Behavior of the homogeneous symmetric polynomial in an angular region

Abstract

1. A sequence o f positive numbers is called a x sequence (x-seq. f o r short) i f its members satisfy the inequality 2 am >1 am+lain-l, m = 1 . . . . . k 1. (2) I f the sequence ao . . . . . ak is such that ao . . . . . ak, then it is called a trivial x-seq. (triv. seq. f o r short). The geometric progression (g.p. for short) is a simple example of a x-seq. We will not regard triv. seq. as g.p. It follows from (2) that lnam ) (lnam+l + l n a m l ) / 2 . This means that a x-seq, is a logarithmic convex sequence. It follows from (2) also that