Hi working on Prove that $\sum_{\mathrm{cyc}} \frac{214x^4}{133x^3 + 81y^3} \ge x + y + z$ for $x, y, z > 0$ I found the following conjecture :
Let $0<a\leq b\leq1< 7/5\leq c $ then it seems we have :
$$\frac{a^{4}}{133a^{3}+81b^{3}}+\frac{b^{3}}{133b^{3}+81c^{3}}+\frac{c^{4}}{133c^{3}+81a^{3}}-\frac{\left(a+\sqrt{b}+c\right)}{214}\ge 0$$
For an attempt of the RiverLi's inequality see the first link . On my profile I show also a similar inequality but fails to get the hard case describe by RiverLi i mean $(a,b,c)=(43/66,1,121/84)$
It's a partial refinement of the strenghened HN_NH inequality .
Attempt :
As the function $f\left(x\right)=\frac{1}{133+81x^{6}}$ is convex on $(7/5,\infty)$ we use Jensen's inequality and the new problem is :
$$g(x,y,z)=\left(x^{2}+1\right)f\left(\frac{\left(\frac{z}{y}+xy\right)}{x^{2}+1}\right)+\frac{z^{8}}{133z^{6}+81x^{6}}-\frac{\left(x^{2}+y+z^{2}\right)}{214}\ge^? 0$$
We use Buffalo's way and for :
$$g\left(\frac{x}{x+y},1,\frac{7}{5}+x\right)\geq 0$$
Where all the coefficients are positives .
Question : How to (dis)prove it ?
Thanks in advance .
