A property of all-pass filters
First, observe that it's sufficient to show the property for a first-order all-pass filter, because any higher order all-pass filter can be written as a cascade of first-order all-pass sections. And if

$$|G_i(z)|&lt;1\quad\text{for}\quad |z|&gt;1$$

then also

$$\left|\prod_iG_i(z)\right|&lt;1\quad\text{for}\quad |z|&gt;1$$

must be satisfied.

Now derive an expression for the squared magnitude of the first-order all-pass transfer function $G_1(z)$:

$$|G_1(z)|^2=\frac{1-2\text{Re}\{\alpha^*z\}+|\alpha|^2|z|^2}{|z|^2-2\text{Re}\{\alpha^*z\}+|\alpha|^2}\tag{1}$$

Using $(1)$, the inequality $|G_1(z)|^2&lt;1$ can be written as

$$\begin{align}1-2\text{Re}\{\alpha^*z\}+|\alpha|^2|z|^2&amp;&lt;|z|^2-2\text{Re}\{\alpha^*z\}+|\alpha|^2\\
1+|\alpha|^2|z|^2&amp;&lt;|z|^2+|\alpha|^2\\0&amp;&lt;(|z|^2-1)(1-|\alpha|^2)\tag{2}\end{align}$$

Note that if the all-pass filter is stable, we must have $|\alpha|&lt;1$, because $\alpha$ is the pole location, and a causal and stable filter must have all its poles inside the unit circle of the complex plane. Consequently, the inequality $(2)$ is satisfied for $|z|&gt;1$ because both factors on the right-hand side are positive.