Maximum Modulus Principle Problem

In summary, the Maximum Modulus Principle (or corollary) can be used to prove that |g(z)| < 1 for all z \in D, where g(z) is a function that is analytic and non-constant on the disk D = {|z| < 1}, and satisfies Max |g(z)| \leq \frac{1}{r} for all 0< r <1, |z| = r. This is because g(z) must be bounded near 0 and the maximum of g(z) is reached on the boundary of g(z0).
  • #1
dark_dingo
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0

Homework Statement



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


Homework Equations



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Maximum Modulus Principle:

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

The Attempt at a Solution



Not exactly sure how to start.
 
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  • #2
Ok, so basically the problem is that g(z) might be something like [tex] \frac{1}{z}[/tex] 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
that the maximum of g(z) is reached on the boundary of g(z0) ?
 

FAQ: Maximum Modulus Principle Problem

What is the maximum modulus principle problem?

The maximum modulus principle problem is a mathematical concept that states that the maximum value of a complex function occurs either on the boundary of a given domain or at a point where the function is not analytic. In other words, the maximum magnitude of a complex function is either on the boundary of the domain or at a critical point where the function is not differentiable.

How is the maximum modulus principle problem used in complex analysis?

The maximum modulus principle problem is an important tool in complex analysis for finding the maximum value of a complex function within a given domain. It is often used to prove the existence and uniqueness of solutions to certain differential equations, as well as to establish the properties of analytic functions.

What is the significance of the maximum modulus principle problem in mathematics?

The maximum modulus principle problem is significant in mathematics as it helps to understand the behavior of complex functions and their properties. It is also used to prove important theorems in complex analysis, such as the Cauchy-Riemann equations and the Cauchy integral formula.

Can the maximum modulus principle be applied to real-valued functions?

No, the maximum modulus principle is only applicable to complex-valued functions. This is because real-valued functions do not have the same properties as complex functions, such as analyticity and differentiability.

Are there any exceptions to the maximum modulus principle?

Yes, there are some exceptions to the maximum modulus principle, such as when the function has a singularity on the boundary of the domain or when the function is constant. In these cases, the maximum value may not occur on the boundary of the domain.

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