Schur-convex function

<h2 id="properties">Properties</h2>
<p>Every function that is <a href="/facts/Convex_function/IbzG5SLF">convex</a> and <a href="/facts/Symmetric_function/IB8N4T3m">symmetric</a> (under permutations of the arguments) is also Schur-convex.  
</p><p>Every Schur-convex function is symmetric, but not necessarily convex.<a class="footnote-ref" id="fnref:1" href="#fn:1"><sup>1</sup></a>
</p><p>If 
  
    
      
        f
      
    
    {\displaystyle f}
  
 is (strictly) Schur-convex and 
  
    
      
        g
      
    
    {\displaystyle g}
  
 is (strictly) monotonically increasing, then 
  
    
      
        g
        ∘
        f
      
    
    {\displaystyle g\circ f}
  
 is (strictly) Schur-convex.
</p><p>If 
  
    
      
        g
      
    
    {\displaystyle g}
  
 is a convex function defined on a real interval, then 
  
    
      
        
          ∑
          
            i
            =
            1
          
          
            n
          
        
        g
        (
        
          x
          
            i
          
        
        )
      
    
    {\displaystyle \sum _{i=1}^{n}g(x_{i})}
  
 is Schur-convex.
</p>
<h3>Schur–Ostrowski criterion</h3>
<p>If <i>f</i> is symmetric and all first partial derivatives exist, then 
<i>f</i> is Schur-convex if and only if
</p>

(
        
          x
          
            i
          
        
        −
        
          x
          
            j
          
        
        )
        
          (
          
            
              
                
                  ∂
                  f
                
                
                  ∂
                  
                    x
                    
                      i
                    
                  
                
              
            
            −
            
              
                
                  ∂
                  f
                
                
                  ∂
                  
                    x
                    
                      j
                    
                  
                
              
            
          
          )
        
        ≥
        0
      
    
    {\displaystyle (x_{i}-x_{j})\left({\frac {\partial f}{\partial x_{i}}}-{\frac {\partial f}{\partial x_{j}}}\right)\geq 0}
  
 for all 
  
    
      
        x
        ∈
        
          
            R
          
          
            d
          
        
      
    
    {\displaystyle x\in \mathbb {R} ^{d}}

<p>holds for all 
  
    
      
        1
        ≤
        i
        ,
        j
        ≤
        d
      
    
    {\displaystyle 1\leq i,j\leq d}
  
.<a class="footnote-ref" id="fnref:2" href="#fn:2"><sup>2</sup></a>
</p>
<h2 id="examples">Examples</h2>
<ul><li>
  
    
      
        f
        (
        x
        )
        =
        min
        (
        x
        )
      
    
    {\displaystyle f(x)=\min(x)}
  
 is Schur-concave while 
  
    
      
        f
        (
        x
        )
        =
        max
        (
        x
        )
      
    
    {\displaystyle f(x)=\max(x)}
  
 is Schur-convex. This can be seen directly from the definition.</li>
<li>The <a href="/facts/Shannon_entropy/NLg4NLvt">Shannon entropy</a> function 
  
    
      
        
          ∑
          
            i
            =
            1
          
          
            d
          
        
        
          
            P
            
              i
            
          
          ⋅
          
            log
            
              2
            
          
          ⁡
          
            
              1
              
                P
                
                  i
                
              
            
          
        
      
    
    {\displaystyle \sum _{i=1}^{d}{P_{i}\cdot \log _{2}{\frac {1}{P_{i}}}}}
  
 is Schur-concave.</li>
<li>The <a href="/facts/R%C3%A9nyi_entropy/SlrvqKNv">Rényi entropy</a> function is also Schur-concave.</li>
<li>
  
    
      
        x
        ↦
        
          ∑
          
            i
            =
            1
          
          
            d
          
        
        
          
            x
            
              i
            
            
              k
            
          
        
        ,
        k
        ≥
        1
      
    
    {\displaystyle x\mapsto \sum _{i=1}^{d}{x_{i}^{k}},k\geq 1}
  
 is Schur-convex if 
  
    
      
        k
        ≥
        1
      
    
    {\displaystyle k\geq 1}
  
, and Schur-concave if 
  
    
      
        k
        ∈
        (
        0
        ,
        1
        )
      
    
    {\displaystyle k\in (0,1)}
  
.</li>
<li>The function 
  
    
      
        f
        (
        x
        )
        =
        
          ∏
          
            i
            =
            1
          
          
            d
          
        
        
          x
          
            i
          
        
      
    
    {\displaystyle f(x)=\prod _{i=1}^{d}x_{i}}
  
 is Schur-concave, when we assume all 
  
    
      
        
          x
          
            i
          
        
        >
        0
      
    
    {\displaystyle x_{i}>0}
  
. In the same way, all the <a href="/facts/Elementary_symmetric_polynomial/9RFXe9b6">elementary symmetric functions</a> are Schur-concave, when 
  
    
      
        
          x
          
            i
          
        
        >
        0
      
    
    {\displaystyle x_{i}>0}
  
.</li>
<li>A natural interpretation of <a href="/facts/Majorization/RSSsgWTP">majorization</a> is that if 
  
    
      
        x
        ≻
        y
      
    
    {\displaystyle x\succ y}
  
 then 
  
    
      
        x
      
    
    {\displaystyle x}
  
 is less spread out than 
  
    
      
        y
      
    
    {\displaystyle y}
  
. So it is natural to ask if statistical measures of variability are Schur-convex. The <a href="/facts/Variance/ULBJKXD1">variance</a> and <a href="/facts/Standard_deviation/qSug9BRl">standard deviation</a> are Schur-convex functions, while the <a href="/facts/Median_absolute_deviation/P6SSMLdt">median absolute deviation</a> is not.</li>
<li>A probability example: If  
  
    
      
        
          X
          
            1
          
        
        ,
        …
        ,
        
          X
          
            n
          
        
      
    
    {\displaystyle X_{1},\dots ,X_{n}}
  
 are <a href="/facts/Exchangeable_random_variables/tHvmhbRy">exchangeable random variables</a>, then the function 
  
    
      
        
          E
        
        
          ∏
          
            j
            =
            1
          
          
            n
          
        
        
          X
          
            j
          
          
            
              a
              
                j
              
            
          
        
      
    
    {\displaystyle {\text{E}}\prod _{j=1}^{n}X_{j}^{a_{j}}}
  
 is Schur-convex as a function of 
  
    
      
        a
        =
        (
        
          a
          
            1
          
        
        ,
        …
        ,
        
          a
          
            n
          
        
        )
      
    
    {\displaystyle a=(a_{1},\dots ,a_{n})}
  
, assuming that the expectations exist.</li>
<li>The <a href="/facts/Gini_coefficient/HledImpA">Gini coefficient</a> is strictly Schur convex.</li></ul>

<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Quasiconvex_function/emHsjeRj">Quasiconvex function</a></li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1"><p>Roberts, A. Wayne; Varberg, Dale E. (1973). Convex functions. New York: Academic Press. p. 258. ISBN 9780080873725. <a href="9780080873725" target="_blank">9780080873725</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li>
<li id="fn:2"><p>E. Peajcariaac, Josip; L. Tong, Y. (3 June 1992). Convex Functions, Partial Orderings, and Statistical Applications. Academic Press. p. 333. ISBN 9780080925226. <a href="9780080925226" target="_blank">9780080925226</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li>
</ol>

Schur-convex function open-in-new

Schur-convex function