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An inequality for mixed power means. (English) Zbl 0938.26010

This paper contains a weighted version of a mixed power means inequality proved by B. Mond and the reviewer [Austral. Math. Soc. Gaz. 23, No. 2, 67-70 (1996; Zbl 0866.26015)].
If \(s>r\) and if \(w= (w_1,w_2,\dots, w_n)\) satisfy \[ W_n w_k- W_k w_n>0\quad\text{for }2\leq k\leq n-1,\tag{\(*\)} \] where \(W_k:= \sum^k_{i=1} w_i\), then \[ m_{r,s}(a;w)\geq m_{s,r}(a; w). \] Proof is by mathematical induction, so the condition \((*)\) should be satisfied for \(n= 3,4,\dots\) . So it can be given in simpler form: the sequence \(\{w_k/W_k\}\) is nonincreasing.

MSC:

26D15 Inequalities for sums, series and integrals
26E60 Means

Citations:

Zbl 0866.26015
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