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Log concavity of $(1+x)^m (1+ x^k)$
Handelman, David...

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Log concavity of $(1+x)^m (1+ x^k)$ by Handelman, David ( Author )
Lemonjar Open Access
Australian National University
27-07-2023
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 $k^4$.
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Article
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29.34 KB
English
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MYR 0.01
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http://arxiv.org/abs/1102.2961
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