On the roots of a hyperbolic polynomial pencil

Abstract

Let ${\nu }_0(t),{\nu }_1(t),\dots ,{\nu }_n(t)$ be the roots of the equation $R(z)=t$, where $R(z)$ is a rational function of the form \[R(z)=z+\sum\limits_{k=1}^n\frac{α_k}{z-μ_k},\] ${\mu }_k$ are pairwise different 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 $-\infty <t<\infty$.

9 pages