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3. Log Concavity

Let $\mathcal{L}\text{-}\mathcal{P}$ denote the class of functions which are locally uniform limits of polynomials having only real zeros, and let the class of functions in $\mathcal{L}\text{-}\mathcal{P}$ which have only non-negative Maclaurin coefficients be denoted $\mathcal{L}\text{-}\mathcal{P}^+$.
    1. Problem 3.1.

      [D. Karp] Let \[Q_n^{(\alpha, \beta)}:=\sum_{k=0}^n\binom{n}{k}f_kf_{n-k}((x+\alpha)_k(x+\beta)_{n-k} - (x+\alpha+\beta)_k(x)_{n-k}),\] where $\alpha, \beta>0$, $f_k^2\ge f_{k-1}f_{k+1}$, and $(x)_k = x(x+1)\cdots(x+k-1)$. Then the Maclaurin coefficients of $Q^{(\,\alpha,\,\beta)}$ are non-negative.
        1. Remark. [D. Karp] I have proved the case $\alpha=\beta=1$ and formulated a number of generalizations of this conjecture in http://arxiv.org/abs/1203.1482
            •     The following conjecture of (McNamara et al.) was proved by P. Brändén. Let $p(x):=\sum_{k=0}^n a_k x^k$, with $a_{-1}:=a_{n+1}:=0$. If $p\in\mathcal{L}\text{-}\mathcal{P}^+$, then \[\sum_{k=0}^n \left|\begin{array}{cc} a_k & a_{k+1} \\ a_{k-1} & a_{k} \end{array}\right|x^k \in\mathcal{L}\text{-}\mathcal{P}.\] S. Fisk generalized this statement to polynomials with coefficients formed from the determinants of $3\times 3$ matrices in the conjecture below.

              Conjecture 3.2.

              [Fisk] If $p\in\mathcal{L}\text{-}\mathcal{P}^+$, then \[\sum_{k=0}^n \left|\begin{array}{ccc} a_k & a_{k+1} & a_{k+2} \\ a_{k-1} & a_k & a_{k+1}\\ a_{k-2} & a_{k-1} & a_k\end{array}\right|x^k\in\mathcal{L}\text{-}\mathcal{P}^+.\]
                • Problem 3.3.

                  [D. Karp] Let $\{\phi_k(x)\}_{k=0}^\infty$ and $\{f_k\}_{k=0}^\infty$ be log-concave. When is $\sum f_k\phi_k(x)$ log-concave ?

                      Cite this as: AimPL: Stability and hyperbolicity, available at http://aimpl.org/hyperbolicpoly.