- #1
phonic
- 28
- 0
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!
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!