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The asymptotic behavior of inhomogeneous means. (English) Zbl 0614.26001

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

MSC:

26A12 Rate of growth of functions, orders of infinity, slowly varying functions
26A48 Monotonic functions, generalizations
26D15 Inequalities for sums, series and integrals
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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
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