import Nevanlinna.specialFunctions_CircleIntegral_affine import Nevanlinna.stronglyMeromorphicOn_eliminate open Real theorem jensen₀ {R : ℝ} (hR : 0 < R) (f : ℂ → ℂ) -- WARNING: Not needed. MeromorphicOn suffices (h₁f : StronglyMeromorphicOn f (Metric.closedBall 0 R)) (h₂f : f 0 ≠ 0) : log ‖f 0‖ = -∑ᶠ s, (h₁f.meromorphicOn.divisor s) * log (R * ‖s‖⁻¹) + (2 * π)⁻¹ * ∫ (x : ℝ) in (0)..(2 * π), log ‖f (circleMap 0 R x)‖ := by have h₁U : IsConnected (Metric.closedBall (0 : ℂ) R) := by constructor · apply Metric.nonempty_closedBall.mpr exact le_of_lt hR · exact (convex_closedBall (0 : ℂ) R).isPreconnected have h₂U : IsCompact (Metric.closedBall (0 : ℂ) R) := isCompact_closedBall 0 R have h'₂f : ∃ u : (Metric.closedBall (0 : ℂ) R), f u ≠ 0 := by use ⟨0, Metric.mem_closedBall_self (le_of_lt hR)⟩ have h'₁f : StronglyMeromorphicAt f 0 := by apply h₁f simp exact le_of_lt hR have h''₂f : h'₁f.meromorphicAt.order = 0 := by rwa [h'₁f.order_eq_zero_iff] have h'''₂f : h₁f.meromorphicOn.divisor 0 = 0 := by unfold MeromorphicOn.divisor simp tauto have h₃f : Set.Finite (Function.support h₁f.meromorphicOn.divisor) := by exact Divisor.finiteSupport h₂U (StronglyMeromorphicOn.meromorphicOn h₁f).divisor have h'₃f : ∀ s ∈ h₃f.toFinset, s ≠ 0 := by by_contra hCon push_neg at hCon obtain ⟨s, h₁s, h₂s⟩ := hCon rw [h₂s] at h₁s simp at h₁s tauto have h₄f: Function.support (fun s ↦ (h₁f.meromorphicOn.divisor s) * log (R * ‖s‖⁻¹)) ⊆ h₃f.toFinset := by intro x contrapose simp intro hx rw [hx] simp rw [finsum_eq_sum_of_support_subset _ h₄f] obtain ⟨F, h₁F, h₂F, h₃F, h₄F⟩ := MeromorphicOn.decompose₃' h₂U h₁U h₁f h'₂f have h₁F : Function.mulSupport (fun u ↦ fun z => (z - u) ^ (h₁f.meromorphicOn.divisor u)) ⊆ h₃f.toFinset := by intro u contrapose simp intro hu rw [hu] simp exact rfl rw [finprod_eq_prod_of_mulSupport_subset _ h₁F] at h₄F let G := fun z ↦ log ‖F z‖ + ∑ᶠ s, (h₁f.meromorphicOn.divisor s) * log ‖z - s‖ have h₁G {z : ℂ} : Function.support (fun s ↦ (h₁f.meromorphicOn.divisor s) * log ‖z - s‖) ⊆ h₃f.toFinset := by intro s contrapose simp intro hs rw [hs] simp have decompose_f : ∀ z ∈ Metric.closedBall (0 : ℂ) R, f z ≠ 0 → log ‖f z‖ = G z := by intro z h₁z h₂z rw [h₄F] simp only [Pi.mul_apply, norm_mul] simp only [Finset.prod_apply, norm_prod, norm_zpow] rw [Real.log_mul] rw [Real.log_prod] simp_rw [Real.log_zpow] dsimp only [G] rw [finsum_eq_sum_of_support_subset _ h₁G] -- intro x hx have : z ≠ x := by by_contra hCon rw [← hCon] at hx simp at hx rw [← (h₁f z h₁z).order_eq_zero_iff] at h₂z unfold MeromorphicOn.divisor at hx simp [h₁z] at hx tauto apply zpow_ne_zero simpa -- Complex.abs (F z) ≠ 0 simp exact h₃F ⟨z, h₁z⟩ -- rw [Finset.prod_ne_zero_iff] intro x hx have : z ≠ x := by by_contra hCon rw [← hCon] at hx simp at hx rw [← (h₁f z h₁z).order_eq_zero_iff] at h₂z unfold MeromorphicOn.divisor at hx simp [h₁z] at hx tauto apply zpow_ne_zero simpa have int_logAbs_f_eq_int_G : ∫ (x : ℝ) in (0)..2 * π, log ‖f (circleMap 0 R x)‖ = ∫ (x : ℝ) in (0)..2 * π, G (circleMap 0 R 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 R a)‖ = G (circleMap 0 R a)} ⊆ (circleMap 0 R)⁻¹' (h₃f.toFinset) := by intro a ha simp at ha simp by_contra C have t₀ : (circleMap 0 R a) ∈ Metric.closedBall 0 R := by apply circleMap_mem_closedBall exact le_of_lt hR have t₁ : f (circleMap 0 R a) ≠ 0 := by let A := h₁f (circleMap 0 R a) t₀ rw [← A.order_eq_zero_iff] unfold MeromorphicOn.divisor at C simp [t₀] at C rcases C with C₁|C₂ · assumption · let B := h₁f.meromorphicOn.order_ne_top' h₁U let C := fun q ↦ B.1 q ⟨(circleMap 0 R a), t₀⟩ rw [C₂] at C have : ∃ u : (Metric.closedBall (0 : ℂ) R), (h₁f u u.2).meromorphicAt.order ≠ ⊤ := by use ⟨(0 : ℂ), (by simp; exact le_of_lt hR)⟩ let H := h₁f 0 (by simp; exact le_of_lt hR) let K := H.order_eq_zero_iff.2 h₂f rw [K] simp let D := C this tauto exact ha.2 (decompose_f (circleMap 0 R a) t₀ t₁) apply Set.Countable.mono t₀ apply Set.Countable.preimage_circleMap apply Set.Finite.countable exact Finset.finite_toSet h₃f.toFinset -- exact Ne.symm (ne_of_lt hR) have decompose_int_G : ∫ (x : ℝ) in (0)..2 * π, G (circleMap 0 R x) = (∫ (x : ℝ) in (0)..2 * π, log (Complex.abs (F (circleMap 0 R x)))) + ∑ᶠ x, (h₁f.meromorphicOn.divisor x) * ∫ (x_1 : ℝ) in (0)..2 * π, log (Complex.abs (circleMap 0 R x_1 - ↑x)) := by dsimp [G] rw [intervalIntegral.integral_add] congr have t₀ {x : ℝ} : Function.support (fun s ↦ (h₁f.meromorphicOn.divisor s) * log (Complex.abs (circleMap 0 R x - s))) ⊆ h₃f.toFinset := by intro s hs simp at hs simp [hs.1] conv => left arg 1 intro x rw [finsum_eq_sum_of_support_subset _ t₀] rw [intervalIntegral.integral_finset_sum] let G' := fun x ↦ ((h₁f.meromorphicOn.divisor x) : ℂ) * ∫ (x_1 : ℝ) in (0)..2 * π, log (Complex.abs (circleMap 0 R x_1 - x)) have t₁ : (Function.support fun x ↦ (h₁f.meromorphicOn.divisor x) * ∫ (x_1 : ℝ) in (0)..2 * π, log (Complex.abs (circleMap 0 R x_1 - x))) ⊆ h₃f.toFinset := by simp intro s contrapose! simp tauto conv => right rw [finsum_eq_sum_of_support_subset _ t₁] simp -- ∀ 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 apply intervalIntegrable_logAbs_circleMap_sub_const linarith -- -- case neg apply Continuous.intervalIntegrable apply continuous_iff_continuousAt.2 intro x have : (fun x => log (Complex.abs ( F (circleMap 0 R x)))) = log ∘ Complex.abs ∘ F ∘ circleMap 0 R := rfl rw [this] apply ContinuousAt.comp apply Real.continuousAt_log simp let A := h₃F ⟨(circleMap 0 R x), circleMap_mem_closedBall 0 (le_of_lt hR) x⟩ exact A -- apply ContinuousAt.comp apply Complex.continuous_abs.continuousAt apply ContinuousAt.comp let A := h₂F (circleMap 0 R x) (circleMap_mem_closedBall 0 (le_of_lt hR) x) apply A.continuousAt exact (continuous_circleMap 0 R).continuousAt -- IntervalIntegrable (fun x => ∑ᶠ (s : ℂ), ↑(↑⋯.divisor s) * log (Complex.abs (circleMap 0 1 x - s))) MeasureTheory.volume 0 (2 * π) --simp? at h₁G have h₁G' {z : ℂ} : (Function.support fun s => (h₁f.meromorphicOn.divisor s) * log (Complex.abs (z - s))) ⊆ ↑h₃f.toFinset := by exact h₁G conv => arg 1 intro z rw [finsum_eq_sum_of_support_subset _ h₁G'] conv => arg 1 rw [← Finset.sum_fn] 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‖ = R -- case pos --exact int'₂ h₂i apply intervalIntegrable_logAbs_circleMap_sub_const (Ne.symm (ne_of_lt hR)) -- case neg apply Continuous.intervalIntegrable apply continuous_iff_continuousAt.2 intro x have : (fun x => log (Complex.abs (circleMap 0 R x - ↑i))) = log ∘ Complex.abs ∘ (fun x ↦ circleMap 0 R 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 R x] simp linarith apply ContinuousAt.comp apply Complex.continuous_abs.continuousAt fun_prop have t₁ : (∫ (x : ℝ) in (0)..2 * Real.pi, log ‖F (circleMap 0 R x)‖) = 2 * Real.pi * log ‖F 0‖ := by let logAbsF := fun w ↦ Real.log ‖F w‖ have t₀ : ∀ z ∈ Metric.closedBall 0 R, HarmonicAt logAbsF z := by intro z hz apply logabs_of_holomorphicAt_is_harmonic exact AnalyticAt.holomorphicAt (h₂F z hz) exact h₃F ⟨z, hz⟩ apply harmonic_meanValue₁ R hR t₀ simp_rw [← Complex.norm_eq_abs] at decompose_int_G rw [t₁] at decompose_int_G have h₁G' : (Function.support fun s => (h₁f.meromorphicOn.divisor s) * ∫ (x_1 : ℝ) in (0)..(2 * π), log ‖circleMap 0 R x_1 - s‖) ⊆ ↑h₃f.toFinset := by intro s hs simp at hs simp [hs.1] rw [finsum_eq_sum_of_support_subset _ h₁G'] at decompose_int_G have : ∑ s ∈ h₃f.toFinset, (h₁f.meromorphicOn.divisor s) * ∫ (x_1 : ℝ) in (0)..(2 * π), log ‖circleMap 0 R x_1 - s‖ = ∑ s ∈ h₃f.toFinset, (h₁f.meromorphicOn.divisor s) * (2 * π) * log R := by apply Finset.sum_congr rfl intro s hs have : s ∈ Metric.closedBall 0 R := by let A := h₁f.meromorphicOn.divisor.supportInU have : s ∈ Function.support h₁f.meromorphicOn.divisor := by simp at hs exact hs exact A this rw [int₄ hR this] linarith rw [this] at decompose_int_G simp at decompose_int_G rw [int_logAbs_f_eq_int_G] rw [decompose_int_G] let X := h₄F nth_rw 1 [h₄F] simp have : π⁻¹ * 2⁻¹ * (2 * π) = 1 := by calc π⁻¹ * 2⁻¹ * (2 * π) _ = π⁻¹ * (2⁻¹ * 2) * π := by ring _ = π⁻¹ * π := by ring _ = (π⁻¹ * π) := by ring _ = 1 := by rw [inv_mul_cancel₀] exact pi_ne_zero --rw [this] rw [log_mul] rw [log_prod] simp rw [add_comm] rw [mul_add] rw [← mul_assoc (π⁻¹ * 2⁻¹), this] simp rw [add_comm] nth_rw 2 [add_comm] rw [add_assoc] congr rw [Finset.mul_sum] rw [← sub_eq_add_neg] rw [← Finset.sum_sub_distrib] rw [Finset.sum_congr rfl] intro s hs rw [log_mul, log_inv] rw [← mul_assoc (π⁻¹ * 2⁻¹)] rw [mul_comm _ (2 * π)] rw [← mul_assoc (π⁻¹ * 2⁻¹)] rw [this] simp rw [mul_add] ring -- exact Ne.symm (ne_of_lt hR) -- simp by_contra hCon rw [hCon] at hs simp at hs exact hs h'''₂f -- intro s hs apply zpow_ne_zero simp by_contra hCon rw [hCon] at hs simp at hs exact hs h'''₂f -- simp only [ne_eq, map_eq_zero] rw [← ne_eq] exact h₃F ⟨0, (by simp; exact le_of_lt hR)⟩ -- rw [Finset.prod_ne_zero_iff] intro s hs apply zpow_ne_zero simp by_contra hCon rw [hCon] at hs simp at hs exact hs h'''₂f theorem jensen {R : ℝ} (hR : 0 < R) (f : ℂ → ℂ) (h₁f : MeromorphicOn f (Metric.closedBall 0 R)) (h₁f' : StronglyMeromorphicAt f 0) (h₂f : f 0 ≠ 0) : log ‖f 0‖ = -∑ᶠ s, (h₁f.divisor s) * log (R * ‖s‖⁻¹) + (2 * π)⁻¹ * ∫ (x : ℝ) in (0)..(2 * π), log ‖f (circleMap 0 R x)‖ := by let F := h₁f.makeStronglyMeromorphicOn have : F 0 = f 0 := by unfold F have : 0 ∈ (Metric.closedBall 0 R) := by simp [hR] exact le_of_lt hR unfold MeromorphicOn.makeStronglyMeromorphicOn simp [this] intro h₁R let A := StronglyMeromorphicAt.makeStronglyMeromorphic_id h₁f' simp_rw [A] rw [← this] rw [← this] at h₂f clear this have h₁F := stronglyMeromorphicOn_of_makeStronglyMeromorphicOn h₁f rw [jensen₀ hR F h₁F h₂f] rw [h₁f.divisor_of_makeStronglyMeromorphicOn] congr 2 have {x : ℝ} : log ‖F (circleMap 0 R x)‖ = (fun z ↦ log ‖F z‖) (circleMap 0 R x) := by rfl conv => left arg 1 intro x rw [this] have {x : ℝ} : log ‖f (circleMap 0 R x)‖ = (fun z ↦ log ‖f z‖) (circleMap 0 R x) := by rfl conv => right arg 1 intro x rw [this] have h'R : R ≠ 0 := by exact Ne.symm (ne_of_lt hR) have hU : Metric.sphere (0 : ℂ) |R| ⊆ (Metric.closedBall (0 : ℂ) R) := by have : R = |R| := by exact Eq.symm (abs_of_pos hR) nth_rw 2 [this] exact Metric.sphere_subset_closedBall let A := integral_congr_changeDiscrete h'R hU (f₁ := fun z ↦ log ‖F z‖) (f₂ := fun z ↦ log ‖f z‖) apply A clear A rw [Filter.eventuallyEq_iff_exists_mem] have A := makeStronglyMeromorphicOn_changeDiscrete'' h₁f rw [Filter.eventuallyEq_iff_exists_mem] at A obtain ⟨s, h₁s, h₂s⟩ := A use s constructor · exact h₁s · intro x hx let A := h₂s hx simp rw [A]