Together with AM-GM, Schur’s inequality, Jensen’s inequality or Hölder’s inequality, this is a fundamental result, with remarkable applications. Trying to present some of the faces of this inequality, we will insist on the diversity of the problems that can be solved using it. The main question is: how do we recognize an inequality that can be solved using this method? It is very hard to say this clearly, but it is definitely good to think of Cauchy-Schwarz whenever we have sums of radicals or sums of squares and especially when we have expressions involving fractions.
First, let us speak about some problems in which it is better to apply the direct form of the Cauchy-Schwarz inequality:
$$ \left( \sum_{i=1}^{n} a_{i}b_{i} \right)^{2} \leq \left(
\sum_{i=1}^{n} a_{i}^{2} \right) \left( \sum_{i=1}^{n} b_{i}^{2}
\right).$$
The main difficulty is to choose \(a_{i}\) and \(b_{i}.\) We will see that in some cases this is trivial, while in some other cases it is very difficult. Let us solve some problems now:
Problems:
Problema 1, Problema 2, Problema 3, Problema 4, Problema 5, Problema 6, Problema 7, Problema 8, Problema 9, Problema 10, Problema 11, Problema 12, Problema 13, Problema 14, Problema 15, Problema 16, Problema 17, Problema 18, Problema 19
