Interesting power sums inequalities

[FarRider]

Here are some interesting inequalities you can try to prove. Let [tex]a_1, a_2, .., a_n[/tex] be real numbers and [tex]S(x)=\sum_{i=1}^n{a_i^x }[/tex]. Then

1. [tex]-n/8<=\frac{S(1)S(3)}{S(4)}<=n[/tex]

2. If [tex]S(1)=0[/tex] and [tex]S(2)=n[/tex] then [tex]S(3)<=\frac{n(n-2)}{ \sqrt{n-1} }[/tex]