Theorems · Theorem · complex analysis

ValueDistribution.logCounting_sum_top_le

∀ {𝕜 : Type u_1} [inst : NontriviallyNormedField 𝕜] [inst_1 : ProperSpace 𝕜] {E : Type u_2}
  [inst_2 : NormedAddCommGroup E] [inst_3 : NormedSpace 𝕜 E] {α : Type u_3} (s : Finset α) (f : α → 𝕜 → E) {r : ℝ},
  (∀ a ∈ s, Meromorphic (f a)) →
    1 ≤ r → ValueDistribution.logCounting (∑ a ∈ s, f a) ⊤ r ≤ (∑ a ∈ s, ValueDistribution.logCounting (f a) ⊤) r

For 1 ≤ r, the logarithmic counting function for the poles of a sum ∑ a ∈ s, f a is less than or equal to the sum of the logarithmic counting functions for the poles of the f ·.

Defined in

Mathlib.Analysis.Complex.ValueDistribution.LogCounting.Basic

Cited by

2 results in Mathlib

Foundations

Depth 212 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NontriviallyNormedFieldProperSpaceNormedAddCommGroupNormedSpace