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Quadratic means. (English) Zbl 1041.26015
The paper considers expressions of the form \( {\mathfrak M} = f_1 \mp\sqrt{f_2}\) where \(f_1, f_2\), is a symmetric form of degree \(1, 2,\) in \(n\) variables. The case where \(f_1=0\) has been considered by the second author [Math. Inequal. Appl. 6, 581–593 (2003; Zbl 1047.26015)]. Using suitable bases for the two spaces of forms these expressions can be given in terms of two real numbers. The authors find necessary and sufficient conditions on these numbers for \({\mathfrak M}\) to be a mean. The 2 basis elements for the space of symmetric forms of degree \(2\) are related in a simple way to the mean and standard deviation and the simple internality of the means leads to an improvement of the Brunk inequalities. Comparability of two means of this type is studied as well as the behaviour of the means under equal increments of the variables.

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