  We determine new bounds on the entries of Gorenstein Hilbert functions, both
in any fixed codimension and asymptotically.
  Our first main theorem is a lower bound for the degree $i+1$ entry of a
Gorenstein $h$-vector, in terms of its entry in degree $i$. This result carries
interesting applications concerning unimodality: indeed, an important
consequence is that, given $r$ and $i$, all Gorenstein $h$-vectors of
codimension $r$ and socle degree $e\geq e_0=e_0(r,i)$ (this function being
explicitly computed) are unimodal up to degree $i+1$. This immediately gives a
new proof of a theorem of Stanley that all Gorenstein $h$-vectors in
codimension three are unimodal.
  Our second main theorem is an asymptotic formula for the least value that the
$i$-th entry of a Gorenstein $h$-vector may assume, in terms of codimension,
$r$, and socle degree, $e$. This theorem broadly generalizes a recent result of
ours, where we proved a conjecture of Stanley predicting that asymptotic value
in the specific case $e=4$ and $i=2$, as well as a result of Kleinschmidt which
concerned the logarithmic asymptotic behavior in degree $i= \lfloor \frac{e}{2}
\rfloor $.
