J. Math. Anal. Appl. 325 (2007) 554–559 www.elsevier.com/locate/jmaa A complete asymptotic expansion of power means Ulrich Abel a,∗ , Mircea Ivan b a Fachbereich MND, Fachhochschule Gießen-Friedberg, University of Applied Sciences, Wilhelm-Leuschner-Straße 13, 61169 Friedberg, Germany b Department of Mathematics, Technical University Cluj-Napoca, Str. C. Daicoviciu 15, 400020 Cluj-Napoca, Romania Received 31 December 2005 Available online 28 February 2006 Submitted by William F. Ames Abstract The complete asymptotic expansion of power means in terms of Bell polynomials is obtained. Some results recently obtained by M. Bjelica are generalized.  2006 Elsevier Inc. All rights reserved. Keywords: Asymptotic expansions; Means 1. Introduction and results Let x i > 0, w i  0, 1  i  n, ∑ n i =1 w i = 1, x = (x 1 ,...,x n ). The power mean of order p is defined as M p (x) :=  n  i =1 w i x p i  1/p ( p ∈ R \{0} ) (1) and in the case p = 0 as M 0 (x) := n  i =1 x w i i . * Corresponding author. E-mail addresses: ulrich.abel@mnd.fh-friedberg.de (U. Abel), mircea.ivan@math.utcluj.ro (M. Ivan). 0022-247X/$ – see front matter  2006 Elsevier Inc. All rights reserved. doi:10.1016/j.jmaa.2006.01.089