nevanlinna/Nevanlinna/holomorphic_JensenFormula2....

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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 : }
{S : Set }
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(h₁S : IsPreconnected S)
(h₂S : IsCompact S)
(hf : ∀ s ∈ S, AnalyticAt f s) :
∃ o : , ∃ F : , ∀ z ∈ S, (AnalyticAt F z) ∧ (F z ≠ 0) ∧ (f z = F z * ∏ᶠ s ∈ S, (z - s) ^ (o s)) := by
let o : := by
intro z
if hz : z ∈ S then
let A := hf z hz
let B := A.order
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exact (A.order : )
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else
exact 0
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sorry
theorem jensen_case_R_eq_one'
(f : )
(h₁f : Differentiable f)
(h₂f : f 0 ≠ 0)
(S : Finset )
(a : S → )
(ha : ∀ s, a s ∈ Metric.ball 0 1)
(F : )
(h₁F : Differentiable 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 : ) in (0)..2 * Real.pi, Real.log ‖f (circleMap 0 1 x)‖ := by
sorry