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一类对称函数的Schur $\bm m$-指数凸性

王文,杨世国   

  1. 合肥师范学院数学与统计学院,   合肥 230601
  • 收稿日期:2013-08-26 出版日期:2014-03-25 发布日期:2014-07-18

王文,杨世国. 一类对称函数的Schur $\bm m$-指数凸性[J]. 系统科学与数学, 2014, 34(3): 367-375.

WANG Wen, YANG Shiguo. ON THE SCHUR m-POWER CONVEXITY FOR A CLASS OF SYMMETRIC FUNCTIONS[J]. Journal of Systems Science and Mathematical Sciences, 2014, 34(3): 367-375.

ON THE SCHUR m-POWER CONVEXITY FOR A CLASS OF SYMMETRIC FUNCTIONS

WANG Wen, YANG Shiguo   

  1. School of Mathematics and Statistics, Hefei Normal University, Hefei 230601
  • Received:2013-08-26 Online:2014-03-25 Published:2014-07-18
证明了在函数$f(x)$为乘凸的条件下, 一类对称函数$\sum_{n}^{r}(f(x))=\sum_{1\leq i_{1}<i_{2}<\cdots<i_{r}\leqn}f\\\bigg(\prod_{j=1}^{r} x^{\frac{1}{r}}_{i_j}\bigg)$  是 Schur$m$-指数凸的,  这里$x\in \mathbb{R}^{n}_{+}$, $r\in\mathbb{N}^{+}=\{1,2,\cdots,n\}$. 此结果包含了近期一些已有的结果.应用该结果, 获得了一些特殊的对称函数的Schur $m$-指数凸性.
In this paper, assume that the function $f(x)$ is multiplicatively convex, it is  proven that the symmetric function $\sum_{n}^{r}(f(x))=\sum_{1\leq i_{1}<i_{2}<\cdots<i_{r}\leq n}f\left(\prod_{j=1}^{r} x^{\frac{1}{r}}_{i_j}\right)$  is Schur $m$-power convex for $x\in R^{n}_{+}$ and $r\in N^{+}=\{1,2,\cdots,n\}$, which generalizes some known results. As its application, the Schur $m$-power convexity  of several special symmetric functions is given.

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[1] 张鑑,顾春,石焕南. 一类对称函数的Schur $m$-指数凸性的注记[J]. 系统科学与数学, 2016, 36(10): 1779-1782.
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