The asymptotic behavior of inhomogeneous means
\textit{L. Hoehn} and \textit{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 \[ (1)\quad \lim_{x\to \infty}(M(a_ 1+x,...,x_ n+x)- x)=(a_ 1+...+a_ n)/n. \] \textit{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. \textit{J. L. Brenner} and \textit{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, \textit{P. S. Bullen} (Averages still on move, to appear in Math. Mag.) and the reviewer and \textit{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 \(\phi)\).
scientific article; zbMATH DE number 3993885
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