for finite monotone sequences
$$a_1\leq\dotsb\leq a_n,\quad b_1\leq\dotsb\leq b_n$$
Chebyshev's inequality for monotone functions $f,g\geq0$ is the inequality
where $f$ and $g$ are either both increasing or both decreasing on $[a,b]$.
The inequalities were established by P.L. Chebyshev in 1882.
It is not important that $f$ and $g$ be non-negative. The proof consists of simply integrating the non-negative function $\tau(x,y)=[f(x)-f(y)][g(x)-g(y)]$ over the square $[a,b]\times[a,b]$.
Chebyshev inequality. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Chebyshev_inequality&oldid=44607