REFINEMENTS ON AN INEQUALITY OF KY FAN

The aim of the present paper is to show that there are monotonic conti nuous functions h(t), k(t), p(t), q(t) such that H-n less than or equa l to h(t)less than or equal to G(n) less than or equal to k(t)less tha n or equal to A(n) and H-n/1-H-n less than or equal to p(t)less than o r equal to G(n)/G(n)'less than or equal to q(t)less than or equal to A (n)/A(n)', where A(n), G(n), and H-n, are arithmetic, geometric, and h armonic means, respectively, of positive numbers x(1),x(2),...,x(n); A (n)' and G(n)' are arithmetic and geometric means of a sequence 1-x(1) ,1-x(2),...,1-x(n), with x(i) is an element of(0, 1/2], i=1,2,...,n. ( C) 1996 Academic Press, Inc.