Schur power convexity of Gini means (Q2856991)

The Schur convexity of the two variable, two parameter Gini means has been discussed by many authors and the results extended to various generalisations of Schur convexity such as Schur geometric and harmonic convexity. In this paper these results are included in a general result by the use of the concept of Schur power convexity that essentially says that \(\phi\) is Schur \(f\)-convex if \(f(\mathbf x)\prec f(\mathbf y) \Longrightarrow \phi(\mathbf x)\leq \phi(\mathbf y)\), and taking \(f(x) = (x^m - 1)/m, m\neq 0,= \log x, m=0 \), gives the concept of \(m\)-power Schur convexity. The author gives a series of conditions on the values of \(m\) and the parameters in the Gini-means that permit the deduction that this mean is \(m\)-power Schur convex or concave.