Given $0 < \mu_1<\gamma_1<\mu_2<\gamma_2$, I would like to show
$ \frac{2(\mu_1+\mu_2)(\gamma_1\gamma_2-\mu_1\mu_2)(\gamma_1+\gamma_2)}{(\gamma_1+\mu_1)(\gamma_1+\mu_2)(\gamma_2+\mu_1)(\gamma_2+\mu_2)} <1.$
Any ideas?
Given $0 < \mu_1<\gamma_1<\mu_2<\gamma_2$, I would like to show
$ \frac{2(\mu_1+\mu_2)(\gamma_1\gamma_2-\mu_1\mu_2)(\gamma_1+\gamma_2)}{(\gamma_1+\mu_1)(\gamma_1+\mu_2)(\gamma_2+\mu_1)(\gamma_2+\mu_2)} <1.$
Any ideas?
The inequality is not correct. Investigate the case where $\gamma_2$ is much larger than the other three quantities. Then the inequality reads $\frac{2(\mu_1+\mu_2)\gamma_1}{(\gamma_1+\mu_1)(\gamma_1+\mu_2)} <1.$ Let next $\mu_2$ become larger than $\mu_1$ and $\gamma_1$ and you end up with $ \frac{2 \gamma_1}{\gamma_1 + \mu_1} < 1.$ Now it is easy to see that the inequality is violated for $\gamma_2 \gg \mu_2 \gg\gamma_1 \gg \mu_1$.
If you don't believe in the asymptotic treatment, put $\mu_1 =1$, $\gamma_1=2$, $\mu_2=3$ and $\gamma_2=100$ in the original inequality.
(Maybe the 1 on the right hand side of the inequality should be a 2?)