This paper studies the behaviour of positive solutions of the recursive equation , , where is the mth elementary symmetric polynomial on k variables, for , and , with . A variant of Newton's inequalities is employed. Included among the results is a generalization of a particular case of Theorem 4.11 in E. A. Grove and G. Ladas, Periodicities in Nonlinear Difference Equations, Chapman & Hall/CRC Press, Boca Raton, 2004.