$(1+x^2y+x^4y^2)^3\le(1+x^3+x^6)^2(1+y^3+y^6)$ 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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Hint: Hölder's inequality. ${}$ |
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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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