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$(1+x^2y+x^4y^2)^3\le(1+x^3+x^6)^2(1+y^3+y^6)$ for any $x,y>0$.

I have no idea about it. could you please give me some hint. I'm not requesting the whole proof, a piece of hint or direction to start up is enough. Thank you!

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up vote 5 down vote accepted

Hint: Hölder's inequality. ${}$

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I got it, Thank you very much. I will post the full answer in a minute. It's a pity I can't upvote you. – user58842 Jan 19 '13 at 5:36
@user58842, Glad to help :) – user27126 Jan 19 '13 at 5:39
upvote, Thank you! :-) – user58842 Jan 19 '13 at 5:53

Hölder's inequality: $|\mathbf{x}\cdot\mathbf{y}|\le\|\mathbf{x}\|_p\|\mathbf{y}\|_q$ where $\mathbf{x},\mathbf{y}\in\mathbb{R}^n$, $p,q>0$ and satisfy $\frac{1}{p}+\frac{1}{q}=1$.

Taking $\mathbf{x}=(1,x^2,x^4),\mathbf{y}=(1,y,y^2), p=\frac{3}{2}, q=3$, we get the inequality.

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