Schur \(m\)-power convexity of generalized Hamy symmetric function (Q2928376)

Let \(r\in \{1, 2, \dots, n\}\). The authors prove that the generalized Hamy symmetric function, NEWLINE\[NEWLINEF^\ast_n ((x_1,x_2,\dots,x_n), r) =\sum_{i_1 +i_2 +\dots+i_n =r,\;i_j>0} (x_1^{i_1} x_2^{i_2} \dots x_n^{i_3})^{\frac{1}{r}},NEWLINE\]NEWLINENEWLINE is Schur \(m\)-power concave -- respectively convex -- on the positive orthant of the real \(n\)-dimensional vector space when \(m\) is greater or equal to 1 -- respectively when \(m\) is less or equal to \(\frac{1}{r}\).