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Analytic function and equality

Prove that if f(z) : H -> H is an analytic function from the upper-half plane to itself,
then:|f(z) &#8722; f(z_0)|/|f(z) &#8722; (f(z_0))bar|<=|z &#8722; z_0|/|z &#8722; (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?

Solution Summary

This shows how to determine when equality holds for an analytic function.