  Assume $R$ is a polynomial ring over a field and $I$ is a homogeneous
Gorenstein ideal of codimension $g\ge3$ and initial degree $p\ge2$. We prove
that the number of minimal generators $\nu(I_p)$ of $I$ that are in degree $p$
is bounded above by $\nu_0={p+g-1\choose g-1}-{p+g-3\choose g-1}$, which is the
number of minimal generators of the defining ideal of the extremal Gorenstein
algebra of codimension $g$ and initial degree $p$. Further, $I$ is itself
extremal if $\nu(I_p)=\nu_0$.
