On the infinite families of zero bounds of complex polynomials

We derive several classes of zero bounds for univariate polynomials with complex coefficients. The derivations are based on the iterative algorithms of Graffe and Jin, and the approximation methods of Newton and Kalantari. All of the proposed bounds are shown to be asymptotically sharp, and the convergence speeds are compared. In particular, we derive an improving factor to tighten Kalantari’s bounds. This factor and Kalantari’s bound can both be readily computed with a precalculated lookup table. We also propose a modified version of Kalantari’s bound that is independent of the table, and we prove that, under some conditions our bound is tighter. Numerical examples are provided to illustrate the effectiveness of our results.