# On the Roots of a Hyperbolic Polynomial Pencil

Research paper by **Victor Katsnelson**

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

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