Ratio Vectors of Polynomial-Like Functions

Let p(x) be a hyperbolic polynomial–like function of the form p(x) = (x − r1)1 · · · (x − rN )N , where m1, . . . ,mN are given positive real numbers and r1 < r2 < · · · < rN . Let x1 < x2 < · · · < xN−1 be the N − 1 critical points of p lying in Ik = (rk, rk+1), k = 1, 2, . . . , N−1. Define the ratios σk = xk−rk rk+1−rk , k = 1, 2, . . . , N−1. We prove that mk mk+···+mN < σk < m1+···+mk m1+···+mk+1 . These bounds generalize the bounds given by earlier authors for strictly hyperbolic polynomials of degree n. For N = 3, we find necessary and sufficient conditions for (σ1, σ2) to be a ratio vector. We also find necessary and sufficient conditions on m1,m2,m3 which imply that σ1 < σ2. For N = 4, we also give necessary and sufficient conditions for (σ1, σ2, σ3) to be a ratio vector and we simplify some of the proofs given in an earlier paper of the author on ratio vectors of fourth degree polynomials. Finally we discuss the monotonicity of the ratios when N = 4.