interesting polynomial type inequality

mathemagics · **Joined:** Sun, 23 Mar 2008 10:55:47 UTC **Posts:** 27

If $n\geq 2$ is even integer and $x\in\mathbb{R}$ prove that

$x^n+1\geq 2\sqrt{\dfrac{x^2\left(x^{2n-2}-1\right)}{\left(x^2-1\right)(n-1)}$

outermeasure · **Posted:** Wed, 25 Jul 2012 08:58:05 UTC

mathemagics wrote:

If $n\geq 2$ is even integer and $x\in\mathbb{R}$ prove that

$x^n+1\geq 2\sqrt{\dfrac{x^2\left(x^{2n-2}-1\right)}{\left(x^2-1\right)(n-1)}$

Hint:

Spoiler:

Edit: correction.

mathemagics · **Joined:** Sun, 23 Mar 2008 10:55:47 UTC **Posts:** 27

Could you please give more details?

I can't see how you get the first inequality by the second one and how you prove it (the second one).

For example the C-S inequality gives that:

$(n-1)\left(x^{2n-2}+x^{2n-4}+\cdots+x^2\right)\geq \left(x^{n-1}+ x^{n-2}+\cdots + x\right)^2$

and if we accept the second inequality then I don't think that this helps to get the first one because (look the opposite boxed inequality below):

$\left(\dfrac{x^n+1}{2}\right)^2\geq \left(\dfrac{x^{n-1}+x^{n-2}+\cdots+x}{n-1}\right)^2 \boxed{\stackrel{C-S}{\leq}} \dfrac{x^{2n-2}+x^{2n-4}+\cdots+x^2}{n-1}$

Shadow · **Posted:** Thu, 26 Jul 2012 21:55:41 UTC

This is a proposed problem, he doesn't have to post more. If you need this for homework or something, it should be on another board.

outermeasure · **Posted:** Fri, 27 Jul 2012 05:18:04 UTC

Oops, I forgot the words "the proof of" after "follows from", now corrected.

Shadow · **Posted:** Fri, 27 Jul 2012 05:22:42 UTC

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interesting polynomial type inequality

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