proof of maximal modulus principle

$f:U\to ℂ$ is holomorphic and therefore continuous, so $|f|$ will also be continuous on $U$.$K\subset U$ is compact and since $|f|$ is continuous on $K$ it must attain a maximum and a minimum value there.

Suppose the maximum of $|f|$ is attained at $z_0$ in the interior of $K$.

By definition there will exist $r>0$ such that the set $S_r=\{z\in ℂ:{|z-z_0|}^2\le r^2\}\subset K$.

Consider $C_r$ the boundary of the previous set parameterized counter-clockwise.Since $f$ is holomorphic by hypothesis, Cauchy integral formula says that

$$f(z_0)=\frac{1}{2\pi i}{\oint }_C\frac{f(z)}{z-z_0}𝑑z$$(1)

A canonical parameterization of $C_r$ is $z=z_0+r{e}^{i\frac{\theta }{r}}$, for $\theta \in [0,2\pi r]$.

$$f(z_0)=\frac{1}{2\pi r}{\int }_0^{2\pi r}f(z_0+r{e}^{i\frac{\theta }{r}})𝑑\theta$$(2)

Taking modulus on both sides and using the estimating theorem of contour integral

$$|f(z_0)|\le {\max }_{z\in C_r}|f(z)|$$

Since $|f(z_0)|$ is a maximum, the last inequality must be verified by having the equality in the $\le$ verified.

The proof of the estimating theorem of contour integral (http://planetmath.org/ProofOfEstimatingTheoremOfContourIntegral) implies that equality is only verified when

$$\frac{f(z_o+r{e}^{i\frac{\theta }{r}})}{r{e}^{i\frac{\theta }{r}}}=\lambda \overline{i{e}^{i\frac{\theta }{r}}}$$

where $\lambda \in ℂ$ is a constant.Therefore, $f(z_o+r{e}^{i\frac{\theta }{r}})$ is constant and to verify equation 2 its value must be $f(z_0)$.

So $f$ is holomorphic and constant on a circumference.It’s a well known result that if 2 holomorphic functions are equal on a curve, then they are equal on their entire domain, so $f$ is constant.

to see this in this particular circumstance is using equation 1 to calculate the value of $f$ on a point $\xi \in$ interior $S_r$ different than $z_0$. Bearing in mind that $f(z)=f(z_0)$ is constant in $C_r$ the formula reads $f(\xi )=\frac{f(z_0)}{2\pi i}{\oint }_{C_r}\frac{1}{z-\xi }𝑑z=f(z_0)$. So $f$ is really constant in the interior of $S_r$ and the only holomorphic function defined in $K$ that is constant in the interior of $S_r$ is the constant function on all $K$.

Thus if the maximum of $|f|$ is attained in the interior of $K$, then $f$ is constant.If $f$ isn’t constant, the maximum must be attained somewhere in $K$, but not in its interior.Since $K$ is compact, by definition it must be attained at $\partial K$.