Chebyshev's sum inequality

Chebyshev's sum inequality states that two similarly ordered real sequences have an average of pairwise products at least as large as the product of their averages.

Covariance: Covariance measures how two random variables vary together. The inequality is a finite-sequence counterpart of nonnegative covariance for similarly ordered values.
Arithmetic mean: The arithmetic mean is the sum of a finite collection of values divided by its size. Both sides compare averages, so the normalization determines the inequality's exact form.
Weighted average: A weighted average combines values using specified weights, usually normalized to sum to one. A weighted form extends the result from equally weighted terms to nonnegative weights.
Reversed Chebyshev inequality: The reversed Chebyshev inequality bounds the average product in the opposite direction for oppositely ordered sequences. Opposite ordering reverses the covariance sign and therefore the inequality direction.
Rearrangement inequality: The rearrangement inequality bounds sums of products by pairing ordered sequences in matching or opposite order. It gives a broader extremal principle behind the advantage of similarly ordered pairings.
Pairwise product: A pairwise product is the product formed from corresponding entries of two sequences. The left side averages these matched products rather than all possible cross-products.
Expected value: The expected value is the probability-weighted average of a random variable's possible values. The finite inequality yields expectation bounds when ordered functions are evaluated on the same variable.
Cauchy–Schwarz inequality: The Cauchy–Schwarz inequality bounds the inner product of two vectors by the product of their lengths. It bounds products without requiring the sequences to share a monotone ordering.
Summation by parts: Summation by parts transforms a sum of products into boundary terms and a sum involving successive differences. Difference-based proofs expose how matching monotonicity makes the inequality hold.
Increasing sequence: An increasing sequence has terms that rise, or at least do not fall, as the index increases. One common hypothesis requires both sequences to be nondecreasing in the same index.

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