Mixed-mean inequality for submatrix

Mathematica Slovaca

Suyun Zhao¹  Lin Si¹ 

关键词: mixed mean;    power mean;    matrix;    arithmetic-geometric mean inequality;   

DOI  :  10.2478/s12175-013-0150-1

学科分类：数学（综合）

来源: Slovenska Akademia Vied * Matematicky Ustav / Slovak Academy of Sciences, Mathematical Institute

【 摘 要 】

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 ($$frac{m}{2}$$,m] and l is an integer in ($$frac{n}{2}$$, n] respectively, then $$left( {prodlimits_{A in Bleft( {k,l} ight)} {a_A } } ight)^{frac{1}{{left( {_k^m } ight)left( {_l^n } ight)}}} geqslant frac{1}{{left( {_k^m } ight)left( {_l^n } ight)}}left( {sumlimits_{A in Bleft( {k,l} ight)} {g_A } } ight),$$ with equality if and only if b ij is a constant for every i, j.

【 授权许可】

Unknown