A complete asymptotic expansion of power means. (English) Zbl 1104.41015
Let \(x_j>0\), \(w_j\geq 0\) \((j=1,\dots,n)\), \(\sum_{j=1}^n w_j=1\) and \(M_p(\mathbf{x}):= (\sum_{j=1}^n w_j x_j^p)^{1/p}\) for \(p\neq 0\) and \(M_0(\mathbf{x}):= \prod_{j=1}^n x_j^{w_j}.\) The authors’ purpose is to give a complete asymptotic expansion of \(M_p({\mathbf x}+{\mathbf a})\) in terms of partial exponential Bell polynomials.
Reviewer: János Aczél (Waterloo/Ontario)
MSC:
| 41A60 | Asymptotic approximations, asymptotic expansions (steepest descent, etc.) |
| 33E50 | Special functions in characteristic \(p\) (gamma functions, etc.) |
References:
| [1] | Bjelica, M., Asymptotic linearity of mean values, Math. Vesnik, 51, 1-2, 15-19 (1999) |
| [2] | Comtet, L., Advanced Combinatorics (1974), Reidel: Reidel Dordrecht · Zbl 0283.05001 |
| [3] | Guo, Bai-Ni; Qi, Feng, Inequalities for generalized weighted mean values of convex function, Math. Inequal. Appl., 4, 2, 195-202 (2001) · Zbl 0987.26019 |
| [4] | Rudin, W., Real and Complex Analysis (1974), McGraw-Hill · Zbl 0278.26001 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
