A strengthening of the Carleman-Hardy-Pólya inequality

Abstract:

As a consequence of a more general statement proved in the paper, it is deduced that, if $n\ge 1$, and $a_j>0, j=1,2,\dotsc ,n$, then \[ \left (\frac {\sum _{j=1}^n \root j\of {a_1a_2\dotsm a_j}}{\sum _{j=1}^n a_j}\right )^{1/n}+\frac {\root n\of {a_1a_2\dotsm a_n}}{\sum _{j=1}^n \root j\of {a_1a_2\dotsm a_j}}\le \frac {n+1}n,\] with equality if and only if $a_1=a_2=\dotsb =a_n$. This is a new refinement of Carleman’s classic inequality.