# Maximum Modulus Principle Problem

1. Nov 10, 2009

### dark_dingo

1. The problem statement, all variables and given/known data

Let g(z) be a function that is analytic and non-constant on D = {|z| < 1}. Suppose that Max |g(z)| $$\leq$$ $$\frac{1}{r}$$ for all 0< r <1, |z| = r. Use the Maximum Modulus Principle (or corollary) to prove that |g(z)| < 1 for all z $$\in$$ D.

2. Relevant equations

http://hphotos-snc3.fbcdn.net/hs104.snc3/15146_524570240458_58700263_31202039_3403055_n.jpg [Broken]
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Maximum Modulus Principle:

http://hphotos-snc3.fbcdn.net/hs104.snc3/15146_524570060818_58700263_31202038_7869593_n.jpg [Broken]
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http://hphotos-snc3.fbcdn.net/hs084.snc3/15146_524570804328_58700263_31202043_1540731_n.jpg [Broken]

3. The attempt at a solution

Not exactly sure how to start.

Last edited by a moderator: May 4, 2017
2. Nov 10, 2009

### Office_Shredder

Staff Emeritus
Ok, so basically the problem is that g(z) might be something like $$\frac{1}{z}$$ But it's analytic so it has to be defined at 0. This means that near 0 it has to be bounded. What can you conclude?

3. Nov 11, 2009

### dark_dingo

that the maximum of g(z) is reached on the boundary of g(z0) ?