How can I show:
Let $Ω$ be a domain and assume that $f ∈ C(\bar \Omega)$ restricts to a holomorphic function on $Ω$.
Prove: The real and imaginary part of $f$ assume their maxima and minima on the boundary of $Ω$.
Maybe I could show that the real and imaginary part are again holomorphic and then I apply the maximum principle on the real and imaginary part?! Is it actually true that those functions are holomorphic?
Any hints would be great!
Decompose $f(z)$ into its real and imaginary parts:
$$f(z) = f(x,y) \equiv u(x,y) + iv(x,y)$$
Apply the Cauchy-Riemann conditions:
$$ \frac{\partial u}{\partial x} = \frac{\partial v}{\partial y} \\\frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x}$$
From these, it easily follows that:
$$\nabla^2u=0\\\nabla^2v=0$$
This implies that both $u$ and $v$ are solutions to Laplace's equation, which implies that they have no local maxima or minima (their maxima/minima must lie on the boundary of the region).