Bounds for analytic map from unit disk to right half-plane

Question

As discussed at Bounds on a mapping from unit disc to left half plane, for analytic map $f$ from unit disk to left or right half plane with $f(0)=\pm 1$ (depends on left or right half-plane), we must have \begin{align*} \dfrac{1-|z|}{1+|z|}\leq |f(z)|\leq \dfrac{1+|z|}{1-|z|}. \end{align*} I understand we should have $\dfrac{|f(z)-1|}{|f(z)+1|}\leq |z|$ in case of right half-plane, but how we can deduce $|f(z)|\leq \dfrac{1+|z|}{1-|z|}$ from there?