Prove $\forall z\in H$, $\frac{|f'(z)|}{Im f(z)} ≤ \frac{1}{Im f(z)}$ given f holomorphic using Schwarz Lemma

Question

Let H stand for the upper half-plane in the complex plane C. Suppose that $f:H\rightarrow H$ holomorphic. How can I use Schwarz Lemma to show that $\forall z\in H$, $\frac{|f'(z)|}{Im \,f(z)} ≤ \frac{1}{Im\, z}$ , and that if equality holds at some $z_0 \in H$, then $\forall z \in H$, $f(z) = \frac{az+b}{cz+d}$, for some $a, b, c, d \in R, ad − bc = 1$.

Thanks a lot!