Originally Posted by

**mushroom98** Dear all,

I am stuck for two weeks on the next problem. All the numerical test I did show that it is true, but I do not find any proof.

let $\displaystyle (a_i)_{i=1,..,n}$ and $\displaystyle (b_i)_{i=1,..,n-1}$, reals numbers in $\displaystyle (0,1)$,

such that :

$\displaystyle \sum_{i=1}^n a_i +\sum_{i=1}^{n-1} b_i =1.$

show that the function :

$\displaystyle \phi(\alpha)=\sum_{i=1}^{n-1} b_i^\alpha + 1 -\sum_{i=1}^n a_i^\alpha$

as at most one root in $\displaystyle (0,1)$.

Any idea is welcome ! Thank you !