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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?

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If you can find some event whose probability you can prove is equal to this, that would do it. – Michael Hardy Jan 10 '12 at 19:24
@Michael: I would call him a magician! – Fabian Jan 10 '12 at 19:39
Are you sure that the minus shouldn't be a plus sign? Or did you make any other mistakes while plotting it here? – T. Eskin Jan 10 '12 at 19:56
up vote 2 down vote accepted

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?)

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