Theorems · Theorem · real analysis

MonovaryOn.sum_smul_sum_le_card_smul_sum

∀ {ι : Type u_1} {α : Type u_2} {β : Type u_3} [inst : Semiring α] [inst_1 : LinearOrder α] [IsStrictOrderedRing α]
  [ExistsAddOfLE α] [inst_4 : AddCommMonoid β] [inst_5 : LinearOrder β] [IsOrderedCancelAddMonoid β]
  [inst_7 : Module α β] [PosSMulMono α β] {s : Finset ι} {f : ι → α} {g : ι → β},
  MonovaryOn f g ↑s → (∑ i ∈ s, f i) • ∑ i ∈ s, g i ≤ s.card • ∑ i ∈ s, f i • g i

Chebyshev's Sum Inequality: When f and g monovary together (e.g. they are both monotone/antitone), the scalar product of their sum is less than the size of the set times their scalar product.

Defined in

Mathlib.Algebra.Order.Chebyshev

Cited by

4 results in Mathlib

Foundations

Depth 88 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

SemiringLinearOrderIsStrictOrderedRingExistsAddOfLEAddCommMonoidLinearOrderIsOrderedCancelAddMonoidModulePosSMulMono