# On the Roots of a Hyperbolic Polynomial Pencil

Research paper by Victor Katsnelson

Indexed on: 18 Aug '16Published on: 02 Aug '16Published in: Arnold Mathematical Journal

#### Abstract

Abstract Let $$\nu _0(t),\nu _1(t),\ldots ,\nu _n(t)$$ be the roots of the equation $$R(z)=t$$ , where R(z) is a rational function of the form \begin{aligned} R(z)=z-\sum \limits _{k=1}^n\frac{\alpha _k}{z-\mu _k}, \end{aligned} $$\mu _k$$ are pairwise distinct real numbers, $$\alpha _k>0,\,1\le {}k\le {}n$$ . Then for each real $$\xi$$ , the function $$e^{\xi \nu _0(t)}+e^{\xi \nu _1(t)}+\,\cdots \,+e^{\xi \nu _n(t)}$$ is exponentially convex on the interval $$-\infty<t<\infty$$ .AbstractLet $$\nu _0(t),\nu _1(t),\ldots ,\nu _n(t)$$ be the roots of the equation $$R(z)=t$$ , where R(z) is a rational function of the form \begin{aligned} R(z)=z-\sum \limits _{k=1}^n\frac{\alpha _k}{z-\mu _k}, \end{aligned} $$\mu _k$$ are pairwise distinct real numbers, $$\alpha _k>0,\,1\le {}k\le {}n$$ . Then for each real $$\xi$$ , the function $$e^{\xi \nu _0(t)}+e^{\xi \nu _1(t)}+\,\cdots \,+e^{\xi \nu _n(t)}$$ is exponentially convex on the interval $$-\infty<t<\infty$$ . $$\nu _0(t),\nu _1(t),\ldots ,\nu _n(t)$$ $$\nu _0(t),\nu _1(t),\ldots ,\nu _n(t)$$ $$R(z)=t$$ $$R(z)=t$$Rz \begin{aligned} R(z)=z-\sum \limits _{k=1}^n\frac{\alpha _k}{z-\mu _k}, \end{aligned} \begin{aligned} R(z)=z-\sum \limits _{k=1}^n\frac{\alpha _k}{z-\mu _k}, \end{aligned} $$\mu _k$$ $$\mu _k$$ $$\alpha _k>0,\,1\le {}k\le {}n$$ $$\alpha _k>0,\,1\le {}k\le {}n$$ $$\xi$$ $$\xi$$ $$e^{\xi \nu _0(t)}+e^{\xi \nu _1(t)}+\,\cdots \,+e^{\xi \nu _n(t)}$$ $$e^{\xi \nu _0(t)}+e^{\xi \nu _1(t)}+\,\cdots \,+e^{\xi \nu _n(t)}$$ $$-\infty<t<\infty$$ $$-\infty<t<\infty$$