ICM 1998, Berlin, Aug. 18–27

Abstracts of Short Communications and Posters

Section:

8. Analysis

1991 MS Classification: 26, 39, 41 Bjelica, Momčilo, University of Novi Sad, Yugoslavia Asymptotic of means Divided differences (M_{p} − M_{q})/(M_{r} − M_{s}) of the power, symmetric and some other mean values of x = (x_{1}, …, x_{n}) have C^{∞} extension (p − q)/(r − s) to a = (a, …, a). For M = φ^{−1}(∑ u_{i}φ(x_{i})) the extension is (ϕ_{1} − ϕ_{2})/(ϕ_{3} − ϕ_{4}), ϕ = φ′′/φ′. Equivalently, the partial derivatives satisfy ∑_{α} M_{ijα}(a) = ϕ^{(k)}(a)u_{i}(δ_{ij}u_{j}), where α denotes sequences of length k from 1, …, n. Asymptotic linearity of the homogeneous means M_{p} ∼ M_{1} + (p − 1)(M_{1} − M_{0}), x → a, can be placed in the infinity x + a, a → ∞, which strengthens Hoehn and Niven asymptotic property. Higher differences, the asymptotic of directional derivative, some asymptotic relations, and other families of means will be presented. References: M. Bjelica, Fixed point and inequalities, thesis, Belgrade, 1990; R. P. Boas, and J. L. Brenner, The asymptotic behaviour of inhomogeneous means, J. Math. Anal. Appl. 123 (1987), 262–264.