On the degree two entry of a Gorenstein h-vector and a conjecture of Stanley

In this short paper we establish a (non-trivial) lower bound on the degree two entry h2 of a Gorenstein h-vector of any given socle degree e and any codimension r. In particular, when e = 4, that is, for Gorenstein h-vectors of the form h = (1, r,h2, r, 1), our lower bound allows us to prove a conjecture of Stanley on the order of magnitude of the minimum value, say f(r), that h2 may assume