sum_mul_sum_le_card_mul_sum — Mathlib

English

Special case of Jensen for sums of powers: an inequality between sums of powers and products.

Русский

Особый случай неравенства Дженсена для сумм степеней: неравенство между sums степеней и произведений.

LaTeX

$$$\\text{pow\_sum\_le\_card\_mul\_sum\_pow}$$$

Lean4

/-- **Chebyshev's Sum Inequality**: When `f` and `g` monovary together (e.g. they are both
monotone/antitone), the product of their sum is less than the size of the set times their scalar
product. -/
theorem sum_mul_sum_le_card_mul_sum (hfg : MonovaryOn f g s) :
    (∑ i ∈ s, f i) * ∑ i ∈ s, g i ≤ #s * ∑ i ∈ s, f i * g i :=
  by
  rw [← nsmul_eq_mul]
  exact hfg.sum_smul_sum_le_card_smul_sum

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