Schur-Convexity of Generalized Heronian Mean

In this paper, the generalized Heronian mean of n-tuple positive real valued function was defined which was the extension of the generalized Heronian mean of n-tuple positive real variable. By so-called Schur’s condition, the Schur-convexity and Schur-geometric convexity and Schur-harmonic convexity are studied for the generalized Heronian mean of n-tuple positive real valued function, and derive some results. Then the Schur-convexity also was discussed for its quotient, and as applications, a inequality is derived.

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