## 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

### Keywords:

convex functions; weighted inequality; mixed power means

Zbl 0866.26015
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