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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!

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# Homework Help: Proof problem

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