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

  ##  [Maximum Modulus Principle](https://mathlogic.quantumdictionary.io/maximum-modulus-principle-0) 

  

 [![Mathematics & Logic Dictionary](/sites/default/files/styles/large/public/2026-01/Mathematics%20%26%20Logic.png.webp?itok=UhtTRPnp)](/topic-specific-dictionaries/natural-formal-sciences/mathematics-logic)

- Natural & Formal Sciences -

**Mathematics & Logic Dictionary**

 







 

 

 

 



 

 

 

 

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.

 

 

 

 

 





 

 



 ##  [Maximum Modulus Principle](https://puremath.quantumdictionary.io/maximum-modulus-principle-1) 

  

 [![Pure Mathematics Dictionary](/sites/default/files/styles/large/public/2026-01/Pure%20Mathematics.png.webp?itok=5pZnFQ59)](/topic-specific-dictionaries/mathematics-logic/pure-mathematics)

- Mathematics & Logic -

**Pure Mathematics Dictionary**

 







 

 

 

 



 

 

 

 

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.