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On nonnegative matrices with given row and column sums. (English) Zbl 1039.15011

Let \(A\) be a non-negative \(n\times n\) matrix with row sums \(r_1,\dots,r_n\) and column sums \(c_1,\dots,c_n\). Order them decreasingly: \(r_1^{\downarrow }\geq \dots \geq r_n^{\downarrow }\) and \(c_1^{\downarrow }\geq \dots \geq c_n^{\downarrow }\). Let \(\text{su\,}A\) be the sum of the entries of a matrix \(B\) and \(m\) is a nonnegative integer. In 1995, J. K. Merikoski and A. Virtanen [Linear Multilinear Algebra 30, No. 4, 257–259 (1991; Zbl 0746.15010)] proved that \[ \text{su\,}A^m\leq \left( \sum_{j=1}^nr_j^m\right) ^{1/2}\left( \sum_{j=1}^nc_j^m\right) ^{1/2}\quad \text{ for all } m \text{ and } n.\tag{*} \] They asked whether the sharper inequality \[ \text{su\,}A^m\leq \sum_{j=1}^n(r_jc_j)^{m/2} \tag{C\(_1\)} \] holds. They posed also the conjecture \[ \text{su\,}A^m\leq \sum_{j=1}^n(r_j^{\downarrow }c_j^{\downarrow })^{m/2}\tag{C\(_2\)} \] which is stronger than (*) and weaker than (C\(_1\)). Laasko in his thesis proved that (C\(_1\)) is true for (i) \(n=1\) (ii) \(m=0\) or \(m=2\) (iii) \(n=2\) and \(m\geq 4\) is even, and false in the remaining cases.
In this article, the authors prove that (C\(_2\)) is true for \(n=2\) and false for (i) \(m=3\) and \(n\geq 8\) (ii) \(m=4\) and \(n\geq 50\) (iii) \(m\geq 3\) and \(n\) (depending on \(m\)) is large enough.
They also prove that (C\(_1\)) is true for \(m\) (depending on \(n\)) large enough. Nothing is said about the validity of (C\(_2\)) whenever \(m=3\) and \(3\leq n\leq 7\) or \(m=4\) and \(3\leq n\leq 49\).

MSC:

15B48 Positive matrices and their generalizations; cones of matrices
15A45 Miscellaneous inequalities involving matrices

Citations:

Zbl 0746.15010
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