A strengthening of the Carleman-Hardy-Pólya inequality

As a consequence of a more general statement proved in the paper, it is deduced that, if $n\ge1$, and $a_j>0,\,j=1,2,\dotsc,n$, then

$\left(\frac{\sum_{j=1}^n \root j\of{a_1a_2\dotsm a_j}}{\sum_{j=1}... ...of{a_1a_2\dotsm a_n}}{\sum_{j=1}^n \root j\of{a_1a_2\dotsm a_j}}\le\frac{n+1}n,$

with equality if and only if $a_1=a_2=\dotsb =a_n$. This is a new refinement of Carleman's classic inequality.