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On the convergence of means. (English) Zbl 0727.26012

This note is the latest generalization of L. Hoehn and I. Niven’s [Math. Mag. 58, 151-156 (1985; Zbl 0601.26011)] theorem that if M is the arithmetic, geometric, harmonic, or quadratic mean, then (for positive \(x_ k)\) the limit of \(M(x_ 1+t,...,x_ n+t)-t\) is the arithmetic mean of \(\{x_ k\}\). The author finds necessary and sufficient conditions for a family of deviation means [Z. Daróczy, Publ. Math. 19, 211-217 (1972; Zbl 0265.26010)] to be convergent, and deduces necessary and sufficient conditions for Hoehn and Niven’s theorem to hold for these means, which include the quasi- arithmetic means.
Reviewer: R.P.Boas (Seattle)

MSC:

26D15 Inequalities for sums, series and integrals
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References:

[1] Aczél, J.; Losonczi, L.; Páles, Zs, The behaviour of comprehensive classes of means under equal increments of their variables, (Walter, W., General Inequalities 5. General Inequalities 5, Internat. Ser. Numer. Math., Vol. 80 (1987), Birkhäuser: Birkhäuser Basel/Boston/Stuttgart), 459-461 · Zbl 0638.26014
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