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Homogeneous mean values: weights and asymptotics. (English) Zbl 0614.26002

The authors prove the following. Let M be a reflexive \((M(\lambda,...,\lambda)=\lambda)\) homogeneous mean (presumably on the Cartesian power of a real interval going to \(\infty)\), with continuous second derivatives in each variable in a neighborhood of (1,...,1) and \(q_ k=(\partial M/\partial x_ k)(1,...,1)\quad (k=1,...,n).\) Then \[ M(a_ 1+x,...,a_ n+x)=x+\sum^{n}_{k=1}q_ ka_ k+0(1/x)\quad as\quad x\to \infty. \] (For the history of the problem see the preceding review.) Several means satisfying these conditions (and one which does not) are enumerated. One of them, the hypergeometric mean [B. C. Carlson, Proc. Am. Math. Soc. 16, 759-766 (1965; Zbl 0137.268)] is further investigated for infinite sequences.
Reviewer: J.Aczél

MSC:

26A12 Rate of growth of functions, orders of infinity, slowly varying functions
26D15 Inequalities for sums, series and integrals
33C05 Classical hypergeometric functions, \({}_2F_1\)
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