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Comparing f(sigma(x)) and sigma(f(x))

  1. Aug 9, 2007 #1
    Hello,

    The function in question has the following properties:

    1. Is monotonically increasing
    2. Is concave downwards
    3. f(0) =0

    I was trying to find a relation between sum(f(x_i)) and f(sum(x_i)) i=1 to i=n and all x_i positive.

    I tried a few things, including this:

    nx>=x
    f'(nx)<=f'(x)
    integrating 0 to x
    (1/n)f(nx)<=f(x)
    taking sigma
    sum((1/n)f(nx))<=sum(f(x))

    i tried using this along with jensen's inequality but couldnt draw any conclusions. Any help would be greatly appreciated.
     
  2. jcsd
  3. Aug 9, 2007 #2

    Dick

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    Think about the linear function L(x)=x*f(sum(x_i))/sum(x_i). L(0)=0, L(sum(x_i))=f(sum(x_i)). But for 0<=x<=sum(x_i), L(x)<=f(x) because of your convexity condition. Can you fill in the rest? You don't need the monotone increasing condition.
     
  4. Aug 9, 2007 #3
    The function is concave downards f''(x)<0 so shouldn't the L(x) and f(x) relation reverse sign?

    Also, how do I introduce a sigma(f(x_i)) into the inequalities?
     
  5. Aug 10, 2007 #4

    Dick

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    Nooo. The chord to a concave downward function is below the function. Like f(x)=(-x^2). sigma(f(x_i))>=sigma(L(x_i))=L(sigma(x_i)).
     
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