A theorem in complex analysis that asserts: if f is a nonconstant holomorphic function on a connected open set, then the absolute value |f| cannot attain a local (or interior) maximum; any global maximum of |f| on a bounded domain occurs on the domain's boundary unless f is constant.

Principle

The modulus of a holomorphic function behaves like a subharmonic function (log |f| is subharmonic), so interior maxima are forbidden for nonconstant holomorphic functions; analyticity enforces rigidity of values.

Demonstration

On a bounded region D, take a holomorphic f with |f(z0)| = max_{D} |f|. The subharmonicity of log|f| implies it is constant, hence f is constant. Concrete example: nonconstant polynomials do not attain their maximum modulus on any bounded open set interior—only on boundary curves enclosing the set.

Misapplication

Applying the principle to non-holomorphic functions, to meromorphic functions without handling poles, or to functions on disconnected domains; for harmonic functions the maximum principle applies differently (to the function itself, not its modulus).

Consequence

Immediate corollaries include the open mapping theorem (nonconstant holomorphic maps are open), uniqueness continuation, and strong restrictions on zeros and level sets; it also underlies many uniqueness and extremal problems in complex analysis.

Reversal

The minimal-modulus statement requires modification: if f has no zeros, then 1/f is holomorphic and the maximum modulus principle applied to 1/f becomes a minimum principle for |f|. A raw inversion claiming interior minima for arbitrary holomorphic f is false unless f has no zeros and one applies the reciprocal trick.

Boundary

Assumes holomorphicity on a connected open set and excludes singularities and branch points inside the domain; statements for meromorphic functions need poles treated separately, and for harmonic or real-analytic functions the structure differs.

Semantic Tension

Close to general maximum principles for harmonic functions but distinct: maximum modulus concerns |f| and uses analyticity (subharmonicity of log|f|), while harmonic maximum principles act directly on harmonic (real-valued) solutions of Laplace's equation.

Synthesis

The Maximum Modulus Principle compresses analyticity into a rigidity statement: a holomorphic function that is not constant cannot have its modulus peak in the interior, which yields openness of holomorphic maps, uniqueness properties, and controls on where extremal behavior can occur.