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By a criterion of Craven--Csordas and McNamara--Sagan it is known that a sequence is $\\infty$-log-concave if it satisfies the stronger inequality $a_k^2 \\geq r a_{k - 1} a_{k + 1}$ for large enough $r$. On the other hand, a recent result of Br\\\"and\\'en shows that $\\infty$-log-concave sequences include sequences whose generating polynomial has only negative real roots. 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Medina","submitted_at":"2014-05-07T22:03:52Z","abstract_excerpt":"Following Boros--Moll, a sequence $(a_n)$ is $m$-log-concave if $\\mathcal{L}^j (a_n) \\geq 0$ for all $j = 0, 1, \\ldots, m$. Here, $\\mathcal{L}$ is the operator defined by $\\mathcal{L} (a_n) = a_n^2 - a_{n - 1} a_{n + 1}$. By a criterion of Craven--Csordas and McNamara--Sagan it is known that a sequence is $\\infty$-log-concave if it satisfies the stronger inequality $a_k^2 \\geq r a_{k - 1} a_{k + 1}$ for large enough $r$. On the other hand, a recent result of Br\\\"and\\'en shows that $\\infty$-log-concave sequences include sequences whose generating polynomial has only negative real roots. 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