Schur convexity and Schur multiplicative convexity for a class of symmetric functions with applications - Wei-feng, Xia (Ukrains’kyi Matematychnyi Zhurnal, 2009)
For $x = (x_1, x_2, …, x_n) ∈ (0, 1 ]^n$ and $r ∈ \{ 1, 2, … , n\}$, a symmetric function $F_n(x, r)$ is defined by the relation$$F_n(x,r) = F_n(x_1, x_2, …, x_n; r) = ∑_{1 ⩽ i_1 < i_2…i_r ⩽n } ∏^r_{j=1}\frac{1−x_{i_j}}{x_{i_j}},$$where $i_1 , i_2 , ... , i_n$ are positive integers. This paper deals with the Schur convexity and Schur multiplicative convexity of $F_n(x, r)$. As applications, some inequalities are established by using the theory of majorization.