Log concavity of $(1+x{)}^m(1+x^k)$

Abstract

Let $m$ and $k\ge 2$ 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 $m\ge k^2-3$; 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 $k^4$.

Better, shorter, and more likely correct proofs; new results for property E