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!
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.