On the Hurwitz stability of noninteger Hadamard powers of stable polynomials

Abstract

Consider a polynomial f(z)=anzn+...+a1z+a0 of positive coefficients that is stable (in the Hurwitz sense), i.e., every root of f lies in the open left half-plane of C. Due to Garloff and Wagner [J. Math. Anal. Appl. 202 (1996)], the pth Hadamard power of f: f[p](z):=anpzn+...+a1pz+a0p is stable if p is a positive integer number. However, it turns out that f[p] does not need to be stable for all real p>1. A counterexample is known for n=8 and p=1.139. On the other hand, f[p] is stable for n=1,2,3,4, and every p>1. In this paper we fill the gap by showing that f[p] is stable for n=5 and constructing counterexamples for n≥6. Moreover, by means of Rouché's Theorem, we give some stability conditions for polynomials and two examples that complete and illustrate the results.