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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Refinements of Ky Fan's inequality


Author: Horst Alzer
Journal: Proc. Amer. Math. Soc. 117 (1993), 159-165
MSC: Primary 26D20
DOI: https://doi.org/10.1090/S0002-9939-1993-1116251-3
MathSciNet review: 1116251
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Abstract: We prove the inequalities

$\displaystyle A_n' /G_n' \leqslant (1 - G_n' )/(1 - A_n' ) \leqslant {A_n}/{G_n}$

and

$\displaystyle A_n' /G_n' \leqslant (1 - {G_n})/(1 - {A_n}) \leqslant {A_n}/{G_n},$

where $ {A_n}$ and $ {G_n}$ (respectively, $ A_n'$ and $ G_n'$) denote the unweighted arithmetic and geometric means of $ {x_1}, \ldots ,{x_n}$ (respectively, $ 1 - {x_1}, \ldots ,\;1 - {x_n}$) with $ {x_i} \in (0,\tfrac{1} {2}](i = 1, \ldots ,n;n \geqslant 2$. Further we show that the ratios $ (1 - G_n' )/(1 - A_n')$ and $ (1 - {G_n})/(1 - {A_n})$ can be compared if and only if $ n = 2$.

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DOI: https://doi.org/10.1090/S0002-9939-1993-1116251-3
Keywords: Fan's inequality, arithmetic means, geometric means
Article copyright: © Copyright 1993 American Mathematical Society

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