On the convergence of means (Q803302)

This note is the latest generalization of \textit{L. Hoehn} and \textit{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 [\textit{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.