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Holomorphic Function and Taylor Expansion

Let f(z) be holomorphic in |z|less than R with Taylor expansion f(z)=sum(a_nz^n) and set
I_2(r)=1/2pi(integral from 0 to 2pi of|f(re^itheta)|^2 d(theta), where 0<=r<R.
Show that
a) I_2(r)=sum(n=0 to 00)|a_n|^2r^2n

b) I_2(r) is increasing.

c) |f(0)|^2<=I_2(r)<=M(r)^2, with M(r)=sup_|z||f(z)|

Solution Summary

A Holomorphic Function and Taylor Expansion are investigated.

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