= Solution
Let
$$
J(z)=\frac1{\overline z}
$$
be reflection in the unit circle, and represent a <Möbius transformation> $g$ by
$$
M=\begin{pmatrix}\alpha&\beta\\ \gamma&\delta\end{pmatrix}.
$$
A direct calculation shows that $JgJ$ is represented by
$$
\tau(M)=\begin{pmatrix}\overline\delta&\overline\gamma\\
\overline\beta&\overline\alpha\end{pmatrix}.
$$
Thus $gJ=Jg$ exactly when $M$ and $\tau(M)$ differ by a nonzero scalar. Applying the conjugate-linear involution $\tau$ twice shows that this scalar has <modulus> one. Rescaling $M$ by a suitable complex scalar then makes $M=\tau(M)$. Consequently all commuting maps, and only those maps, have the form
$$
\boxed{g(z)=\frac{az+b}{\overline b z+\overline a}},
\qquad |a|^2-|b|^2\ne0.
$$
This is the <Möbius maps commuting with reflection in the unit circle> classification.
For such a map,
$$
|az+b|^2-|\overline b z+\overline a|^2
=(|a|^2-|b|^2)(|z|^2-1).
$$
It follows that $|g(z)|<1$ whenever $|z|<1$ precisely when
$$
\boxed{|a|>|b|}.
$$
The inverse has the same property, so these and only these maps preserve the unit disc.
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