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Prove that there are $a, b,$ positive numbers such that:

$f(x,y)=x^4+y^4-2(x-y)^2 \geq a(x^2+y^2)-b$

I've tried using the fact that $x^4 + y^4 \geq 2(xy)^2$ and also that $xy\geq -(1/2)(x^2+y^2)$

I've also tried that:

$f(x,y)\gt 2(xy)^2 -2(x-y)^2 = 2(xy)^2 -2(x^2+y^2) +4xy$

I factorized $xy$ and it didn't work so i don't know if i'm on the right way

I'll appreciate any help or hint.

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Ayoub Boumchich
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    Welcome to MSE. It will be more likely that you will get an answer if you show us that you made an effort. This should be added to the question rather than in the comments. – José Carlos Santos Sep 30 '18 at 11:36
  • You can prove this for all polynomial $h(x,y)$ and $g(x,y)$ of degree less than or equal to $3$. In other words you can find $a,b$ positive numbers such that $$x^4+y^4-h(x,y)\ge ag(x,y)-b$$ Note that in your example $h(x,y)=2(x-y)^2$ and $g(x,y)=x^2+y^2$.

    This is because $LHS$ tends to infinity faster than $RHS$. It might be useful for you to reason about this comment.

     – Piquito Sep 30 '18 at 14:39

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