An Inequality for Elementary Symmetric Functions

Extract

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

1

E₀ = 1 and E_r=0(r>m).

We define the rth symmetric mean by

2

where denote the binomial coefficient. If α₁ α₂,…,α_m 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.