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Holomorphic functions

1. Let f(z) be a holomorphic function in the disc |z| < R1 and set
M(r) = sup|f(z)|(|z|=r), A(r) = supR(f(z)) (|z|<r) where 0<=r<R_1

(a) Show that M(r) is monotonic and, in fact, strictly increasing, unless f is a
constant.
(b) Show that A(r) is monotonic and, in fact, strictly increasing, unless f is constant.

Solution Preview

Proof:
(a) We consider r1<r2. We want to show that M(r1)<=M(r2).
Since f(z) is non-constant holomorphic in the disc |z|<=r2, then |f(z)| can not obtain its
maximum value inside the disc. In another word, |f(z)| obtains its maximum value
only on the border |z|=r2. So ...

Solution Summary

This provides an example of working with an increasing holomorphic function.

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