##  [Maximum Modulus Principle](/maximum-modulus-principle-0) 

 Definition

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

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

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

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

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

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

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

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

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.