A principle in complex analysis which states that a nonconstant holomorphic function on a connected open domain cannot attain a strict local maximum of its modulus in the interior; if the modulus attains a global maximum on the domain, the function is constant. Equivalently, |f| is subharmonic and achieves its maximum on the boundary.

Principle

Holomorphicity forces the magnitude of a nonconstant analytic function to behave so that maxima occur only at the boundary of a domain; interior extrema of the modulus imply constancy.

Demonstration

For f(z)=z on the unit disc, |f| attains its maximum on the unit circle, never strictly inside. If a holomorphic f on a connected domain has |f(z0)| ≥ |f(z)| for all z in a neighborhood of z0, then f is constant by the maximum modulus principle.

Misapplication

Applying the principle to functions that are merely differentiable or to meromorphic functions with poles; the result requires holomorphicity on the domain (or subharmonicity of log|f| for nonvanishing functions).

Consequence

Underpins uniqueness results, the open mapping theorem, and rigidity properties of holomorphic maps; it gives control of zeros and growth and is a key tool in boundary value problems.

Reversal

A corresponding minimum modulus statement holds for nonvanishing holomorphic functions (the minimum of |f| is attained on the boundary), but interior minima may exist at zeros; reversing the principle without the nonvanishing hypothesis fails.

Boundary

Requires a connected open domain and holomorphicity on that domain; does not apply at singularities, for merely continuous complex functions, or to harmonic functions without modification.

Semantic Tension

Nearby meanings include the maximum principle for harmonic functions and the open mapping theorem; one must distinguish modulus maxima from real/complex component extrema and note subharmonic vs harmonic contexts.

Synthesis

The maximum modulus principle asserts that the modulus of a nonconstant holomorphic function on a connected domain cannot have interior strict maxima, linking holomorphicity, subharmonicity and boundary behavior into a single rigidity statement used throughout complex analysis.