Mixed-Mean Inequality for Submatrix

Abstract

For a $m\times n$ matrix $B=(b_{ij}{)}_{m\times n}$ with nonnegative entries $b_{ij}$ and any $k\times l-$submatrix $B_{ij}$ of $B$, let $a_{B_{ij}}$ and $g_{B_{ij}}$ denote the arithmetic mean and geometric mean of elements of $B_{ij}$ respectively. It is proved that if $k$ is an integer in $(\frac{m}{2},m]$ and $l$ is an integer in $(\frac{n}{2},n]$ respectively, then $$(\prod _{\genfrac{}{}{0ex}{}{i=k,j=l}{B_{ij}\subset B}}{a}_{B_{ij}}{)}^{\frac{1}{C_m^k\cdot C_n^l}}\ge \frac{1}{C_m^k\cdot C_n^l}(\sum _{\genfrac{}{}{0ex}{}{i=k,j=l}{B_{ij}\subset B}}{g}_{B_{ij}}),$$ with equality if and only if $b_{ij}$ is a constant for every $i,j$.

6 pages