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Prove $$\sum\limits_{cyc}\frac{1}{x^{2017}+ x^{2015}+ 1} \geq 1$$ with $x,\,y,\,z>0,\,xyz= 1$

I try to use Jensen inequality, then:

$$f\left ( x \right )+ f\left ( y \right )+ f\left ( z \right )\geqq 3f\left (\sqrt[3]{xyz} \right )$$

But is that true? Help me! And give some interesting solutions! Thanks!

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closed as off-topic by Saad, Carl Mummert, JonMark Perry, Jose Arnaldo Bebita-Dris, user223391 Jun 21 '18 at 18:12

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  • $\begingroup$ This is not a genuine try. $\endgroup$ – Saad Jun 18 '18 at 1:59
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Jensen and Karamata for the function $f(t)=\frac{1}{e^{2015t}+e^{2017t}+1}$ helps here, but it's very ugly.

I think it's better to use the following way.

We need to prove that $$\sum_{cyc}\left(x^{2017}+x^{2015}+1\right)\left(y^{2017}+y^{2015}+1\right)\geq\prod_{cyc}\left(x^{2017}+x^{2015}+1\right)$$ or $$\sum_{cyc}\left(\left(x^{2017}+x^{2015}\right)\left(x^{2017}+x^{2015}\right)+2\left(x^{2017}+x^{2015}\right)+1\right)\geq$$ $$\geq\prod_{cyc}\left(x^{2017}+x^{2015}\right)+\sum_{cyc}\left(x^{2017}+x^{2015}\right)\left(x^{2017}+x^{2015}\right)+\sum_{cyc}\left(x^{2017}+x^{2015}\right)+1$$ or $$2+\sum_{cyc}\left(x^{2017}+x^{2015}\right)\geq\prod_{cyc}\left(x^{2017}+x^{2015}\right)$$ or $$2+\sum_{cyc}\left(x^{2017}+x^{2015}\right)\geq\prod_{cyc}\left(x^2+1\right)$$ or $$\sum_{cyc}\left(x^{2017}+x^{2015}\right)\geq\sum_{cyc}(x^2y^2+x^2),$$ which is true by Muirhead.

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