From 8b0d0f5c051ff55b4af2606767f65d391afb3400 Mon Sep 17 00:00:00 2001
From: Stefan Kebekus <kebekus@users.noreply.github.com>
Date: Mon, 12 Aug 2024 13:05:55 +0200
Subject: [PATCH] Update holomorphic_JensenFormula2.lean

---
 Nevanlinna/holomorphic_JensenFormula2.lean | 29 ++++++++++++++++++++++
 1 file changed, 29 insertions(+)

diff --git a/Nevanlinna/holomorphic_JensenFormula2.lean b/Nevanlinna/holomorphic_JensenFormula2.lean
index e69de29..e413143 100644
--- a/Nevanlinna/holomorphic_JensenFormula2.lean
+++ b/Nevanlinna/holomorphic_JensenFormula2.lean
@@ -0,0 +1,29 @@
+import Mathlib.Analysis.Complex.CauchyIntegral
+import Nevanlinna.harmonicAt_examples
+import Nevanlinna.harmonicAt_meanValue
+import Mathlib.Analysis.Analytic.IsolatedZeros
+
+
+lemma xx
+  {f : ℂ → ℂ}
+  {S : Set ℂ}
+  {R : ℝ}
+  (h₁ : DifferentiableOn ℂ f (Metric.ball z₀ R)) :
+  ∃ o : ℂ → ℕ, ∃ F : ℂ → ℂ, ∀ z ∈ (Metric.ball z₀ R), (DifferentiableAt ℂ F z) ∧ (F z ≠ 0) ∧ (f z = F z * ∏ᶠ s ∈ (Metric.ball z₀ R), (z - s) ^ (o s)) := by
+  sorry
+
+
+theorem jensen_case_R_eq_one'
+  (f : ℂ → ℂ)
+  (h₁f : Differentiable ℂ f)
+  (h₂f : f 0 ≠ 0)
+  (S : Finset ℕ)
+  (a : S → ℂ)
+  (ha : ∀ s, a s ∈ Metric.ball 0 1)
+  (F : ℂ → ℂ)
+  (h₁F : Differentiable ℂ F)
+  (h₂F : ∀ z, F z ≠ 0)
+  (h₃F : f = fun z ↦ (F z) * ∏ s : S, (z - a s))
+  :
+  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
+  sorry