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.
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.
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:
We know also that
which is the two-variable Cauchy-Schwarz Inequality
Lets begin with an easy problem.
Show that for , we have
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