• Power sum and elementary symmetric polynomial

Let $x_1,. . . , x_n$ be variables, denote for $k \ge 1$ by $p_k(x_1,\dots,x_n)$ the $k-th$ power sum:

$$ p_k(x_1,\dots,x_n)=\sum\nolimits_{i=1}^nx_i^k = x_1^k+\cdots+x_n^k,$$

and for $k \ge 1$ denote by $e_k(x_1,\dots, x_n)$ the elementary symmetric polynomial (that is, the sum of all distinct products of k distinct variables), so

: \begin{align} e_0(x_1, \ldots, x_n) &= 1,\\ e_1(x_1, \ldots, x_n) &= x_1 + x_2 + \cdots + x_n,\\ e_2(x_1, \ldots, x_n) &= \textstyle\sum_{1\leq i<j\leq n}x_ix_j,\\ e_n(x_1, \ldots, x_n) &= x_1 x_2 \cdots x_n,\\ e_k(x_1, \ldots, x_n) &= 0, \quad\text{for}\ k>n.\\ \end{align}

• Majorizes

If $x_1,. . . , x_n$ and $y_1, . . . , y_n$ are numbers, such that $(x_1,. . . , x_n)$ majorizes $(x_1,. . . , x_n)$ if only if

$x_1+x_2+\dots+x_n = y_1+y_2+\dots+y_n$ and $x_{1}\geq x_{2}\geq \cdots \geq x_{n}$ and $y_{1}\geq y_{2}\geq \cdots \geq y_{n}$

$ x_{1}+\cdots +x_{i}\geq y_{1}+\cdots +y_{i}$ for all $i \in \{1,..., n − 1\}$.

I am looking for a proof of the inequality related to Power sum and elementary symmetric polynomial and majorizes as follows:

Let $n$ be the interger number $n \ge 2$ and $x_1,. . . , x_n$ and $y_1, . . . , y_n$ are nonegative real numbers such that $x_1+x_2+\dots+x_n = y_1+y_2+\dots+y_n$ then $(x_1,. . . , x_n)$ majorizes $(x_1,. . . , x_n)$ if only if

$$e_k(x_1, \ldots, x_n) \leq e_k(y_1, \ldots, y_n)$$

and $$ p_k(x_1,\dots,x_n) \ge p_k(y_1,\dots,y_n)$$

for all $k \in \{2, \cdots, n \}$