On efficient computation and asymptotic sharpness of Kalantari's bounds for zeros of polynomials

We study an infinite family of lower and upper bounds on the modulus of zeros of complex polynomials derived by Kalantari. We first give a simple characterization of these bounds which leads to an efficient algorithm for their computation. For a polynomial of degree $n$ our algorithm computes the first $m$ bounds in Kalantari's family in $O(mn)$ operations. We further prove that for every complex polynomial these lower and upper bounds converge to the tightest annulus containing the roots, and thus settle a problem raised in Kalantari's paper.