9. Given $a_{1}, a_{2}, \cdots, a_{n} ; b_{1}, b_{2}, \cdots, b_{n}$ are two sets of positive real numbers, and $x_{1}, x_{2}, \cdots, x_{n}$ are positive variables, find the minimum value of the function $y=b_{1}^{k} x_{1}^{k}+b_{2}^{k} x_{2}^{k}+\cdots+b_{n}^{k} x_{n}^{k}$.
9. By Hölder's inequality, we have
$$\begin{array}{l}
\left(b_{1}^{k} x_{1}^{k}+b_{2}^{k} x_{2}^{k}+\cdots+b_{n}^{k} x_{n}^{k}\right)^{\frac{1}{k}}\left[\left(\frac{a_{1}}{b_{1}}\right)^{\frac{k}{k-1}}+\left(\frac{a_{2}}{b_{2}}\right)^{\frac{k}{k-1}}+\cdots+\left(\frac{a_{n}}{b_{n}}\right)^{\frac{k}{k-1}}\right] \geqslant \\
a_{1} x_{1}+a_{2} x_{2}+\cdots+a_{n} x_{n}=p
\end{array}$$

Therefore,
$$\begin{array}{l}
y^{\frac{1}{k}}\left[\left(\frac{a_{1}}{b_{1}}\right)^{\frac{k}{k-1}}+\left(\frac{a_{2}}{b_{2}}\right)^{\frac{k}{k-1}}+\cdots+\left(\frac{a_{n}}{b_{n}}\right)^{\frac{k}{k-1}}\right] \geqslant p \\
y \geqslant \frac{p^{k}}{\left[\left(\frac{a_{1}}{b_{1}}\right)^{\frac{k}{k-1}}+\left(\frac{a_{2}}{b_{2}}\right)^{\frac{k}{k-1}}+\cdots+\left(\frac{a_{n}}{b_{n}}\right)^{\frac{k}{k-1}}\right]^{k-1}}
\end{array}$$

Thus, the minimum value of \( y \) is
$$\frac{p^{k}}{\left[\left(\frac{a_{1}}{b_{1}}\right)^{\frac{k}{k-1}}+\left(\frac{a_{2}}{b_{2}}\right)^{\frac{k}{k-1}}+\cdots+\left(\frac{a_{n}}{b_{n}}\right)^{\frac{k}{k-1}}\right]^{k-1}}$$