Complex Analysis: Maximum Modulus Principle

[tylerc1991]

Homework Statement

Let f(z) = u(x,y) + iv(x,y) be a continuous, non-constant function that is analytic on some closed bounded region R. Prove that the component function u(x,y) reaches a minimum value on the boundary of R.

The Attempt at a Solution

By the minimum modulus principle, f(z) reaches its minimum value on the boundary of R. => u(x,y) reaches its minimum value on the boundary of R or v(x,y) reaches its minimum value on the boundary of R (or both). Assume that v(x,y) reaches its minimum value on the boundary of R (and that u(x,y) does not). Then u(x,y) is constant throughout R (since f(z) is bounded on R and hence u(x,y) is bounded on R. I have shown in another problem that if this case arises that u(x,y) is constant). Consider the function exp(z). If u(x,y) is constant then |exp(z)| is constant and hence a contradiction is reached.

 

[mighty2000]

Thank you for your post. I would like to provide some feedback and suggestions on your solution.

Firstly, your statement that "f(z) reaches its minimum value on the boundary of R" is not entirely accurate. The minimum modulus principle states that the modulus of an analytic function reaches its minimum value on the boundary of a region, not the function itself. In other words, |f(z)| reaches its minimum value on the boundary, not f(z) itself.

Secondly, your assumption that v(x,y) reaches its minimum value on the boundary of R (and u(x,y) does not) is not justified. It is possible for both u(x,y) and v(x,y) to reach their minimum values on the boundary of R. In fact, this is the case for most analytic functions.

Thirdly, your reasoning for why u(x,y) must be constant if v(x,y) reaches its minimum value on the boundary is not clear. It would be helpful to provide a more detailed explanation or proof for this statement.

Lastly, your example of the function exp(z) does not directly relate to the problem at hand and may be confusing for readers. It would be better to provide a specific counterexample that shows why the assumption that v(x,y) reaches its minimum value on the boundary leads to a contradiction.

Overall, your solution shows potential but could benefit from some clarification and further explanation. I hope my feedback has been helpful and I encourage you to continue exploring and learning about analytic functions.