Past exam of the geometry course of the University of Oxford / 2007 / B / 8 / iii / Solution

Choose a disk on which $|f(z)|\le M=|f(a)|$. If $M=0$ the assertion is immediate. Otherwise multiply $f$ by a fixed unit scalar so that its value at $a$ is the positive real number $M$. The mean-value identity shows the average of $\mathrm{Re}f$ on every sufficiently small circle is $M$. Since $\mathrm{Re}f\le |f|\le M$ there, continuity forces equality at every point on each circle. Equality of real part to $M$ together with modulus at most $M$ forces the imaginary part to vanish. Varying the radius covers a neighborhood of $a$. Thus both $|f|$ and $f$ are constant near $a$, proving the local maximum-modulus assertion.