证明了在函数$f(x)$为乘凸的条件下, 一类对称函数$$\begin{gathered} \sum _n^r(f(x))=\sum _{1\le i_1<i_2<\cdots <i_r\text{leqn}}f \\ (\prod _{j=1}^r{x}_{i_j}^{\frac{1}{r}}) \end{gathered}$$  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 $\sum _n^r(f(x))=\sum _{1\le i_1<i_2<\cdots <i_r\le n}f\left(\prod _{j=1}^r{x}_{i_j}^{\frac{1}{r}}\right)$  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.