On the zeros of self-inversive polynomials

Abstract

A classical result due to Cohn states that a self-inversive polynomial has all its zeros on the unit circle if and only if all the zeros of its derivative lie in the closed unit disk. A more flexible necessary and sufficient condition than that of Cohns was given by Chen. However those results do not give any information on the simplicity of zeros of a self-inversive polynomial. This paper modifies the above results so that they serve as necessary and sufficient conditions for the simplicity as well as the unimodularity of zeros of a self-inversive polynomial.