the maximum modulus principle
If a nonconstant holomorphic function lives on a region, its size |f| cannot reach a peak anywhere in the interior. Any largest value of |f| on a closed bounded region is achieved only on the boundary. A holomorphic function has no interior mountaintops of magnitude — to find where it is biggest, you look around the edge.
The mechanism is the mean-value property. Since f(z_0) equals the average of f over a small circle about z_0, the modulus |f(z_0)| is at most the average of |f| around that circle. If |f(z_0)| were a strict interior maximum, every neighboring value would be no larger, yet their average equals the center — which can only happen if they are all exactly equal to |f(z_0)|. Pushing this everywhere forces |f| to be constant on a neighborhood, and the identity theorem then forces f itself to be constant on the whole connected region. So a nonconstant f simply cannot have such a peak inside.
This principle is one of the great rigidity tools. It gives uniqueness for boundary-value problems: two holomorphic functions agreeing on a boundary must agree inside (apply the principle to their difference). It underlies the Schwarz lemma, Phragmen-Lindelof estimates, and the three-lines and three-circles theorems. The honest caveat: it bounds |f|, the modulus, not the real or imaginary part alone in the same simple way, and the boundary maximum is attained only when the region is bounded and f extends continuously to it.
On the closed unit disk, f(z) = z has |f(z)| = |z| at most 1, with the maximum 1 attained only on the boundary circle |z| = 1 and never at an interior point.
The modulus of a nonconstant holomorphic function peaks on the boundary.
The principle constrains |f|, not the real part Re f or Im f directly — though those satisfy their own maximum principle as harmonic functions; and on an unbounded region the boundary maximum can fail without an extra growth condition (Phragmen-Lindelof).