A new condition for a polynomial p(s) to be a local convex direction for a Hurwitz stable polynomial q(s) is derived. The condition is in terms of polynomials associated with the even and odd parts of p(s) and q(s), and constitutes a generalization of Rantzer's phase-growth condition for global convex directions. It is used to determine convex directions for certain subsets of Hurwitz stable polynomials.