On the zeros of convex combinations of polynomials

Abstract

Given monic nth degree polynomials P0(z) and P1(z), let PA(Z) = (1 − A)P0(z)+AP1(z). If the zeros of P0and P1all lie in a circle C or on a line L, necessary and sufficient conditionsare given for the zeros of PA(0≤A≤ l) to all lie onCor L. This describes certain convex sets of monic nth degree polynomials having zeros in C or L. If the zeros of P0andP1lie in the unit disk and P0andP1have real coefficients, then the zeros of PA(0≤A≤ l) lie in the disk |z|