Boas, R. P.; Brenner, J. L. The asymptotic behavior of inhomogeneous means. (English) Zbl 0614.26001 J. Math. Anal. Appl. 123, 262-264 (1987). L. Hoehn and I. Niven [Math. Mag. 58, 151-156 (1985; Zbl 0601.26011)] have proved (by individual proofs for each mean) that the (symmetric) arithmetic, geometric, harmonic means and the root-mean- square satisfy \[ (1)\quad \lim_{x\to \infty}(M(a_ 1+x,...,x_ n+x)- x)=(a_ 1+...+a_ n)/n. \] J. L. Brenner [Pi Mu Epsilon J. 8, 160-163 (1985; Zbl 0601.26012)] proved the same for all root-mean-powers by use of the binomial expansion for positive integer exponents and by monotonicity for noninteger ones. J. L. Brenner and B. C. Carlson [J. Math. Anal. Appl. 123, 265-280 (1987; following review)] have proved the analogue of (1) for homogeneous nonsymmetric means M twice continuously differentiable in each variable in a neighborhood of \((1,...,1),\) with \(q_ k=(\partial M/\partial x_ k)(1,...,1)\quad (k=1,...,n),\) viz. \[ (2)\quad \lim_{x\to \infty}(M(a_ 1+x,...,a_ n+x)-x)=q_ 1a_ 1+...+q_ na_ n. \] Independently, P. S. Bullen (Averages still on move, to appear in Math. Mag.) and the reviewer and Zs. Páles (The behavior of means under equal increments of their variables, to appear in Am. Math. Mon.) gave simple proofs of (2) for homogeneous means for which M is differentiable at \((1,...,1)\) [with \(q_ k=(\partial M/\partial x_ k)(1,...,1)\quad (k=1,...,n)].\) The present paper extends (2) to not necessarily homogeneous quasilinear means \[ M(a_ 1,...,a_ n)=\phi^{-1}[q_ 1\phi (a_ 1)+...+q_ n\phi (a_ n)]\quad (q_ k>0,\quad \sum q_ k=1), \] where \(\phi\) is positive, strictly increasing, differentiable and satisfies \[ \lim_{x\to \infty}\phi (x)=\infty,\quad \lim_{x\to \infty}(\phi '(x+y)/\phi '(x))=1, \] uniformly with respect to y on each finite positive interval, \(\phi '(x)/\phi (x)=o(1)\quad as\quad x\to \infty\) and \(\phi^{-1}\) has the same properties as \(\phi\) (there are also sufficient conditions for decreasing \(\phi)\). Reviewer: J.Aczél Cited in 3 ReviewsCited in 3 Documents MSC: 26A12 Rate of growth of functions, orders of infinity, slowly varying functions 26A48 Monotonic functions, generalizations 26D15 Inequalities for sums, series and integrals Keywords:homogeneous means; inhomogeneous means; asymptotic behavior; quasilinear means; ordo; harmonic means; root-mean-square; root-mean-powers Citations:Zbl 0614.26002; Zbl 0601.26011; Zbl 0601.26012 PDFBibTeX XMLCite \textit{R. P. Boas} and \textit{J. L. Brenner}, J. Math. Anal. Appl. 123, 262--264 (1987; Zbl 0614.26001) Full Text: DOI References: [1] Brenner, J. L., Limits of means for large values of the variables, Pi Mu Epsilon J., 8, 160-163 (1985) · Zbl 0601.26012 [2] Brenner, J. L.; Carlson, B. C., Homogeneous mean values: Weights and asymptotics, J. Math. Anal. Appl., 123, 265-280 (1987) · Zbl 0614.26002 [3] Hoehn, L.; Niven, I., Averages on the move, Math. Mag., 58, 151-156 (1985) · Zbl 0601.26011 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.