Some inequalities for elementary mean values
Abstract: Upper and lower bounds for the difference between the arithmetic and harmonic means of n positive numbers are obtained in terms of n and the largest and smallest of the numbers. Also, results of S. H. Tung , are used to obtain upper and lower bounds for the elementary mean values of Hardy, Littlewood, and Pólya.
-  G. H. Hardy, J. E. Littlewood, and G. Pólya, Inequalities, Cambridge, at the University Press, 1952. 2d ed. MR 0046395
-  S. H. Tung, On lower and upper bounds of the difference between the arithmetic and the geometric mean, Math. Comp. 29 (1975), 834–836. MR 0393393, https://doi.org/10.1090/S0025-5718-1975-0393393-9