# Prove that $\left (\frac{a^2 + b^2 +c^2}{a+b+c} \right) ^ {(a+b+c)} > a^a b^b c^c$

Prove that $\left (\dfrac{a^2 + b^2 +c^2}{a+b+c} \right) ^ {(a+b+c)} > a^a b^b c^c$ if $a$, $b$ and $c$ are distinct natural numbers. Is it possible using induction?

I recommend to read the answer from here. section Weighted AM-GM Inequality :) it is a good one .

• nice link! (+1) Aug 23, 2012 at 19:14
• @downvoter why (-1)?
– Iuli
Feb 5, 2013 at 12:56
• wasn't me but I guess because the linked page is no longer available Feb 10, 2013 at 0:46
• @Iuli Answers that only include a link are bad, for exactly this reason - very often the page that was linked to disappears. Can you edit your answer so that it's useful again? Feb 13, 2013 at 12:19

Rewrite it as: $$\left( a \frac{a}{a+b+c} + b \frac{b}{a+b+c} + c \frac{c}{a+b+c} \right) > a^\frac{a}{a+b+c} \cdot b^\frac{b}{a+b+c} \cdot c^\frac{c}{a+b+c}$$ This is Jensen's inequality: $$\log\left(\mathsf{E}\left(X\right)\right) > \mathsf{E}\left(\log\left(X\right)\right) \quad \text{or}\quad \mathsf{E}\left(X\right) > \exp \left( \mathsf{E}\left(\log\left(X\right)\right) \right)$$ where $X$ is the random variable which can assume one of three possible values $\{a,b,c\}$ with respective probabilities $\{ \frac{a}{a+b+c}, \frac{b}{a+b+c}, \frac{c}{a+b+c} \}$.

• @Fixee Thanks for the edits. Aug 23, 2012 at 16:45
• Nice solution! (+1) Aug 23, 2012 at 19:15