Quantcast

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 $$\begin{aligned} R(z)=z-\sum \limits _{k=1}^n\frac{\alpha _k}{z-\mu _k}, \end{aligned}$$ $$\begin{aligned} R(z)=z-\sum \limits _{k=1}^n\frac{\alpha _k}{z-\mu _k}, \end{aligned}$$ \(\mu _k\) \(\mu _k\) \(\alpha _k>0,\,1\le {}k\le {}n\) \(\alpha _k>0,\,1\le {}k\le {}n\) \(\xi \) \(\xi \) \(e^{\xi \nu _0(t)}+e^{\xi \nu _1(t)}+\,\cdots \,+e^{\xi \nu _n(t)}\) \(e^{\xi \nu _0(t)}+e^{\xi \nu _1(t)}+\,\cdots \,+e^{\xi \nu _n(t)}\) \(-\infty<t<\infty \) \(-\infty<t<\infty \)