424 lines
12 KiB
Plaintext
424 lines
12 KiB
Plaintext
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
|
||
|
||
|
||
sorry
|