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Proof problem

  1. Feb 9, 2006 #1
    Dear all,

    I want to prove that the following inequalities are true. I hope you can give some hints. Thanks a lot!

    Define

    [tex]
    c_{\beta}=\sum_{j=1}^n
    \sigma_j^{\frac{2\beta}{\beta+1}}\sum_{1\leq i<k \leq n}\Big(
    \sigma_k^{\frac{2}{3(\beta+1)}} +
    \sigma_i^{\frac{2}{3(\beta+1)}} \Big)^3 , \mbox{\hspace{1.5cm}}0\leq\sigma_i^2\leq 1, \forall i\in \{1,2,\cdots,n\}
    [/tex]


    Prove that:

    1)
    [tex]
    n \sum_{1\leq i <k \leq n}\Big(
    \sigma_k^{\frac{2}{3}} +
    \sigma_i^{\frac{2}{3}} \Big)^3 \geq \sum_{j=1}^{n}\sigma_j \sum_{1\leq i <k \leq n}\Big(
    \sigma_k^{\frac{1}{3}} +
    \sigma_i^{\frac{1}{3}} \Big)^3
    [/tex]


    and
    2)
    [tex]
    4n(n-1) \sum_{j=1}^n \sigma_j^2 \geq \sum_{j=1}^{n} \sigma_j \sum_{1\leq i <k \leq n}\Big(
    \sigma_k^{\frac{1}{3}} +
    \sigma_i^{\frac{1}{3}} \Big)^3
    [/tex]

    I.e.:
    [tex]
    c_0 \geq c_1
    [/tex]

    [tex]
    c_{\infty} \geq c_1
    [/tex]


    It is easy to prove when n=2 by taking the dirivative with respect to [tex]\sigma_1[/tex], and showing that the dirivative switches the sign at point [tex]\sigma_1 = \sigma_2[/tex]. How to prove when n>2?

    Thanks a lot!
     
  2. jcsd
  3. Feb 10, 2006 #2
    1) So, what happens if you let B=0, then B=1 ? Base on your given condition C0>C1, you can prove 1.
    2) Things are just the same, you pay attention to 4n(n-1) which is kind of a sum's result, and let B->infinity, then you will have C_infinity that again, under given condition C_inf > C1. You can prove 2.
     
    Last edited: Feb 10, 2006
  4. Feb 14, 2006 #3
    Thanks for your sugestion. I want to prove that both C_0 and C_inf are larger than C_1. Could you give more detailed hints? Thanks a lot!

     
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