Maximum Modulus Principle

  • #1
Complex Analysis: Maximum Modulus Principle

Homework Statement


Let f(z)=u(x,y)+iv(x,y) be a function that is continuous on a closed bounded region R and analytic and not constant throughout the interior of R. Prove that the component function u(x,y) has a minimum value in R which occurs on the boundary of R and never in the interior.


Homework Equations


-Maximum Modulus Principle

-Corollary: Suppose that a function f is continuous on a closed bounded region R and that it is analytic and not constant in the interior of R. Then the maximum value of |f(z)| in R, which is always reached, occurs somewhere on the boundary of R and never in the interior.

-Also the last thing the author notes before the homework is "When the function f in the corollary is written f(z)=u(x,y)+iv(x,y), the component function u(x, y) also has a maximum value in R which is assumed on the boundary of R and never in the interior, where it is harmonic. This is because the composite function g(z)=exp[f(z)] is continuous in R and analytic and not constant in the interior. Hence its modulus |g(z)|=exp[u(x,y)], which is continuous in R, must assume its maximum value in R on the boundary. In view of the increasing nature of the exponential function it follows that the maximum value of u(x,y) also occurs on the boundary.


The Attempt at a Solution


Can I just make the same argument but for the minimum? I guess I do not fully understand this last point that the author makes. If someone could explain the reasoning for his argument, I would appreciate it.

Why does he raise e to the function? So for maximizing it, you are just trying to find the furthest point from the origin, which would always be on the boundary? So for the minimum you would try to find the closest point to the origin. But couldn't R circle the origin? So the origin is the closest point to the origin but this is on the interior, not on the boundary?
 
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