Some inequalities for elementary mean values

Upper and lower bounds for the difference between the arithmetic and harmonic mcans of it positive numbers are obtained in terms of n and the largest and smallest of the numbers. Also, results of S. H. Tung [21, are used to obtain upper and lower bounds for the elementary mean values M,, of Hardy, Littlewood, and Polya. 1. In 1975, S. H. Tung proved the following theorem [21: Let O 0, we may, without loss of generality, assume xl = 1. THEOREM 1. Let I = x < X2*** < Xn = B. Then (B 1)2 12 n(B + )< A(I,..., B) H(I,..., B) < (B'!2-1)2. Proof. For each k, 2 < k < n, let Ak= A(x, X2,..., qXk.,,Xn) and Hk= H(xj, x2,..., xk-1, xn). Fix xl, x2, .. ,P Xn2, x,, and let x" = x vary in (1, B]. Let D(x)=An-Hn = ( I)An-_ +x _ nxHn_ n ~~n (n -1)x +H6 Computation of D'(x) shows that x = Hn, is its only positive zero, and standard methods of analysis show that a minimum for D(x) is attained at x = Hn.. Received November 2, 1982; revised February 11, 1983. 1980 Mathematics Subject Classification. Primary 26D20. ?1984 American Mathematical Society 0025-5718/84 $1.00 + $.25 per page 193 This content downloaded from 207.46.13.51 on Sun, 19 Jun 2016 07:18:29 UTC All use subject to http://about.jstor.org/terms