It's a new inequality that I have created this is the following :
Let $x,y,z$ be real positive numbers such that $xyz=1$ then we have : $$\sum_{cyc}^{}\frac{(\frac{z}{x})^{\frac{1}{2}}}{(z+1+\frac{1}{x})^{\frac{1}{2}}(7+\frac{1}{x})}\leq \frac{\sqrt{3}}{8}$$
I have tried to apply Jensen's inequality to the function $f(x)=\frac{1}{7+\frac{1}{x}}$ wich is concave but we don't get a good expression . I also tried to expand the expression but it fails . After this I have no idea to prove this...
Any hints would be appreciable. Thanks in advance
Let $x=\frac{a}{b}$ and $y=\frac{b}{c},$ where $a$, $b$ and $c$ are positives.
Thus, $z=\frac{c}{a}$ and we need to prove that $$\sum_{cyc}\frac{1}{7a+b}\leq\frac{1}{8}\sqrt{\frac{3(a+b+c)}{abc}},$$ which was here:
If $a+b+c=abc$ then $\sum\limits_{cyc}\frac{1}{7a+b}\leq\frac{\sqrt3}{8}$