Consider the iterative equation , where are reals and its characteristic polynomial . Assume that: (i) has a unique positive root , (ii) is a simple root, (iii) the absolute values of other complex roots of are greater than . The main result (Theorem 1) states that if either , or , and satisfies then and for .
This is a generalization of a result of the reviewer [On an equation of linear iteration. Aequationes Math. (to appear)] where instead of conditions (i)–(iii) it is assumed that , and the greatest common divisor of the support of is equal to 1.
The following lemma being a consequence of the Theorem of Kronecker serves as the main tool in the proof of Theorem 1: Let , , and let for . If then for .
A number of examples shows that the assumptions of Theorem 1 are essential and answers some questions asked by the reviewer [loc. cit.].