  Let $m$ and $k \geq 2$ be positive integers. We show that polynomial $P =
(1+x)^m(1+x^k)$ is strongly unimodal (frequently known as {\it log concave\/})
if and only if $m \geq k^2 -3$; this is also the criterion for $P$ to be merely
unimodal (that is, for $P$ of this form, unimodality implies strong
unimodality).{ }In section 2, we investigate an analogous question, concerning
the property $\EE$ of functions $f$ analytic on a neighbourhood of the unit
circle [H2], and show that the corresponding minimal $m$ is rather surprisingly
of order $k^4$.
