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Inequality of exponent

  1. Jan 31, 2006 #1
    Dear all,

    Does anyone know how to determine which of the following sum is larger? Thanks a lot!
    [tex]
    \sum_{i=1}^n a_i^{1/3}\sum_{i=1}^n a_i
    [/tex]

    [tex]
    n\sum_{i=1}^n a_i^{4/3}
    [/tex]

    [tex]
    a_i>0 \forall i
    [/tex]

    This is two polynomial of the same order. It is not clear to determine which one is larger, if I take the derivatives.
     
  2. jcsd
  3. Jan 31, 2006 #2

    JasonRox

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    Homework Helper
    Gold Member

    Try proving by induction.
     
  4. Jan 31, 2006 #3

    mathman

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    Science Advisor
    Gold Member

    There is no general answer. It is easy to compare the two sums if all a's are the same. If =1, the sums are the same. If a>1, the first sum is larger. If a<1, the second sum is larger.
     
  5. Jan 31, 2006 #4

    JasonRox

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    That's exactly what I was thinking.

    Basically it depends on the sequence of [itex]a_i[/itex]. Like whether or not it has certain boundaries like you mentionned.
     
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