Bernstein's theorem is an inequality relating the maximum modulus of a complex polynomial function on the unit disk with the maximum modulus of its derivative on the unit disk. It was proven by Sergei Bernstein while he was working on approximation theory.
 Let 
  
    
      
        
          max
          
            
              |
            
            z
            
              |
            
            =
            1
          
        
        
          |
        
        f
        (
        z
        )
        
          |
        
      
    
    {\displaystyle \max _{|z|=1}|f(z)|}
  
 denote the maximum modulus of an arbitrary
function 
  
    
      
        f
        (
        z
        )
      
    
    {\displaystyle f(z)}
  
 on 
  
    
      
        
          |
        
        z
        
          |
        
        =
        1
      
    
    {\displaystyle |z|=1}
  
, and let 
  
    
      
        
          f
          ′
        
        (
        z
        )
      
    
    {\displaystyle f'(z)}
  
 denote its derivative.
Then for every polynomial 
  
    
      
        P
        (
        z
        )
      
    
    {\displaystyle P(z)}
  
 of degree 
  
    
      
        n
      
    
    {\displaystyle n}
  
 we have
 The inequality is best possible with equality holding if and only if
 
 Let 
  
    
      
        P
        (
        z
        )
      
    
    {\displaystyle P(z)}
  
 be a polynomial of degree 
  
    
      
        n
      
    
    {\displaystyle n}
  
, and let 
  
    
      
        Q
        (
        z
        )
      
    
    {\displaystyle Q(z)}
  
 be another polynomial of the same degree with no zeros in 
  
    
      
        
          |
        
        z
        
          |
        
        ≥
        1
      
    
    {\displaystyle |z|\geq 1}
  
. We show first that if 
  
    
      
        
          |
        
        P
        (
        z
        )
        
          |
        
        <
        
          |
        
        Q
        (
        z
        )
        
          |
        
      
    
    {\displaystyle |P(z)|<|Q(z)|}
  
 on 
  
    
      
        
          |
        
        z
        
          |
        
        =
        1
      
    
    {\displaystyle |z|=1}
  
, then 
  
    
      
        
          |
        
        
          P
          ′
        
        (
        z
        )
        
          |
        
        <
        
          |
        
        
          Q
          ′
        
        (
        z
        )
        
          |
        
      
    
    {\displaystyle |P'(z)|<|Q'(z)|}
  
 on 
  
    
      
        
          |
        
        z
        
          |
        
        ≥
        1
      
    
    {\displaystyle |z|\geq 1}
  
. 
 By Rouché's theorem, 
  
    
      
        P
        (
        z
        )
        +
        ε
         
        Q
        (
        z
        )
      
    
    {\displaystyle P(z)+\varepsilon \ Q(z)}
  
 with 
  
    
      
        
          |
        
        ε
        
          |
        
        ≥
        1
      
    
    {\displaystyle |\varepsilon |\geq 1}
  
 has all
its zeros in 
  
    
      
        
          |
        
        z
        
          |
        
        <
        1
      
    
    {\displaystyle |z|<1}
  
. By virtue of the Gauss–Lucas theorem,

  
    
      
        
          P
          ′
        
        (
        z
        )
        +
        ε
         
        
          Q
          ′
        
        (
        z
        )
      
    
    {\displaystyle P'(z)+\varepsilon \ Q'(z)}
  
 has all its zeros in 
  
    
      
        
          |
        
        z
        
          |
        
        <
        1
      
    
    {\displaystyle |z|<1}
  
 as well.
It follows that 
  
    
      
        
          |
        
        
          P
          ′
        
        (
        z
        )
        
          |
        
        <
        
          |
        
        
          Q
          ′
        
        (
        z
        )
        
          |
        
      
    
    {\displaystyle |P'(z)|<|Q'(z)|}
  
 on 
  
    
      
        
          |
        
        z
        
          |
        
        ≥
        1
      
    
    {\displaystyle |z|\geq 1}
  
,
otherwise we could choose an 
  
    
      
        ε
      
    
    {\displaystyle \varepsilon }
  
 with 
  
    
      
        
          |
        
        ε
        
          |
        
        ≥
        1
      
    
    {\displaystyle |\varepsilon |\geq 1}
  
 such that

  
    
      
        
          P
          ′
        
        (
        z
        )
        +
        ε
        
          Q
          ′
        
        (
        z
        )
      
    
    {\displaystyle P'(z)+\varepsilon Q'(z)}
  
 has a zero in 
  
    
      
        
          |
        
        z
        
          |
        
        ≥
        1
      
    
    {\displaystyle |z|\geq 1}
  
.
 For an arbitrary polynomial 
  
    
      
        P
        (
        z
        )
      
    
    {\displaystyle P(z)}
  
 of degree 
  
    
      
        n
      
    
    {\displaystyle n}
  
, we obtain Bernstein's Theorem by applying the above result to the polynomials 
  
    
      
        Q
        (
        z
        )
        =
        C
        
          z
          
            n
          
        
      
    
    {\displaystyle Q(z)=Cz^{n}}
  
, where 
  
    
      
        C
      
    
    {\displaystyle C}
  
 is an arbitrary constant exceeding 
  
    
      
        
          max
          
            
              |
            
            z
            
              |
            
            =
            1
          
        
        
          |
        
        P
        (
        z
        )
        
          |
        
      
    
    {\displaystyle \max _{|z|=1}|P(z)|}
  
.
 In mathematical analysis, Bernstein's inequality states that on the complex plane, within the disk of radius 1, the degree of a polynomial times the maximum value of a polynomial is an upper bound for the similar maximum of its derivative. Taking the k-th derivative of the theorem,
 Paul Erdős conjectured that if 
  
    
      
        P
        (
        z
        )
      
    
    {\displaystyle P(z)}
  
 has no zeros in 
  
    
      
        
          |
        
        z
        
          |
        
        <
        1
      
    
    {\displaystyle |z|<1}
  
, then 
  
    
      
        
          max
          
            
              |
            
            z
            
              |
            
            =
            1
          
        
        
          |
        
        
          P
          ′
        
        (
        z
        )
        
          |
        
        ≤
        
          
            n
            2
          
        
        
          max
          
            
              |
            
            z
            
              |
            
            =
            1
          
        
        
          |
        
        P
        (
        z
        )
        
          |
        
      
    
    {\displaystyle \max _{|z|=1}|P'(z)|\leq {\frac {n}{2}}\max _{|z|=1}|P(z)|}
  
. This was proved by Peter Lax.
 M. A. Malik showed that if 
  
    
      
        P
        (
        z
        )
      
    
    {\displaystyle P(z)}
  
 has no zeros in 
  
    
      
        
          |
        
        z
        
          |
        
        <
        k
      
    
    {\displaystyle |z|<k}
  
 for a given 
  
    
      
        k
        ≥
        1
      
    
    {\displaystyle k\geq 1}
  
, then 
  
    
      
        
          max
          
            
              |
            
            z
            
              |
            
            =
            1
          
        
        
          |
        
        
          P
          ′
        
        (
        z
        )
        
          |
        
        ≤
        
          
            n
            
              1
              +
              k
            
          
        
        
          max
          
            
              |
            
            z
            
              |
            
            =
            1
          
        
        
          |
        
        P
        (
        z
        )
        
          |
        
      
    
    {\displaystyle \max _{|z|=1}|P'(z)|\leq {\frac {n}{1+k}}\max _{|z|=1}|P(z)|}
  
.