Generating edges of D-stable polynomials

It is shown that if a polynomial P is D-stable, where D is convex and contains the origin, then all convex linear combinations of P and its normalized derivative, zP'/n, are also D-stable. It is also shown that convex linear combinations of the logarithmic derivatives of a D-stable polynomial with a convex D have both their poles and zeros in D. Both theorems provide an example of how to generate edges and polytopes of D-stable polynomials and rational functions from a given finite set of D-stable polynomials.<>