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|

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

*|f(z)|≤1*for all*z**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

Step 3

For every *z ∈ D, 0 ≤ |f(z)| &...*

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