We are already familiar with Bernoulli's inequality: $(1+px)\le(1+x)^p$ for $x\ge-1, p\ge1$.
Can this be generalized to say something useful about $(1+p_1x_1+\cdots+p_nx_n)$? For instance, one would hope that something like this could hold: $(1+p_1x_1+\cdots+p_nx_n) \le (1+x_1+\cdots+x_n)^{p_1+\cdots+p_n}$ However, this doesn't seem obvious. If you have any other ideas than using Bernoulli, that's great too, of course.