For \(a\in\DD\) let \(\psi_a(z)=\frac{a-z}{1-\bar az}\) be the Blaschke factor. Then:
\(\psi_a\in\Aut(\DD)\), and \(\psi_a\) maps the unit circle \(S^1\) onto itself;
\(\psi_a(0)=a\) and \(\psi_a(a)=0\);
\(\psi_a\circ\psi_a=\id_\DD\), so \(\psi_a^{-1}=\psi_a\);
\(\psi_a'(z)=\frac{\abs{a}^2-1}{(1-\bar az)^2}\);
\(\psi_a(\lambda z)=\lambda\,\psi_{\bar\lambda a}(z)\) for every \(\lambda\in\CC\) with \(\abs\lambda=1\).
The denominator \(1-\bar az\) is nonzero for \(\abs{z}\le1\), since \(\abs{\bar az}<1\). For \(\abs{z}=1\), \(\abs{1-\bar az}=\abs{\bar z-\bar a}=\abs{a-z}\), so \(\abs{\psi_a(z)}=1\). By the maximum modulus principle \(\abs{\psi_a}<1\) on \(\DD\) (\(\psi_a\) is nonconstant because \(\abs a<1\)). The values at \(0\) and \(a\) are immediate. A direct computation gives \[ \psi_a(\psi_a(z))=\frac{a(1-\bar az)-(a-z)}{(1-\bar az)-\bar a(a-z)}=\frac{z(1-\abs a^2)}{1-\abs a^2}=z, \] so \(\psi_a\) is a bijection of \(\DD\) equal to its own inverse, hence in \(\Aut(\DD)\), and likewise a bijection of \(S^1\). By the quotient rule, \[ \psi_a'(z)=\frac{-(1-\bar az)+\bar a(a-z)}{(1-\bar az)^2}=\frac{\abs a^2-1}{(1-\bar az)^2}. \] Finally, for \(\abs\lambda=1\), \(\lambda\bar\lambda=1\) gives \[ \lambda\,\psi_{\bar\lambda a}(z)=\lambda\,\frac{\bar\lambda a-z}{1-\lambda\bar az}=\frac{a-\lambda z}{1-\bar a\lambda z}=\psi_a(\lambda z). \]