Example 2 Given that $x, y, z$ are positive real numbers, and satisfy $x^{4}+y^{4}+z^{4}=1$, find the minimum value of $\frac{x^{3}}{1-x^{8}}+\frac{y^{3}}{1-y^{8}}$ + $\frac{z^{3}}{1-z^{8}}$. (1999 Jiangsu Province Mathematical Winter Camp Problem)
Solve by Cauchy-Schwarz inequality
$$\begin{array}{l}
\left(\frac{x^{3}}{1-x^{8}}+\frac{y^{3}}{1-y^{8}}+\frac{z^{3}}{1-z^{8}}\right)\left[x^{5}\left(1-x^{8}\right)+y^{5}\left(1-y^{8}\right)+z^{5}\left(1-z^{8}\right)\right] \geqslant \\
\left(x^{4}+y^{4}+z^{4}\right)^{2}
\end{array}$$

By the AM-GM inequality, we get

So
$$8 \cdot \sqrt[4]{\left(\frac{1}{3}\right)^{9}} x^{4} \geqslant x^{5}-x^{13}$$

Similarly,
$$\begin{array}{l}
8 \cdot \sqrt[4]{\left(\frac{1}{3}\right)^{9}} y^{4} \geqslant y^{5}-y^{13} \\
8 \cdot \sqrt[4]{\left(\frac{1}{3}\right)^{9}} z^{4} \geqslant z^{5}-z^{13}
\end{array}$$

Adding the above three inequalities, we get
$$8 \sqrt[4]{\left(\frac{1}{3}\right)^{9}} \geqslant x^{5}\left(1-x^{8}\right)+y^{5}\left(1-y^{8}\right)+z^{5}\left(1-z^{8}\right)$$
(2) Substituting (2) into (1) gives
$$\frac{x^{3}}{1-x^{8}}+\frac{y^{3}}{1-y^{8}}+\frac{z^{3}}{1-z^{8}} \geqslant \frac{9}{8} \sqrt[4]{3}$$

Thus, the minimum value of $\frac{x^{3}}{1-x^{8}}+\frac{y^{3}}{1-y^{8}}+\frac{z^{3}}{1-z^{8}}$ is $\frac{9}{8} \sqrt[4]{3}$. This minimum value is achieved when $x=y=z=\sqrt[4]{\frac{1}{3}}$.