Update holomorphic_JensenFormula2.lean

Nevanlinna/holomorphic_JensenFormula2.lean

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+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