On the roots of a hyperbolic polynomial pencil

TL;DR

This paper investigates the roots of a specific rational function polynomial pencil and proves that a certain exponential sum of these roots is exponentially convex over the entire real line.

Contribution

It establishes the exponential convexity of the sum of exponentials of roots of a rational function polynomial pencil, extending understanding of root behavior in such functions.

Findings

The roots are well-defined for real parameters.

The exponential sum of roots exhibits exponential convexity.

The result holds for all real values of the parameter.

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 \[R(z)=z+\sum\limits_{k=1}^n\frac{\alpha_k}{z-\mu_k},\] $\mu_k$ are pairwise different real numbers, $\alpha_k>0,\,1\leq{}k\leq{}n$. 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 $-\infty<t<\infty$.