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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Rearrangements


Author: Henryk Minc
Journal: Trans. Amer. Math. Soc. 159 (1971), 497-504
MSC: Primary 15.58; Secondary 26.00
DOI: https://doi.org/10.1090/S0002-9947-1971-0283002-4
MathSciNet review: 0283002
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Abstract: It is shown that if $ {a^{(t)}} = (a_1^{(t)},a_2^{(t)}, \ldots ,a_n^{(t)}),t = 1, \ldots ,m$, are nonnegative $ n$-tuples, then the maxima of $ \sum\nolimits_{i = 1}^n {a_i^{(1)}a_i^{(2)} \cdots a_i^{(m)}} $ of $ \prod\nolimits_{i = 1}^n {{{\min }_t}(a_i^{(t)})} $ and of $ \Sigma _{i = 1}^n$ min $ (a_i^{(t)})$, and the minima of $ \prod\nolimits_{i = 1}^n {(a_i^{(1)} + a_i^{(2)} + } \cdots + a_i^{(m)})$, of $ \prod\nolimits_{i = 1}^n {{{\max }_t}(a_i^{(t)})} $ and of $ \sum\nolimits_{i = 1}^n {{{\max }_t}(a_i^{(t)})} $ are attained when the $ n$-tuples $ {a^{(1)}},{a^{(2)}}, \ldots ,{a^{(m)}}$ are similarly ordered. Necessary and sufficient conditions for equality are obtained in each case. An application to bounds for permanents of $ (0,1)$-matrices is given.


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DOI: https://doi.org/10.1090/S0002-9947-1971-0283002-4
Keywords: Inequalities, rearrangements, sums and products of nonnegative $ n$-tuples, Hardy-Littlewood-Pólya rearrangement theorem, permanents of $ (0,1)$-matrices
Article copyright: © Copyright 1971 American Mathematical Society

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