Let $\Delta$ be a shellable simplicial complex. It is known that the $h$-vector of $\Delta$ is positive, and there are several useful inequalities for them. I am particularly stuck with the following, which appears as an exercise on a article of Bjorner but that does not seems obvious at all to me.
If $i\geq 1$ and $h_i\geq 2$ then $h_{i-1}\geq 2$.
I could manage to prove that if $h_i\geq 1$ then $h_{i-1}\geq 1$, but the inequality above is a bit harder I think.