Proving Inequalities with n > 2: A Challenge

[phonic]

Dear all,

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

Define

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


Prove that:

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


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

I.e.:
$$c_0 \geq c_1$$

$$c_{\infty} \geq c_1$$


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

Thanks a lot!

 

[Emieno]

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.

 

[phonic]

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!

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.