the minimum modulus principle
The maximum-modulus principle forbids interior peaks of |f|; the minimum-modulus principle is the matching statement for valleys, but with a crucial caveat. If a nonconstant holomorphic function has no zero inside a region, then |f| cannot attain a minimum in the interior either — the smallest value of |f| also sits on the boundary. The catch is the no-zero condition, because where f vanishes, |f| obviously hits its lowest possible value, zero, and that can happen inside.
The trick is to look at 1 / f. If f is holomorphic and never zero on the region, then 1 / f is holomorphic there too. An interior minimum of |f| would be an interior maximum of |1 / f| = 1 / |f|, which the maximum-modulus principle outlaws. So |f| has no interior minimum unless f is constant. Concretely, to apply it you first check f has no zeros in the closed region, then conclude the minimum of |f| is on the boundary; if f does have a zero, the minimum is simply that zero.
This pair sandwiches a nonvanishing holomorphic function: both its largest and smallest moduli live on the boundary, so the interior modulus is squeezed between the boundary extremes. The principle is used to prove sharper forms of the open mapping theorem and to locate where functions can be small. The honest qualifier worth repeating: drop the no-zero assumption and the minimum statement breaks, since a zero is a legitimate interior minimum of |f|.
On the closed unit disk, f(z) = z + 2 never vanishes (its values lie near 2), so |f| has no interior minimum; the smallest |z + 2| = 1 is attained at the boundary point z = -1.
Without zeros, the smallest modulus also sits on the boundary.
The no-zero hypothesis is essential: a holomorphic function does attain an interior minimum of |f| precisely at any zero, where |f| = 0, so the principle only governs the zero-free case.