The aim of this paper is to present newupper bounds for the distance between a properly normalized permanent of a rectangular complex matrix and the product of the arithmetic means of the entries of its columns. It turns out that the bounds improve those from earlier work. Our proofs are based on some new identities for the above-mentioned difference and also for related expressions for matrices over a rational associative commutative unital algebra. Some of our identities are generalizations of results in [J. Dougall, Quantitative proofs of certain algebraic inequalities, Proc. Edinb. Math. Soc. , 24:61–77, 1905]. Second-order results are also included.

中文翻译：

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通过恒等式的新永久近似不等式

本文的目的是为矩形复矩阵的适当归一化永久矩阵与其列的条目的算术平均值的乘积之间的距离提供新的上界。事实证明，边界改进了早期工作的边界。我们的证明基于上述差异的一些新恒等式以及有理结合交换单位代数上矩阵的相关表达式。我们的一些身份是 [J. Dougall，某些代数不等式的定量证明，Proc。爱丁堡 数学。社会。, 24:61–77, 1905]。二阶结果也包括在内。