Stability of a polynomial and convexity of a frequency response arc

A frequency response arc associated with a polynomial p(s) is the plot of Ψ(w)=p(jw). A proper frequency response arc associated with p(s) is part of the plot which does not pass through the origin, and the net phase change of Ψ(w) does not exceed 180 degrees. Recently Hamann and Barmish [HB] proved convexity of all proper frequency response arcs associated with a Hurwitz polynomial. In this paper we suggest an alternative proof of this result, and discuss connections between stability of a polynomial and convexity of the associated frequency response arc. Our proof is analytical and complements the geometric proof given by HB. It is hoped that the analytic approach will provide an insight into convexity of frequency response arcs associated with Hurwitz quasipolynomials