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Geometric means of two positive numbers. (English) Zbl 1102.26011
In this interesting paper the authors observe that many classical two variable means are of one of the following types: \( \mu=L\pm\sqrt{Q}, Q/L, C/L\), where \(L, Q, C\) are symmetric forms of degree one, two and three, respectively. They ask what restrictions must be placed on the coefficients of these forms in order that \( \mu\) is a mean: in particular in order that they are internal, that is \(\min\{x.y\}\leq \mu(x,y)\leq \max\{x,y\}\). They answer this question and proceed to discuss the comparability of the resulting classes of means. They further discuss the Gauss compounding and the behaviour under equal increments of the variables in the first two cases, that is the quadratic and root quadratic cases.

26D05 Inequalities for trigonometric functions and polynomials
26E60 Means
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