An Inequality for Complete Symmetric Functions

Consider the identitywhere aj, …, am are positive real numbers. Then for r = 1, 2, 3, … Tr =Tr(a1, …, am) is called the rth complete symmetric function in a1, …, am (T0=l).

Two inequalities for the complete symmetric functions

In (l) Hunter proved that the complete symmetric functions of even order are positive definite by obtaining the inequalitywhere ht denotes the complete symmetric function of order t. In this note we show that the inequality can be strengthened, which, in turn, enables theorem 2 of (l) to be sharpened. We also obtain a special case of an inequality conjectured by McLeod(2).

Inequalities for Generalized Symmetric Functions

The generating series for the elementary symmetric function Er, the complete symmetric function Hr, are defined byrespectively.