This is not an answer, this is a message to @Gjergji Zaimi. Thank You very much. (And thank to dear  Wolfgang very much). Your answer is true with the version above. But if You see my comment to You and Wolfgang. I said that when $m=n$ the inequality 1 is [A Muirhead Like Inequality](https://mathoverflow.net/questions/303013/a-muirhead-like-inequality). The inequality 2 is [A Muirhead](https://www.cut-the-knot.org/arithmetic/algebra/MuirheadInequality.shtml). But You and Wolfgang said no.

So I see detail my question again (with Wolfgang help me formulate). I think the old version is not formulate true with my ideas, so may I re-formulate my ideas again as follows: 


>**Inequality 1:** Let $n>2$ and $1 \le m \le n$ be integers. Let $x_1, \dots, x_n$ and $y_1,\dots, y_n$ be nonnegative real numbers such that $(x_1,\dots, x_n)$ [majorizes](https://en.wikipedia.org/wiki/Majorization) $(y_1,\dots, y_n)$. Then for all reals $0 \leq a_1, a_2,\cdots,a_n \leq 1$, 

>$$\sum\limits_{sym}\left(  \sum\limits_{sym} x_{i_1}^{a_{p_1}} \cdots x_{i_m}^{a_{p_m}} \right) \leq \sum\limits_{sym}\left(  \sum\limits_{sym} y_{i_1}^{a_{p_1}} \cdots y_{i_m}^{a_{p_m}} \right) $$

* When $m=n$ this inequality is [A Muirhead Like Inequality](https://mathoverflow.net/questions/303013/a-muirhead-like-inequality)

>**Inequality 2:** Let $n>2$ and $1 \le m \le n$ be integers. Let $x_1, \dots, x_n$ and $y_1,\dots, y_n$ be nonnegative real numbers such that $(x_1,\dots, x_n)$ [majorizes](https://en.wikipedia.org/wiki/Majorization) $(y_1,\dots, y_n)$. Then for all reals $ a_1, a_2,\dots,a_n \geq 0$, 

> $$\sum\limits_{sym}\left(  \sum\limits_{sym} a_{i_1}^{x_{p_1}} \cdots a_{i_m}^{x_{p_m}} \right) \geq \sum\limits_{sym}\left(  \sum\limits_{sym} a_{i_1}^{y_{p_1}} \cdots a_{i_m}^{y_{p_m}} \right)$$

* When $m=n$ this inequality is [A Muirhead](https://www.cut-the-knot.org/arithmetic/algebra/MuirheadInequality.shtml)

**Example for Inequality 1 with $n=3$, $m=2$.** 

$${x_1}^{a_1}.{x_2}^{a_2}+{x_1}^{a_1}.{x_3}^{a_2}+{x_2}^{a_1}.{x_3}^{a_2}+{x_1}^{a_2}.{x_2}^{a_3}+{x_1}^{a_2}.{x_3}^{a_3}+{x_2}^{a_2}.{x_3}^{a_3}+{x_1}^{a_1}.{x_2}^{a_3}+{x_1}^{a_1}.{x_3}^{a_3}+{x_2}^{a_1}.{x_3}^{a_3}+{x_1}^{a_2}.{x_2}^{a_1}+{x_1}^{a_2}.{x_3}^{a_1}+{x_2}^{a_2}.{x_3}^{a_1}+{x_1}^{a_3}.{x_2}^{a_2}+{x_1}^{a_3}.{x_3}^{a_2}+{x_2}^{a_3}.{x_3}^{a_2}+{x_1}^{a_3}.{x_2}^{a_1}+{x_1}^{a_3}.{x_3}^{a_1}+{x_2}^{a_3}.{x_3}^{a_1} \leq {y_1}^{a_1}.{y_2}^{a_2}+{y_1}^{a_1}.{y_3}^{a_2}+{y_2}^{a_1}.{y_3}^{a_2}+{y_1}^{a_2}.{y_2}^{a_3}+{y_1}^{a_2}.{y_3}^{a_3}+{y_2}^{a_2}.{y_3}^{a_3}+{y_1}^{a_1}.{y_2}^{a_3}+{y_1}^{a_1}.{y_3}^{a_3}+{y_2}^{a_1}.{y_3}^{a_3}+{y_1}^{a_2}.{y_2}^{a_1}+{y_1}^{a_2}.{y_3}^{a_1}+{y_2}^{a_2}.{y_3}^{a_1}+{y_1}^{a_3}.{y_2}^{a_2}+{y_1}^{a_3}.{y_3}^{a_2}+{y_2}^{a_3}.{y_3}^{a_2}+{y_1}^{a_3}.{y_2}^{a_1}+{y_1}^{a_3}.{y_3}^{a_1}+{y_2}^{a_3}.{y_3}^{a_1}
$$

**Example Inequality 2 with $n=3$, $m=2$.**

$${a_1}^{x_1}.{a_2}^{x_2}+{a_1}^{x_1}.{a_3}^{x_2}+{a_2}^{x_1}.{a_3}^{x_2}+{a_1}^{x_2}.{a_2}^{x_3}+{a_1}^{x_2}.{a_3}^{x_3}+{a_2}^{x_2}.{a_3}^{x_3}+{a_1}^{x_1}.{a_2}^{x_3}+{a_1}^{x_1}.{a_3}^{x_3}+{a_2}^{x_1}.{a_3}^{x_3}+{a_1}^{x_2}.{a_2}^{x_1}+{a_1}^{x_2}.{a_3}^{x_1}+{a_2}^{x_2}.{a_3}^{x_1}+{a_1}^{x_3}.{a_2}^{x_2}+{a_1}^{x_3}.{a_3}^{x_2}+{a_2}^{x_3}.{a_3}^{x_2}+{a_1}^{x_3}.{a_2}^{x_1}+{a_1}^{x_3}.{a_3}^{x_1}+{a_2}^{x_3}.{a_3}^{x_1} \geq {a_1}^{y_1}.{a_2}^{y_2}+{a_1}^{y_1}.{a_3}^{y_2}+{a_2}^{y_1}.{a_3}^{y_2}+{a_1}^{y_2}.{a_2}^{y_3}+{a_1}^{y_2}.{a_3}^{y_3}+{a_2}^{y_2}.{a_3}^{y_3}+{a_1}^{y_1}.{a_2}^{y_3}+{a_1}^{y_1}.{a_3}^{y_3}+{a_2}^{y_1}.{a_3}^{y_3}+{a_1}^{y_2}.{a_2}^{y_1}+{a_1}^{y_2}.{a_3}^{y_1}+{a_2}^{y_2}.{a_3}^{y_1}+{a_1}^{y_3}.{a_2}^{y_2}+{a_1}^{y_3}.{a_3}^{y_2}+{a_2}^{y_3}.{a_3}^{y_2}+{a_1}^{y_3}.{a_2}^{y_1}+{a_1}^{y_3}.{a_3}^{y_1}+{a_2}^{y_3}.{a_3}^{y_1}
$$