Necessary and sufficient conditions for the Cauchy mean value to be Schur- power convex or concave are given. As applications, Schur- power convexity of the exponential mean is discussed, and a comparative inequality between the Gini mean and the Stolarsky mean is given.
"Schur-$m$ power convexity of Cauchy means and its application." Rocky Mountain J. Math. 50 (5) 1859 - 1869, October 2020. https://doi.org/10.1216/rmj.2020.50.1859