_mul]
|
||||
rw [inv_mul_cancel₀ (Ne.symm (ne_of_lt hR))]
|
||||
rw [one_smul]
|
||||
· apply Function.rightInverse_iff_comp.mpr
|
||||
funext x
|
||||
dsimp only [e, e', id_eq, eq_mp_eq_cast, Function.comp_apply]
|
||||
conv =>
|
||||
left
|
||||
arg 1
|
||||
rw [← smul_assoc, smul_eq_mul]
|
||||
rw [mul_inv_cancel₀ (Ne.symm (ne_of_lt hR))]
|
||||
rw [one_smul]
|
||||
|
||||
intro x
|
||||
simp
|
||||
by_cases hx : x = (0 : ℂ)
|
||||
rw [hx]
|
||||
simp
|
||||
|
||||
rw [log_mul, log_mul, log_inv, log_inv]
|
||||
have : R = |R| := by exact Eq.symm (abs_of_pos hR)
|
||||
rw [← this]
|
||||
simp
|
||||
left
|
||||
congr 1
|
||||
|
||||
dsimp [AnalyticOn.order]
|
||||
rw [← AnalyticAt.order_comp_CLE ℓ]
|
||||
|
||||
--
|
||||
simpa
|
||||
--
|
||||
have : R = |R| := by exact Eq.symm (abs_of_pos hR)
|
||||
rw [← this]
|
||||
apply inv_ne_zero
|
||||
exact Ne.symm (ne_of_lt hR)
|
||||
--
|
||||
exact Ne.symm (ne_of_lt hR)
|
||||
--
|
||||
simp
|
||||
constructor
|
||||
· assumption
|
||||
· exact Ne.symm (ne_of_lt hR)
|
||||
Loading…
Reference in New Issue