Maximum modulus principle

<p>In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, the maximum modulus principle in <a href="/facts/Complex_analysis/hgCoQlH0">complex analysis</a> states that if 
  
    
      
        f
      
    
    {\displaystyle f}
  
 is a <a href="/facts/Holomorphic_function/kVzthtEf">holomorphic function</a>, then the <a href="/facts/Absolute_value/dwJBpEs5">modulus</a>

|
        
        f
        
          |
        
      
    
    {\displaystyle |f|}
  
 cannot exhibit a strict <a href="/facts/Maximum/ADlgnyzV">maximum</a> that is strictly within the <a href="/facts/Domain_of_a_function/FQvoeUpX">domain</a> of 
  
    
      
        f
      
    
    {\displaystyle f}
  
. 
</p><p>In other words, either 
  
    
      
        f
      
    
    {\displaystyle f}
  
 is locally a <a href="/facts/Constant_function/BmPx6dn7">constant function</a>, or, for any point 
  
    
      
        
          z
          
            0
          
        
      
    
    {\displaystyle z_{0}}
  
 inside the domain of 
  
    
      
        f
      
    
    {\displaystyle f}
  
 there exist other points arbitrarily close to 
  
    
      
        
          z
          
            0
          
        
      
    
    {\displaystyle z_{0}}
  
 at which 
  
    
      
        
          |
        
        f
        
          |
        
      
    
    {\displaystyle |f|}
  
 takes larger values.
</p>

Maximum modulus principle open-in-new

Maximum modulus principle