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Radical and rational means of degree two. (English) Zbl 1047.26015
The author gives conditions for $$\sqrt{Q}, Q/S$$ to be internal, where $$Q$$ is a symmetric quadratic form and $$S=\sum_{i=1}^nx_i$$. In the case of two variables many well-known means are of one of these types. Precisely, $$\sqrt{Q}$$ is internal if and only if $$Q= {2\over n(n-1)}(s \Delta+P),\, 0\leq s\leq 1$$, and $$Q/S$$ is internal if and only if $$Q={t \Delta + 2P\over S},\, 0\leq t\leq 1$$; $$\Delta=\sum_{1\leq i<j\leq n}(x_j-x_i)^2, P = \sum_{1\leq i<j\leq n}x_ix_j$$. This gives two one-parameter families of means and also comparability results are given. Some comments are made on the higher degree cases.

##### MSC:
 2.6e+61 Means
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