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Problems with using C-S (Cauchy-Schwarz inequality)

Problems about Cauchy–Schwarz inequality.

  1. C-S inequality it's the following.

Let $a_1$, $a_2$,..., $a_n$, $b_1$, $b_2$,..., $b_n$ be real numbers. Prove that: $$(a_1^2+a_2^2+...+a_n^2)(b_1^2+b_2^2+...+b_n)^2\geq(a_1b_1+a_2b_2+...+a_nb_n)^2.$$

  1. C-S inequality in the Engel form it's the following.

Let $a_1$, $a_2$,..., $a_n$ be real numbers and $b_1$, $b_2$,..., $b_n$ be positive numbers. Prove that: $$\frac{a_1^2}{b_1}+\frac{a_2^2}{b_2}+...+\frac{a_n^2}{b_n}\geq\frac{(a_1+a_2+...+a_n)^2}{b_1+b_2+...+b_n}$$

  1. C-S inequality in the integral form.

    Let $f$ and $g$ be integrable functions on $[a,b]$. Prove that: $$\int\limits_a^bf(x)^2dx\int\limits_a^bg(x)^2dx\geq\left(\int\limits_a^bf(x)g(x)dx\right)^2$$

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