Stieltjes Functions and Hurwitz Stable Entire Functions

Abstract

The concept of stability, originally introduced for polynomials, will be extended to apply to the class of entire functions. This generalization will be called Hurwitz stablility and the class of Hurwitz stable functions will serve as the main focus of this paper. A first theorem will show how, given a function of either of the Stieltjes classes, a Hurwitz stable function might be constructed. A second approach to constructing Hurwitz stable functions, based on using additional functions from the Laguerre-P\'{o}lya class, will be presented in a second theorem.