Maximum modulus principle in several variables

Remarks

For $z$ near $a$, the map $t \mapsto f(a + t(z - a))$ is a holomorphic function of one variable on a disc around $t = 0$ whose modulus has a maximum at $t = 0$. By the one-variable maximum modulus principle it is constant, so $f$ is constant near $a$, and the identity theorem makes $f$ constant on $D$. Consequence: if $D$ is bounded and $f$ is continuous on the closure $\overline{D}$ and holomorphic on $D$, then $\max_{\overline{D}} \abs{f}$ is attained on the boundary of $D$. Connectedness is needed: on two disjoint balls, a function equal to $0$ on one and $1$ on the other attains its maximum modulus without being constant. Example: on the unit polydisc of $\C^2$, $f(z) = z_1 z_2$ satisfies $\abs{f} < 1$ inside and reaches $\abs{f} = 1$ only where $\abs{z_1} = \abs{z_2} = 1$, on the distinguished boundary.

Going deeper

For a student

The proof is a typical way to pass from one to several variables: restrict to complex lines through a point and use the one-variable result. For special domains the maximum is found on a much smaller part of the boundary. On a polydisc it is the distinguished boundary, an $n$-dimensional torus. The study of these smallest sets leads to the Shilov boundary and to peak points.