Geodesic
On limit problems associated with some inequalities
Glasgow mathematical journal, Tome 14 (1973) no. 2, pp. 123-127

Voir la notice de l'article provenant de la source Cambridge University Press

Let {an} be a sequence of non-negative real numbers. Suppose thatThen M1,n is the arithmetic mean, MO,n the geometric mean, and Mr,n the generalized mean of order r, of a1, a2, ..., an. By a result of Everitt [1] and McLaughlin and Metcalf [5], {n(Mr,n–Ms,n)}, where r ≧ l ≧ s, is a monotonic increasing sequence. It follows that this sequence tends to a finite or an infinite limit as n → ∞. Everitt [2, 3] found a necessary and sufficient condition for the finiteness of this limit in the cases r, s = 1, 0 and r ≧ 1 > s > 0. His results are included in the following theorem. 
Diananda, P. H. On limit problems associated with some inequalities. Glasgow mathematical journal, Tome 14 (1973) no. 2, pp. 123-127. doi: 10.1017/S0017089500001865
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[1] 1.Everitt, W. N., On an inequality for the generalized arithmetic and geometric means, Amer. Math. Monthly 70 (1963), 251–255. Google Scholar

[2] 2.Everitt, W. N., On a limit problem associated with the arithmetic–geometric mean inequality, J. London Math. Soc. 42 (1967), 712–718. Google Scholar | DOI

[3] 3.Everitt, W. N., Corrigendum to [2], J. London Math. Soc. (2) 1 (1969), 428–430. Google Scholar | DOI

[4] 4.Hardy, G. H., J. E. Littlewood and G. Pólya, Inequalities (Cambridge, 1934). Google Scholar

[5] 5.McLaughlin, H. W. and Metcalf, F. T., An inequality for generalized means, Pacific J. Math. 22 (1967), 303–311. Google Scholar | DOI

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