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

TL;DR

This paper establishes a precise criterion for the log concavity and unimodality of the polynomial $(1+x)^m(1+x^k)$ based on the parameters $m$ and $k$, and explores related properties of analytic functions.

Contribution

It provides a necessary and sufficient condition for the polynomial's log concavity and unimodality, and investigates analogous properties for analytic functions near the unit circle.

Findings

Polynomial $P = (1+x)^m(1+x^k)$ is strongly unimodal if and only if $m \,\geq\, k^2 - 3$.

Unimodality implies strong unimodality for this polynomial form.

The minimal $m$ for the property $ ext{EE}$ of certain analytic functions is of order $k^4$.

Abstract

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 $k^4$.