Combine my first previous question and second previous question with the Muirhead inequality. I have posed conjectures of two inequalities 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 $(y_1,\dots, y_n)$. Then for all reals $0 \leq a_1, a_2,\cdots,a_n \leq 1$,$$\sum\limits_{1 \le p_1 <\cdots < p_m \le n}\left( \sum\limits_{1 \le i_1 <\cdots < i_m \le n} x_{i_1}^{a_{p_1}} \cdots x_{i_m}^{a_{p_m}} \right) \leq \sum\limits_{1 \le p_1 <\cdots < p_m \le n}\left( \sum\limits_{1 \le i_1 <\cdots < i_m \le n} y_{i_1}^{a_{p_1}} \cdots y_{i_m}^{a_{p_m}} \right) $$

**Example for inequality 1 with $n=3, m=2$:**

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

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 $(y_1,\dots, y_n)$. Then for all reals $ a_1, a_2,\dots,a_n \geq 0$,$$\sum\limits_{1 \le p_1 <\cdots < p_m \le n}\left( \sum\limits_{1 \le i_1 <\cdots < i_m \le n} a_{i_1}^{x_{p_1}} \cdots a_{i_m}^{x_{p_m}} \right) \geq \sum\limits_{1 \le p_1 <\cdots < p_m \le n}\left( \sum\limits_{1 \le i_1 <\cdots < i_m \le n} a_{i_1}^{y_{p_1}} \cdots a_{i_m}^{y_{p_m}} \right)$$

**Example for inequality 2 with $n=3, m=2$:**

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

My question:I am looking for a proof of two inequalities above.

symmetricalsums, i.e. adding all the permutations of the $a_i$'s! E.g. if there is a term ${a_1}^{x_1}{a_2}^{x_2}$ on the LHS, there should also be ${a_2}^{x_1}{a_1}^{x_2}$. $\endgroup$7more comments