= Solution
For $a\in\mathbb D$, the <automorphism of the unit disk> $T_a(z)=(z-a)/(1-\bar a z)$ sends $a$ to zero. The <holomorphic function> $F=T_{f(a)}\circ f\circ T_a^{-1}$ maps the <unit disc> into itself and satisfies $F(0)=0$. The <Schwarz lemma> therefore gives $|F(w)|\leq|w|$. Evaluate at $w=T_a(b)$ to obtain the <Schwarz-Pick theorem>:
$$
\boxed{\left|\frac{f(b)-f(a)}{1-f(b)\overline{f(a)}}\right|\leq\left|\frac{b-a}{1-b\bar a}\right|}.
$$
The denominators do not vanish because both their products have <modulus> less than one.
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