Not difficult but interesting inequality

[manlio]

Let x_k (k=1,2,..,n) be real positive numbers such that x_1+..+x_n=1.
Prove that

\sum (k=1,..,n)(x_k+1/x_k)^2 \geq (n+1)^a/n^(a-1)

for every real number a>0.

[pbornsztein]

This is not true for x_ = 1/n for i=1,...,n, and for a sufficientely large since in that case the LHS is (1+n ² ) ² /n and the RHS is n( 1+1/n) ^a .
Thus the RHS is a non bounded from above function of a....

Pierre.

[manlio]

Sorry, the correct inequality is



\sum (k=1,..,n)(x_k+1/x_k)^a \geq (n^2+1)^a/n^(a-1)



Sorry, again.

[pbornsztein]

Hmmm...are you sure about the fact that it holds for all a > 0?
I would say a \geq 1.

Pierre.

[manlio]

This inequality is valid for a \geq 0 and it is reversed for 0>a \geq -1.

[Mamat]

I think we can solve this inequality by using Jensen's inequality.