## 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

Zbl 0746.15010
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