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Is this inequality true? If so, what is its name and how to get it?

For $x_p\gt0$, $\prod\limits_{p=1}^{n}x_p=1$, and $b^2-4c\lt0$. $$ \prod_{p=1}^{n}\left(x_p^2+bx_p+c\right)\leqslant\sum_{p=1}^{n}\left(x_p^2\prod_{q=1,q\neq p}^{n}\left(x_q^2+bx_q+c\right)\right)\Large{?} $$

Caleb Stanford
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x.y.z...
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  • did not make the calculations but i think it holds, in order to prove it, use the given ineq. and add terms and variations of the sequence product and at the last step use sth like Jensen's inequality – Nikos M. Jun 13 '14 at 20:55

1 Answers1

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(might be wrong) This doesn't seem true to me. Take $x_i = 1$, then we want to show that

$$ ( 1 + b + c)^n \leq n ( 1 + b + c)^{n-1}$$

Clearly, this is not true for a large enough $c$.

Calvin Lin
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