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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. Add remark

      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.

              Add remark

              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}^+.


                • Add remark

                  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.