Mixed-Mean Inequality for Submatrix

TL;DR

This paper establishes a new inequality relating the geometric and arithmetic means of submatrices within a nonnegative matrix, extending the classical mean inequalities to specific submatrix sizes.

Contribution

It introduces a novel mixed-mean inequality for submatrices of a nonnegative matrix, with conditions on submatrix sizes and equality characterization.

Findings

Proves a new inequality linking geometric and arithmetic means of submatrices.

Identifies conditions under which equality holds, i.e., all entries are equal.

Extends classical mean inequalities to structured submatrices.

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 $$\Big(\prod_{i=k,j=l\atop B_{ij}\subset B}a_{B_{ij}}\Big)^{\frac{1}{C_m^k\cdot C_n^l}} \geq\frac{1}{C_m^k\cdot C_n^l}\Big(\sum_{i=k,j=l\atop B_{ij}\subset B}g_{B_{ij}}\Big),$$ with equality if and only if $b_{ij}$ is a constant for every $i,j$.