I'm working on a problem comparing the width of two confidence intervals. The problem boils down to proving this inequality:
$$p \cdot F_{p, \,n-p} \,(1-\alpha) > F_{1, \,n-p} \,(1-\alpha),$$
where $F_{p,\,q} \,(\alpha)$ denotes the $\alpha$-th quantile of an F-distribution with $p$ and $q$ degrees of freedoom.
Also, I think there is a (relatively) well-known fact about the quantiles of F-distribution is that $F_{p_1, \,n-p} \,(1-\alpha) < F_{p_2, \,n-p} \,(1-\alpha)$ for any $p_1 > p_2$. However, this is exactly what makes this hard to prove.
Any ideas or hints are greatly welcomed. Thanks!