Overview

Cauchy-Schwarz provides a clean upper or lower bound for sums of products. It is a staple in inequality and geometry problems.

Key Ideas

• $(a_1^2+\cdots+a_n^2)(b_1^2+\cdots+b_n^2) \ge (a_1b_1+\cdots+a_nb_n)^2$.
• Equality holds when the vectors are proportional.
• Use it to bound dot products or sum-of-squares expressions.

Core Skills

Match Terms Carefully

Decide which expressions should be the $a_i$ and $b_i$ lists so that the right side becomes what you want to bound.

Use the Engel Form

For sums like $\sum \frac{x_i^2}{a_i}$, use $\left(\sum \frac{x_i^2}{a_i}\right) \left(\sum a_i\right) \ge (\sum x_i)^2$.

Check Equality

Equality happens when $a_i/b_i$ is constant. Use it to identify where an extremum is achieved.

Worked Example

Show that $a^2+b^2 \ge \frac{(a+b)^2}{2}$.

Apply Cauchy-Schwarz to $(a,b)$ and $(1,1)$: $(a^2+b^2)(1^2+1^2) \ge (a+b)^2$. So $2(a^2+b^2) \ge (a+b)^2$, or $a^2+b^2 \ge \frac{(a+b)^2}{2}$.

More Examples (NOTE: As more practice, try to find equality cases as well!)

Example 1: Two-Term Bound

Show that $(x+y)^2 \le 2(x^2+y^2)$.

This is the same inequality with $a=x,b=y$.

Example 2: Engel Form

Prove $\frac{1}{a} + \frac{1}{b} \ge \frac{4}{a+b}$ for $a,b>0$.

Use $\left(\frac{1}{a}+\frac{1}{b}\right)(a+b) \ge (1+1)^2 = 4$.

Example 3: Dot Product Bound

If $a_1^2+a_2^2=13$ and $b_1^2+b_2^2=5$, find the maximum of $a_1b_1+a_2b_2$.

By Cauchy-Schwarz, it is at most $\sqrt{13\cdot 5} = \sqrt{65}$.

Example 4: Radical Sum!

Let $a,b,c$ be real numbers $>0$ that satisfy $a+b+c=6$. Your job is to find the MAXIMUM value of the following expression: $\sqrt{a+ab}+\sqrt{b+bc}+\sqrt{c+ca}$.

First, we can rewrite the expression as $\sqrt{a(1+b)}+\sqrt{b(1+c)}+\sqrt{c(1+a)}$, and this is the key step as it allows for the next crucial step. We can exploit this expression via Cauchy Schwarz as follows:

This then becomes $6(9)=54$, and thus we get that the maximum is $3\sqrt6$.

Example 5: Integer Triples!

You are given that positive integers $x,y,z$ satisfy the equation $x+y+z=12$. Your job is to find the MINIMUM value of the following expression: $\frac1x+\frac1y+\frac1z$.

This problem is easier than the last one as we can directly apply Cauchy-Schwarz,

Thus, we get:

Notice that equality occurs at $x=y=z=4$.

Example 6: Squares!

Over all positive real numbers $a,b$ that satisfy $a^2+b^2=10$, your job is to find the MAXIMUM value of $a+2b$.

Again, we can diredctly tackle this via Cauchy-Schwarz as follows,

Thus, we get that the maximum value is $5\sqrt2$.

Example 7: Symmetry

You are given that there are positive reals $a,b,c$ that satisfy $ab+bc+ca=3$. Your job is to find the MINIMUM value of the following expression: $\frac{a+b+c}{ab+bc+ca}$.

First off, notice that since the denominator is FIXED, all we need to do is minimize $a+b+c$. Then, by Cauchy-Schwarz, we have:

And thus, $a+b+c\ge3$, so from here we get our answer quite easily:

Example 8: Fractions!

For all positive reals $x,y$ that satisfy $x+y=1$, your job is to find the minimum value of the following expression: $\frac1x+\frac4y$.

We can use Titu's lemma here to get:

And so, the minimum value is $9$.

Example 9: Square Root Sum

For all positive reals $a,b,c$ that satisfy $a+b+c=1$. your job is to find the maximum value of the following expression $\sqrt{ab}+\sqrt{bc}+\sqrt{ca}$.

First off, by Cauchy-Schwarz, we get that

Thus the maximum value is $1$.

Strategy Checklist

• Decide which terms to pair as $a_i$ and $b_i$.
• Consider the Engel form for sums of fractions.
• Use equality conditions to confirm the extremum.

Common Pitfalls

• Using Cauchy-Schwarz when AM-GM or Jensen is simpler.
• Missing the equality condition in optimization problems.
• Choosing a pairing that complicates the right-hand side.

Practice Problems