Theorems · Theorem · real analysis

Monovary.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] {f : ι → α} {g : ι → β} [inst_9 : Fintype ι],
  Monovary f g → (∑ i, f i) • ∑ i, g i ≤ Fintype.card ι • ∑ i, 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

0 results in Mathlib

Foundations

Depth 89 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

SemiringLinearOrderIsStrictOrderedRingExistsAddOfLEAddCommMonoidLinearOrderIsOrderedCancelAddMonoidModulePosSMulMonoFintype