In mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, Chebyshev's sum inequality, named after Pafnuty Chebyshev

Pafnuty Chebyshev

Pafnuty Lvovich Chebyshev was a Russian mathematician. His name can be alternatively transliterated as Chebychev, Chebysheff, Chebyshov, Tschebyshev, Tchebycheff, or Tschebyscheff .-Early years:One of nine children, Chebyshev was born in the village of Okatovo in the district of Borovsk,...

, states that if


and


then


Similarly, if


and


then

Proof

Consider the sum


The two sequences are non-increasing, therefore and have the same sign for any . Hence .

Opening the brackets, we deduce:


whence


An alternative proof is simply obtained with the rearrangement inequality.

Continuous version

There is also a continuous version of Chebyshev's sum inequality:

If f and g are real-valued, integrable functions over [0, 1], both increasing or both decreasing, then