Let's say I have $N$ groups that are mutually independent ($X_1,\ldots,X_N$). What I am looking for is a p-value to test the null hypothesis "$\mu_1=\mu_2\vee\mu_1=\mu_3\vee\ldots\vee\mu_1=\mu_N$", if that makes any sense (my apologies if that's an abuse of notation; I'm a bit of a newbie when it comes to statistics).
What I mean is, I would like to test whether the mean of $X_1$ in particular is significantly different from the means of $X_2,\ldots,X_N$. Actually, it would be ideal if I could test whether it's not just "different", but less than the rest (i.e., $\mu_1<\mu_2\wedge\ldots\wedge\mu_1<\mu_N$).
My problem is that I can't assume $\mu_2=\mu_3=\ldots=\mu_N$, or else I could just perform an ANOVA on the $X_1,\ldots,X_N$. My Wikipedia searching has told me that my problem comes from the fact that ANOVA is an "omnibus test", when I really need some type of "contrast test" or something. I'm sure there's a standard test of for this problem, but I just haven't found the right thing to search for yet. If anyone could point me in the right direction I'd really appreciate it (if that direction happens to involve a Matlab function, that would be amazing). Thanks!