a) Let $\displaystyle y_1 ,...,y_n$ be an reordering of $\displaystyle x_1 ,...,x_n>0$. Prove that:
$\displaystyle \sum_{k=1}^{n}\frac{x_k}{y_k}\geq n$

b) Assuming
$\displaystyle x_{n+1}=x_1$ prove also:
$\displaystyle \sum_{k=1}^{n}\frac{x_{k+1}}{x_k}\leq \sum_{k=1}^{n}\left(\frac{x_k}{x_{k+1}}\right)^n$


Hello,

I have shown a) by using the
inequality of arithmetic and geometric means.
But I have problems with b). I failed to use any inequalities of means.


I tried to define$\displaystyle z_k=x_k/x_{k+1}$. Then I have

$\displaystyle \sum_{k=1}^{n}\frac{1}{y_k}\leq \sum_{k=1}^{n}y_k^n$,

given the additional $\displaystyle \prod_{k=1}^n y_k=1$. Though it seemed smart, I couldn't make use of it.

Is there anybody able to help?

Thanks in advance,

mili