Theorem 13.11 Let $u_{1}, u_{2}, \cdots, u_{n}, t$ be non-negative real numbers, $2 \leqslant m, n \in \mathbf{N}$, and let the polynomial $f(y)$ in $y$ be defined as
$$f(y)=\prod_{j_{1}, j_{2}, \cdots, j_{n-1}} \prod_{Y_{0,1, \cdots, m-1}}\left(y-\left[\sqrt[m]{u_{1}}+\zeta_{m}^{\left(j_{1}\right)} \sqrt[m]{u_{2}}+\cdots+\zeta_{m}^{j_{n-1} m} \sqrt[m]{u_{n}}\right]^{m}\right)$$

Then the inequality
$$\sqrt[m]{u_{1}}+\sqrt[m]{u_{2}}+\cdots+\sqrt[m]{u_{n}} \leqslant \sqrt[m]{t}$$

holds if and only if the following $r=m^{n-1}$ real numbers $c_{0}, c_{1}, \cdots, c_{r-1}$ satisfy
$$c_{0}=f(t) \geqslant 0, c_{1}=f^{(1)}(t) \geqslant 0, \cdots, c_{r-1}=f^{(r-1)}(t) \geqslant 0$$

where $f^{(k)}$ is the $k$-th derivative of $f$, $k=1,2, \cdots, r-1$.