Hello guys, how you doing? This time let’s talk about Cauchy-Schwarz inequality, it’s a well known inequality and useful across multiple branches of mathematics. Actually, it’s a special case of Hölder’s inequality. In math contest, it’s really useful, so let’s start with the definition.

Definition

Let , for be real numbers. Then:

Equality occurs if and only if there exists such that , for

An easy proof for Cauchy-Schwarz Inequality

There a lot of proofs about it, I think that below is the easiest way.

Let:

Being a sum of squares, is always non-negative. Now we expand it and collect terms:

The discriminant is , computing the discriminant we have:

Dividing by 4 and rearranging yields Cauchy-Schwarz Inequality. It holds when has a real root (repeated of course). From the first form of and using the fact that the sum of squares equal to 0 only when each square equals to 0, we have for all that is when is constant for all .

A Trigonometric Proof

For the graph we know the following:

,

,

Applying trigonometric identities:

Then:

We know also that

So:

Therefore

which is the two-variable Cauchy-Schwarz Inequality

Solving problems

Lets begin with an easy problem.

Show that for , we have

Let’s make

Then, for Cauchy-Schwarz Inequality

(APMO’ 1991) For positive reals such that , show that

We assume as true the last sentence and make an artifice.

We know that , then instead of we write Also, let’s make for

Let’s say that before the last sentence, there was a replacement of and for , then the inequality could have been read as:

We notice that the last sentence is in fact the Cauchy-Schwarz Inequality, then we can conclude that:

But there is something else, we notice that

is a consequence of Cauchy-Schwarz Inequality, this lemma is known as Titu’s Lemma. Which is a special form helps tackle a lot of optimization problems involving squares in the numerator, immediately.

(China Mathematical Olympiad 2004) For a given integer , suppose positive integers satisfy and . Prove that, for any real number , the following inequality holds,

For , from we have

For , using Cauchy-Schwarz Inequality, we have

Futher, for positive integers , we have and

For . So