In short how does this:
Landau's Proof of the Maximum Modulus Theorem
prove the maximum modulus theorem?
It is not clear to me that this proof shows that the maximum of an analytic function must occur on the boundary of domain. Infact it seems to me that all we have done is rearrived at the assumption $f(z) \leq M$, just with $z= z_0$.
Can anyone clarify. Many thanks!
Best Answer
We need that $C$ is open an $\partial C$ has finite length $L$, we assume $f$ is analytic on an open containing $C\cup \partial C$, obtaining for $z_0\in C$ $$2i\pi f(z_0)^n = \int_{\partial C}\frac{f(z)^n}{z-z_0}dz, \qquad 2\pi |f(z_0)|^n \le \frac{L}{dist(z_0,\partial C)} M^n$$
where $M^n = \sup_{z\in \partial C} |f(z)^n|$
If $|f(z_0)|>M$ this wouldn't be satisfied for $n$ large enough