  We investigate log-concavity in the context of level Hilbert functions and
pure $O$-sequences, two classes of numerical sequences introduced by Stanley in
the late Seventies whose structural properties have since been the object of a
remarkable amount of interest in combinatorial commutative algebra. However, a
systematic study of the log-concavity of these sequences began only recently,
thanks to a paper by Iarrobino.
  The goal of this note is to address two general questions left open by
Iarrobino's work: 1) Given the integer pair $(r,t)$, are all level Hilbert
functions of codimension $r$ and type $t$ log-concave? 2) How about pure
$O$-sequences with the same parameters?
  Iarrobino's main results consisted of a positive answer to 1) for $r=2$ and
any $t$, and for $(r,t)=(3,1)$. Further, he proved that the answer to 1) is
negative for $(r,t)=(4,1)$.
  Our chief contribution to 1) is to provide a negative answer in all remaining
cases, with the exception of $(r,t)=(3,2)$, which is still open in any
characteristic. We then propose a few detailed conjectures specifically on
level Hilbert functions of codimension 3 and type 2.
  As for question 2), we show that the answer is positive for all pairs
$(r,1)$; negative for $(r,t)=(3,4)$; and negative for any pair $(r,t)$ with
$r\ge 4$ and $2\le t\le r+1$. Interestingly, the main case that remains open is
again $(r,t)=(3,2)$. Further, we conjecture that, in analogy with the behavior
of arbitrary level Hilbert functions, log-concavity fails for pure
$O$-sequences of any codimension $r\ge 3$ and type $t$ large enough.
