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Many properties of extended mean values have been considered in [10].
For and, extended mean values were defined in [9] by Stolarsky as follows: (216).
For fixed, (i if or, then the extended mean values are Schur-convex with respect to, (ii)if, then the extended mean values are Schur-concave with respect to.
For fixed, (i the extended mean values are Schur-convex with respect to if and only if, (ii the extended mean values are Schur-concave with respect to if and only if.
Schur-convexity of the extended mean values with respect to and was considered in [4, 5, 11].
Shi et al. in [5], using Theorem A1 and Lemma A3 obtained the following condition for the Schur-convexity of the extended mean values with respect to.
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Justyna Jupowicz-Kozak
CEO of Professional Science Editing for Scientists @ prosciediting.com