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I am trying to prove that $\forall x_1,\dots,x_n,y_1,\dots,y_n \in \mathbb R$: $(\sum_{i=1}^{n}x_i\ y_i)^2 \leq \ (\sum_{i=1}^{n}i \ x_i^2) \cdot (\sum_{i=1}^{n} \frac{y_i^2}{i})$

I know the cauchy-schwarz inequality is $(\sum_{i=1}^{n}x_i\ y_i)^2 \leq \ (\sum_{i=1}^{n}x_i^2) \cdot (\sum_{i=1}^{n} y_i^2)$

I am trying to show that $(\sum_{i=1}^{n}x_i^2) \cdot (\sum_{i=1}^{n} y_i^2) \leq \ (\sum_{i=1}^{n}i \ x_i^2) \cdot (\sum_{i=1}^{n} \frac{y_i^2}{i})$ but I don't know how to continue from there..

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  • $\begingroup$ well, $x_iy_i = \tilde x_i \tilde y_i$ with $\tilde x_i = \sqrt ix_i$ and $\tilde y_i = \frac{y_i}{\sqrt i}$... so just use CS on $\tilde x, \tilde y$ $\endgroup$ – Surb Jun 9 '17 at 15:16
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$\displaystyle\left(\sum_{i=1}^nx_iy_i\right)^2=\left(\sum_{i=1}^n\left(\sqrt ix_i\right)\left(\frac{y_i}{\sqrt i}\right)\right)^2\leqslant\left(\sum_{i=1}^ni{x_i}^2\right)\left(\sum_{i=1}^n\frac{{y_i}^2}i\right)$

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Hint: Write $$\left(\sum_{i=1}^n x_iy_i\right)^2 = \left(\sum_{i=1}^n (x_i \sqrt{i}) \left( \frac{y_i}{\sqrt{i}}\right)\right)^2.$$ What happens when you apply CS to this?

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