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Gnan mean for two variables. (English) Zbl 1149.26032

The Gnan means, the name is unexplained, are generalizations of a generalization of the Heron mean. For example: \[ gn(a,b,k;\alpha, \beta)= \Biggl(1/(k+1)\biggr(\sum_{i=0}^k\Bigl(\bigl((k-i)a^{ \alpha} + ib^{ \beta}\bigr)/k\Bigr)\biggr)^{\beta/ \alpha}\Biggr)^{1/ \beta} \] which reduces to the Heron mean if \( \alpha= 0,\beta = 1, k=2\). Monotonicity results with respect to \(\alpha , \beta \) are immediate from those for the power means. In addition, monotonicity results with respect to \(k\) are also obtained. Limit results are given and weighted means of this type are also considered.

MSC:

26D15 Inequalities for sums, series and integrals
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