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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:
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