2024-08-12 13:05:55 +02:00
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