What is Cauchy Schwarz in 8th grade terms?

Solution 1:

In geometry terms that you can understand, the Cauchy-Schwarz inequality says that:

Among all the parallelograms with sides a and b, the rectangle is the one with the largest area.

Usually you can use this inequality when you are looking for an upper (or lower) bound of an expression.

I wanted to give you an example, but my geometry studies are too far to remember an easy demonstration. Maybe you can ask somebody to give you a compass-and-straightedge demonstration of the equivalence between Cauchy-Schwarz and triangle inequalities.

Solution 2:

Since you said you browse AoPS, maybe you're interested in math contests. In math contests, the following forms of Cauchy-Schwarz (C-S) inequality are used:

For all $a_i,b_i\in\Bbb R$:

$$\left(a_1^2+a_2^2+\cdots+a_n^2\right)\left(b_1^2+b_2^2+\cdots+b_n^2\right)\ge (a_1b_1+a_2b_2+\cdots+a_nb_n)^2$$

Equality holds if and only if either $a_1=b_1k, a_2=b_2k,\ldots, a_n=b_nk$ for some $k\in\Bbb R$ or $a_1=a_2=\cdots=a_n=0$ or $b_1=b_2=\cdots=b_n=0$.

For all $a_i,b_i\in\Bbb R^+$:

$$\sqrt{a_1+a_2+\cdots+a_n}\sqrt{b_1+b_2+\cdots+b_n}\ge \sqrt{a_1b_1}+\sqrt{a_2b_2}+\cdots+\sqrt{a_nb_n}$$

Equality holds if and only if $\frac{a_1}{b_1}=\frac{a_2}{b_2}=\cdots=\frac{a_n}{b_n}$.

The following is also called Titu's lemma, or Engel's form of C-S inequality (to prove it, multiply both sides by $b_1+b_2+\cdots+b_n$ and apply the first form of C-S inequality):

For $b_i\in\Bbb R^+, a_i\in\Bbb R$:

$$\frac{a_1^2}{b_1}+\frac{a_2^2}{b_2}+\cdots+\frac{a_n^2}{b_n}\ge \frac{(a_1+a_2+\cdots+a_n)^2}{b_1+b_2+\cdots+b_n}$$

with equality if and only if $\frac{a_1}{b_1}=\frac{a_2}{b_2}=\cdots=\frac{a_n}{b_n}$.

The more general is Hölder's inequality:

For all $a_i,b_i,c_i\in\Bbb R$:

$$\left(a_1^3+a_2^3+\cdots+a_n^3\right)\left(b_1^3+b_2^3+\cdots+b_n^3\right)\left(c_1^3+c_2^3+\cdots+c_n^3\right)$$

$$\ge (a_1b_1c_1+a_2b_2c_2+\cdots+a_nb_nc_n)^3$$

The same holds for all $a_i,b_i,c_i,d_i\in\Bbb R$, etc. I.e., any number of arrays $a_i,b_i,c_i,d_i,\ldots$. In general, all these inequalities are called Hölder's inequalities.