### Analytic function

Prove that if f(z) : H -> H is an analytic function from the upper-half plane to itself, then:|f(z) − f(z_0)|/|f(z) − (f(z_0))bar|<=|z − z_0|/|z − (z_0)bar| where z,z_0 are in H and |f'(z)|/Im(f(z))<=1/Im(z) where z is in H. When does equality hold?