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## 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.