Suppose $x_1\ge x_2\ge \cdots \ge x_n\ge 0$ and $y_1\ge y_2\ge\cdots\ge y_n\ge0$ be reals such that for any positive **integer** $p$,
$$
\sum_{i=1}^n x_i^p \geq \sum_{i=1}^n y_i^p.
$$

Question:Is there a constant $c\ge 1$ such that for all $x,y$ satisfying the above, we have $cx\succ_w y$.

Here $cx$ denotes the sequence $cx_1, cx_2,\ldots, cx_n$ and $a \succ_w b$ means that $a$ weakly majorizes $b$ from below.

Recall that a sequence $a_1\ge a_2\ge \cdots\ge a_n$ is said to weakly majorize from below another sequence $b_1\ge b_2\ge\cdots\ge b_n$ if for all $1\le k \le n$ $$ \sum_{i=1}^k a_i\ge \sum_{i=1}^k b_i. $$

**Update:** user35593's example $x=(4,1,1)$ and $y=(3,3,0)$ shows that if there is such a $c$, then $c\ge 1.2$.

**Remark 1:** If $\sum f(x_i)\ge \sum f(y_i)$ for all convex non-decreasing functions, then we have $x\succ_w y$, a result due to Karamata; Hardy, Littlewood and Pólya. In fact testing against all functions of the form $z\mapsto (z-t)^+$ is enough.