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