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I want to prove this, where $x_1,...,x_n$ are positive real numbers: $$(x_1^2+...+x_n^2)^2 \leq (x_1+...+x_n)(x_1^3+...+x_n^3)$$

I have written a proof but I am not very happy with it, using the Cauchy-Schwarz inequality ($|<x,y>| \leq \|x\|\|y\|$):

$$ (x_1^2+...+x_n^2)^2 = \langle (x_1^{\frac{1}{2}} ,..., x_n^{\frac{1}{2}}),(x_1^{\frac{3}{2}} ,..., x_n^{\frac{3}{2}}) \rangle^2 \leq \|(x_1^{\frac{1}{2}} ,..., x_n^{\frac{1}{2}})\|^2 \|(x_1^{\frac{3}{2}} ,..., x_n^{\frac{3}{2}})\|^2 = (x_1+...+x_n)(x_1^3+...+x_n^3)$$

Is there a better proof?

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Why aren't you happy with your proof involving CS? In any proof, at some stage you need to use the fact that a square is non-negative, which is what CS precisely uses as well. – user17762 Feb 2 '13 at 22:38
Your proof is very compact, effectively the inequality is a special form of CS. I doubt there is a shorter proof. – sdcvvc Feb 2 '13 at 22:39
Your proof is great. – 1015 Feb 2 '13 at 22:39
Okay, thanks! I found it somewhat artificial, but I will stick with it. – Angel Alonso Feb 2 '13 at 22:44
up vote 2 down vote accepted

$$\sum_i x_i\sum_j x_j^3 - \sum_i x_i^2\sum_j x_j^2 = \sum_{i< j} (x_i x_j^3 + x_i^3 x_j - 2 x_i^2 x_j^2) = \sum_{i< j} x_i x_j (x_i-x_j)^2 \ge 0$$

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Very nice, thanks! – Angel Alonso Feb 2 '13 at 23:53
@Alderz Just in case you haven't seen it before, this is a particular case of an identity of Lagrange, who used it to prove the Cauchy-Schwarz inequality. – Andrés E. Caicedo Feb 3 '13 at 0:30

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