Schur Convexity and Inequalities for a Multivariate Symmetric Function

Abstract

In the article, we provide the Schur, Schur multiplicative, and Schur harmonic convexities properties for the symmetry function $F_n\left(x,r\right)=F_n\left(x_1,x_2,\cdots ,x_n;r\right)={\prod }_{1\le i_1<i_2<\cdots <i_r\le n}{\left({\sum }_{j=1}^r\left(x_{i_j}/1-x_{i_j}\right)\right)}^{1/r}$ on ${\left[0,1\right)}^n$ and find several new analytical inequalities by use of the majorization theory, where $x=\left(x_1,\cdots ,x_n\right)\in {\left[0,1\right)}^n$, $r=1,2,\cdots ,n$ and $i_1,i_2,\cdots ,i_n$ are positive integers.