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 |
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