Improvement of a convergence condition for the Durand-Kerner iteration

The Durand-Kerner iteration is a well-known simultaneous method for approximation of (simple) zeros of a polynomial. By relating Weierstrassʹ correction and the minimal distance between approximations practical conditions for convergence have been obtained. These conditions also ensure the existence of isolating discs for the polynomial roots, i.e. each iteration step gives a refined set of inclusion discs. In this paper refined conditions of convergence are given.