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# Maximum Modulus Theorem : Let f be analytic in the disk B(0;R) and for 0 =< r < R define A(r) = max { Re f(z) : |z| = r}. Show that unless f is a constant, A(r) is a strictly increasing function of r.

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Let f be analytic in the disk B(0;R) and for 0 =< r < R define
A(r) = max { Re f(z) : |z| = r}. Show that unless f is a constant, A(r) is a strictly increasing function of r.

Please justify every step and claim and show how you used all what is given. Also refer to theorems or lemmas used in the proof. The section where I got this problem from, talks about The Maximum Principle. The Maximum Modulus Theorem ( first, second, and 3rd versions).

##### Solution Summary

The maximum principle and maximum modulus theorem are investigated.

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Proof:
We consider g(z)=exp(f(z)). Since f is analytic in the disk B(0;R), then g(z) is also analytic in the disk B(0;R).
Now suppose f ...

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