Convex directions for nested Hurwitz polynomials

A nested polynomial family is a polynomial function of a standard polynomial family. Such families arise in stability analysis of uncertain uniform systems. This paper considers the convex direction for nested Hurwitz polynomial families. A necessary and sufficient condition is obtained for a polynomial to be such a convex direction. This condition is on the phase growth rate of the difference polynomial, and it constitutes a generalization of Rantzer's phase-growth condition for convex directions of real Hurwitz polynomials.