##  [Maximum Modulus Principle](/maximum-modulus-principle-1) 

 Definition

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

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

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

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

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

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

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

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

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.