Let f(z) be a non-constant analytic function on the region D = {z ∈ C||z| ≤ 1}. Suppose |f(z)| ≤ 1 for all z∈D.

Question

Let f(z) be a non-constant analytic function on the region D = {z ∈ C||z| ≤ 1}. Suppose |f(z)| ≤ 1 for all z∈D.

Show that if |w|<1, then $\frac{|f′(w)|}{1-|f(w)|^2}$ ≤ $\frac{1}{1-|w|^2}$

Here is my working:

May I know where did I gone wrong?