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Given the following inequality:

$$\sqrt[N]{\prod_i^N x_i} < \frac{1}{N} \sum_i^N x_i$$

for $x_i \in \mathbb{R^+_0}$ (positive reals) and $N \in \mathbb{N}^+$ (positive integer).

How can I prove that it is valid for the above conditions?

BTW: what is the english word for the $\prod$ operation? It has a name in spanish (multiplicatoria or productoria) but I have not been able to find a name for it in english other than "product".

Gabriel
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  • Product is the commonly used term for $\prod$. – Arkady Jul 03 '17 at 16:08
  • Yea, this is AM-GM proof, see thread that @Ravi gives. – 高田航 Jul 03 '17 at 16:10
  • Thanks! I wasn't aware that this was a known mathematical problem. And thank you for clarifying the name of $\prod$. I just assumed that there would be a unique name for the operation. And yes, this is indeed a duplicate of that question. – Gabriel Jul 03 '17 at 16:10
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    if ${x_i}$ are all equal then the two means are equal. You must change the $<$ symbol with the $\leq$ symbol – Raffaele Jul 03 '17 at 16:11

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