Let Er denote the rth elementary symmetric function on α1 α2,…,αm which is defined by

1

E0 = 1 and Er=0(r>m).

We define the rth symmetric mean by

2

where denote the binomial coefficient. If α1 α2,…,αm are positive reals then

we have two well-known inequalities

3

and

4

In this paper we consider a generalization of these inequalities. The inequality (4) is known as Newton's inequality which contains the arithmetic and geometric mean inequality.