Theorems · Theorem · real analysis
Real.harm_mean_le_geom_mean_weighted
∀ {ι : Type u} (s : Finset ι) (w z : ι → ℝ),
s.Nonempty → (∀ i ∈ s, 0 < w i) → ∑ i ∈ s, w i = 1 → (∀ i ∈ s, 0 < z i) → (∑ i ∈ s, w i / z i)⁻¹ ≤ ∏ i ∈ s, z i ^ w iHM-GM inequality: The harmonic mean is less than or equal to the geometric mean, weighted version for real-valued nonnegative functions.
- Defined in
- Mathlib.Analysis.MeanInequalities
- Cited by
- 1 results in Mathlib
- Foundations
- Depth 205 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites24
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Realstatement and proof · cited by 25,697
- Finsetstatement and proof · cited by 13,712
- Finset.sumstatement and proof · cited by 5,195
- Finset.prodstatement and proof · cited by 2,356
- Finset.sum_congrproof · cited by 2,323
- le_reflproof · cited by 2,061
- le_of_ltproof · cited by 1,175
- Finset.Nonemptystatement and proof · cited by 1,001
- le_transproof · cited by 985
- Finset.prod_congrproof · cited by 646
- one_divproof · cited by 624
- inv_invproof · cited by 494
Cited by1
Results whose statement or proof uses this declaration.
- Real.harm_mean_le_geom_meanproof · cited by 0