Example 8 Let $0<a<1$, the function $f(x)=\log _{a} \frac{x-3}{x+3}$ and $g(x)=1+\log _{a}(x-1)$ have a common domain $D$. If $[m, n] \subset D, f(x)$ has a range of $[g(n), g(m)]$ on $[m, n](m<n)$, find the range of values for $a$.
Solve $D=\{x \mid x>3\}$. It is easy to see that $g(x)$ is monotonically decreasing on $D$.
When $33$. That is, $m$ and $n$ are two distinct real roots of the equation $a x^{2}+(2 a-1) x+3-3 a=0$ that are both greater than 3.
Let $h(x)=a x^{2}+(2 a-1) x+3-3 a$, then we have
$$
\left\{\begin{array}{l}
00 \\
-\frac{2 a-1}{2 a}>3, \\
h(3)>0,
\end{array}\right.
$$

Solving this, we get $0<a<\frac{2-\sqrt{3}}{4}$