maximum modulus principle
The maximum modulus principle says that a non-constant holomorphic function can never have a 'peak' of size strictly inside its domain — the largest value of its magnitude is always pushed out to the boundary. Picture the modulus |f| as the height of a stretched membrane over the region; the principle forbids any interior bump. If the highest point appeared in the interior, the function would be forced to be a flat constant everywhere.
Precisely: if f is holomorphic on a connected open set and |f| attains a local maximum at some interior point, then f is constant on the whole set. Equivalently, on a bounded domain where f is holomorphic inside and continuous up to the boundary, the maximum of |f| is attained on the boundary. The mechanism is the mean value property from Cauchy's formula: |f| at a center cannot exceed the average of |f| around a circle, and a strict interior maximum would violate that average unless everything is constant.
There is a matching minimum modulus principle, but with a caveat: a non-constant holomorphic function attains no interior minimum of |f| EXCEPT where f itself is zero (a zero is, trivially, a minimum of the modulus). The principle is the workhorse behind uniqueness arguments, the Schwarz lemma, and many estimates; it expresses, once more, that holomorphic functions are too rigid to have interior local extrema of magnitude.
On the closed unit disc, f(z) = z has |f(z)| = |z|, which is at most 1 inside and equals 1 exactly on the boundary circle — the maximum is attained only on the boundary, never strictly inside, exactly as the principle predicts. There is no interior point where |z| beats its boundary values.
The modulus of z peaks on the boundary, not inside.