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On limit problems associated with some inequalities
Published online by Cambridge University Press: 18 May 2009
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Let {an} be a sequence of non-negative real numbers. Suppose that
Then 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.
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- Copyright © Glasgow Mathematical Journal Trust 1973
References
REFERENCES
1.Everitt, W. N., On an inequality for the generalized arithmetic and geometric means, Amer. Math. Monthly 70 (1963), 251–255.Google Scholar
2.Everitt, W. N., On a limit problem associated with the arithmetic–geometric mean inequality, J. London Math. Soc. 42 (1967), 712–718.CrossRefGoogle Scholar
3.Everitt, W. N., Corrigendum to [2], J. London Math. Soc. (2) 1 (1969), 428–430.CrossRefGoogle Scholar
5.McLaughlin, H. W. and Metcalf, F. T., An inequality for generalized means, Pacific J. Math. 22 (1967), 303–311.CrossRefGoogle Scholar
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