Mixed-mean inequality for submatrix

For an m × n matrix B = (b _ij ) _m×n with nonnegative entries b _ij , let B(k, l) denote the set of all k × l submatrices of B. For each A ∈ B(k, l), let a _A and g _A denote the arithmetic mean and geometric mean of elements of A respectively. It is proved that if k is an integer in ( $\tfrac{m} {2}$ ,m] and l is an integer in ( $\tfrac{n} {2}$ , n] respectively, then $$\left( {\prod\limits_{A \in B\left( {k,l} \right)} {a_A } } \right)^{\tfrac{1} {{\left( {_k^m } \right)\left( {_l^n } \right)}}} \geqslant \frac{1} {{\left( {_k^m } \right)\left( {_l^n } \right)}}\left( {\sum\limits_{A \in B\left( {k,l} \right)} {g_A } } \right),$$ with equality if and only if b _ij is a constant for every i, j.