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On a generalization of \(u\)-means. (English) Zbl 0751.26012

The author offers a “weighted” generalization of H. Bauer’s \(u\)-means [H. Bauer, Manuscr. Math. 55, 199-211 (1986; Zbl 0592.26019)]. Again an interval \(I\subset]0,\infty[\) {in the paper \([0,\infty[\}\) is called contractive if \(x,y\in I\Rightarrow xy\in I\). Let \(I\) and \(I'\) be two contractive intervals, let \(\varphi:I\to I'\) be continuous, strictly monotonic and such that \(x\mapsto\varphi(\alpha x)/\varphi(x)\) is, for all \(\alpha\in I\), monotonic in the same sense as \(\varphi\) and let \(u:I'\to\mathbb{R}_ +\) be continuous, positive and either (“type \(A\)”) decreasing or (“type \(B\)”) such that \(x\mapsto u(x)/x\) is strictly increasing. Then, for all \(a_ 1,\ldots,a_ n\in I\quad(n\geq 2)\), the equation \[ u(\varphi(x^{n-1})/\varphi(x))=\sum^ n_{k=1}u(\varphi(a_ 1a_ 2\cdots a_{k-1}a_{k+1}\cdots a_ n))/\sum^ n_{k=1}\varphi(a_ k) \] has a unique solution \(x=:M_{u,\varphi}(a_ 1,\ldots,a_ n)\).
Bauer’s \(u\)-means are the \(M_{u,\text{id}}\), but the author points out that \(M_{1,1/\text{id}}\) is the harmonic mean \(n/(a_ 1^{- 1}+\cdots+a_ n^{-1})\), which is not a \(u\)-mean.
The author offers, as result, the inequalities \[ M_{u,\text{id}}(a_ 1,\ldots,a_ n)\leq(a_ 1+\cdots+a_ n)/n, \]
\[ M_{u,1/\text{id}}(a_ 1,\ldots,a_ n)\leq n/(a_ 1^{-1}+\cdots+a_ n^{-1}) \] if \(u\) is convex and of type \(A\) or concave and of type \(B\).

MSC:

26D15 Inequalities for sums, series and integrals
26A48 Monotonic functions, generalizations
26A51 Convexity of real functions in one variable, generalizations

Citations:

Zbl 0592.26019
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