Example 9 Given a positive integer $k$, when $x^{k}+y^{k}+z^{k}=1$, find the minimum value of $x^{k+1}+y^{k+1}+z^{k+1}$.
From the assumption and the power mean inequality, we have
$$\left(\frac{x^{k+1}+y^{k+1}+z^{k+1}}{3}\right)^{\frac{1}{k+1}} \geqslant\left(\frac{x^{k}+y^{k}+z^{k}}{3}\right)^{\frac{1}{k}}=\left(\frac{1}{3}\right)^{\frac{1}{k}},$$

Therefore,
$$x^{k+1}+y^{k+1}+z^{k+1} \geqslant 3\left(\frac{1}{3}\right)^{\frac{k+1}{k}}=3^{-\frac{1}{k}}$$

Equality holds when $x=y=z=3^{-\frac{1}{k}}$, so the minimum value is $3^{-\frac{1}{k}}$.