I am searching for the first proof of (or counterexample to) the following conjecture.

(The sum of squared logarithms conjecture) For all natural numbers $n$ and positive numbers $x_1,x_2, \ldots , x_n, y_1,y_2,\ldots, y_n>0$ such that for all $k\in\{1,\ldots, n-1\}$ it holds

$\sum_{i_1<\ldots<i_k} x_{i_1}\, x_{i_2}\ldots x_{i_k}\le \sum_{i_1<\ldots<i_k} y_{i_1}\, y_{i_2}\ldots y_{i_k}$

and $x_1\, x_2\, x_3 \ldots x_n=y_1\, y_2 \,y_3\ldots y_n$

it follows

$\sum_{i=1}^n (\log x_i)^2\le \sum_{i=1}^n (\log y_i)^2$

Replacing the assumption $x_1\, x_2\, x_3 \ldots x_n=y_1\, y_2\, y_3\ldots y_n$ by $x_1\, x_2\, x_3 \ldots x_n\le y_1\, y_2\, y_3\ldots y_n$ easily admits counterexamples.

Proofs are known for $n\in \{1,2,3,4\}$. More information can be found at

https://www.uni-due.de/mathematik/ag_neff/log_conjecture

Immediately after Lev Borisov's sketch of a proof idea below, Lev Borisov, Suvrit Sra, Christian Thiel and myself agreed to work out the details and to write together a complete and self-contained paper on the sum of squared logarithm conjecture and relations to other topics which can be found at: http://arxiv.org/abs/1508.04039 ${}{}{}$

As announced in my first post, the prize winner is Lev Borisov.