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AM-GM Inequality problem.

  1. Nov 1, 2009 #1
    1. The problem statement, all variables and given/known data
    Let X1,X2,...,Xn be positive real numbers. Show that

    ((x15+...+xn5 )/ n)1/5 >= ((x14+...+xn4 )/ n)1/4


    2. Relevant equations



    3. The attempt at a solution
    I have tried by taking logarithms. Is it right approach?
    Or.. It can be applied to AM-GM-HM inequality?
    how?
     
    Last edited: Nov 1, 2009
  2. jcsd
  3. Nov 1, 2009 #2

    jbunniii

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    I think you can prove it using Jensen's inequality, if you know it.
     
  4. Nov 1, 2009 #3
    I know Jensen's inequality, but I am not sure how it will work for my question.
    Can you give me some hints?
    Thanks
     
  5. Nov 1, 2009 #4

    jbunniii

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    OK, one form of Jensen's inequality is

    [tex]\phi\left(\frac{\sum y_i}{n}\right) \leq \frac{\sum \phi(y_i)}{n}[/tex]

    which is true provided that [itex]\phi[/itex] is a convex function.

    Hint: let

    [tex]y_i = x_i^4[/tex].

    Then what convex function [itex]\phi[/itex] would work? Hint: You need something that will change powers of 4 into powers of 5.
     
    Last edited: Nov 1, 2009
  6. Nov 1, 2009 #5

    jambaugh

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    You could raise both sides to the 20th power and expand!?! Your first step however would be to factor out the n's. They should disappear.

    Try maybe some variable substitutions? [tex]x_k = y_k^p[/tex] for some strategically chose value of [tex]p[/tex]

    (I'm guessing here so my suggestions may not be helpful. But I don't think the logarithms will be helpful due to the sums inside.)
     
  7. Nov 1, 2009 #6
    Thanks for your hints, but I am not sure yet.
    If I have x^4 = y and x^5 = y^(5/4), then
    I get the inequality which is just the same with the problem.
    how do I get the inequality of Jensen? (without logarithms?)

    Thanks
     
  8. Nov 1, 2009 #7

    jbunniii

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    I'm not sure that I understand your question. Let

    [tex]y_i = x_i^4[/tex]

    Now what [itex]\phi[/itex] did you choose? If you choose the right one, then after a little algebraic manipulation you can prove that your inequality is true. I just worked it out here at my desk; it took 4 lines.

    If you are asking how to prove Jensen's inequality, the finite case is easy (yet very clever) and can be found under the heading "Proof 1 (finite form)" here:

    http://en.wikipedia.org/wiki/Jensen's_inequality
     
  9. Nov 1, 2009 #8
    You mean I have to make some convex function
    and then apply into the inequality.
    Finally, I will get the answer. Right?

    hmm, If so, I will think about the function.
     
  10. Nov 1, 2009 #9
    I think f(x) = x^(5/4) is fine as your hint.
    And also f(x) is convex as condition of x_{i} --> positive real numbers.
    Right?
     
  11. Nov 1, 2009 #10

    jbunniii

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    Yes, that's correct. (In fact, x^(5/4) isn't even defined for negative x, unless you allow it to take on imaginary values.)
     
  12. Nov 1, 2009 #11

    jambaugh

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    I was thinking more along the lines x = y^(5/4) or x = y^(4/5). Substitute then apply an appropriate power to both sides of the equation and you should get the Jensen form.
     
  13. Nov 1, 2009 #12
    I got the answer.
    It's only 4-5lines.

    Thanks!
     
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