Chebyshev's sum inequality

In mathematics, Chebyshev's sum inequality, named after Pafnuty Chebyshev, states that if a 1 ≥ a 2 ≥ ⋯ ≥ a n {\displaystyle a_{1}\geq a_{2}\geq \cdots \geq a_{n}\quad } and b 1 ≥ b 2 ≥ ⋯ ≥ b n , {\displaystyle \quad b_{1}\geq b_{2}\geq \cdots \geq b_{n},} then 1 n ∑ k = 1 n a k b k ≥ ( 1 n ∑ k = 1 n a k ) ( 1 n ∑ k = 1 n b k ) . {\displaystyle {1 \over n}\sum _{k=1}^{n}a_{k}b_{k}\geq \left({1 \over n}\sum _{k=1}^{n}a_{k}\right)\! \!

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