Chebyshev inequality

for finite monotone sequences

$$a_1\leq\dotsb\leq a_n,\quad b_1\leq\dotsb\leq b_n$$

The inequality

$$\sum_{k=1}^na_k\sum_{k=1}^nb_k\leq n\sum_{k=1}^na_kb_k.$$

Chebyshev's inequality for monotone functions $f,g\geq0$ is the inequality

$$\int\limits_a^bf(x)dx\int\limits_a^bg(x)dx\leq(b-a)\int\limits_a^bf(x)g(x)dx,$$

where $f$ and $g$ are either both increasing or both decreasing on $[a,b]$.

The inequalities were established by P.L. Chebyshev in 1882.

Comments

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 $[a,b]\times[a,b]$.