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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:
26E60 Means
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