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Suppose f(z) is analytic in the unit disk D={|z| < 1} and |f(z)|<= 1 for |z| <1, Prove that |f' (z) | <=  1/1-|z|

Question

Suppose f(z) is analytic in the unit disk D={|z| < 1} and |f(z)|<= 1 for |z| <1, Prove that |f' (z) | <=  1/1-|z|

check_circleAnswer
Step 1

Assume that f(z) is analytic on the unit disk D ={z: |z| < 1}.

Also assume that |f(z)|≤1 for |z|<1.

To prove the given problem, use Schwarz - Pick Lemma which states that:

"Let f be analytic on the unit circle D = {z: |z|<1} and assume that

  1. |f(z)|≤1 for all z
  2. f(a) = b, for some a,b∈ D

then,

                          |f'(a)| ≤ (1-|b|2)/(1-|a|2)"

Step 2

Here in this problem, the assumptions satisfies the hypothesis of Schwartz-Pick lemma.

Thus we have 

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Step 3

For every z ∈ D, 0 ≤ |f(z)| &...

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