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More inequalities

Cauchy(-Schwartz-Buniakowski) inequality:
If $x_1,\dots,x_n,y_1,\dots,y_n$ are real numbers, then

$\displaystyle (x_1^2+\dots+x_n^2)(y_1^2+\dots+y_n^2) \ge (x_1 y_1 + \dots + x_n y_n)^2.$

Chebychev's inequality:
If $x_1 \ge \dots \ge x_n \ge 0$ and $y_1 \ge \dots \ge y_n \ge 0$ , then

$\displaystyle \frac{x_1 y_1 + \dots + x_n y_n}{n} \ge \left(\frac{x_1 + \dots + x_n}{n} \right) \left(\frac{y_1 + \dots + y_n}{n} \right)$

with equality if and only if one of the sequences is constant.

Chebychev's inequality with three sequences:
If $x_1 \ge \dots \ge x_n \ge 0$ , $y_1 \ge \dots \ge y_n \ge 0$ , and $z_1 \ge \dots \ge z_n \ge 0$ , then

$\displaystyle \frac{x_1 y_1 z_1 + \dots + x_n y_n z_n}{n} \ge \left(\frac{x_1 +... ...t(\frac{y_1 + \dots + y_n}{n} \right) \left(\frac{z_1 + \dots + z_n}{n} \right)$

with equality if and only if at least two of the three sequences are constant or one of the sequences is all zero.

You can probably guess what the four-sequence Chebychev inequality looks like.

Hölder's inequality:
Let $a_1,\dots,a_n,b_1,\dots,b_n,\alpha,\beta>0$ and suppose that $\alpha+\beta=1$ . Then

$\displaystyle (a_1+\dots+a_n)^\alpha (b_1+\dots+b_n)^\beta \ge (a_1^\alpha b_1^\beta + \dots + a_n^\alpha b_n^\beta),$

with equality if and only if

$\displaystyle \frac{a_1}{b_1} = \frac{a_2}{b_2} = \dots = \frac{a_n}{b_n}.$

Jensen's extension of Hölder's Inequality:
Suppose $a_1,\dots,a_n,b_1,\dots,b_n,\dots,\ell_1,\dots,\ell_n, \alpha,\beta,\dots,\lambda>0$ , and $\alpha+\beta+\dots+\lambda \ge 1$ . Then

$\displaystyle \left(\sum_{i=1}^n a_i \right)^\alpha \left(\sum_{i=1}^n b_i \rig... ...mbda \ge \sum_{i=1}^n \left( a_i^\alpha b_i^\beta \dots \ell_i^\lambda \right).$

Rearrangement inequality:
Suppose $a_1 \ge \dots \ge a_n$ and $b_1 \ge \dots \ge b_n$ are real numbers. If $\pi$ is a permutation of $1,2,\dots,n$ , then