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Proceedings of the American Mathematical Society

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

 

 

A generalization of the arithmetic-geometric means inequality


Authors: A. M. Fink and Max Jodeit
Journal: Proc. Amer. Math. Soc. 61 (1976), 255-261
MSC: Primary 26A86
MathSciNet review: 0427564
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Abstract: It is shown that the arithmetic mean of $ {x_1}{w_1}, \ldots ,{x_n}{w_n}$ exceeds the geometric mean of $ {x_1}, \ldots ,{x_n}$ unless all the $ x$'s are equal, where $ {w_1}, \ldots ,{w_n}$ depend on $ {x_1}, \ldots ,{x_n}$ and satisfy $ 0 \leqslant {w_i} < 1$ unless $ {x_i} = \min {x_k}$. This inequality is then applied to an integral inequality for functions $ y$ defined on $ [0,\;\infty )$ with $ {y^{(k)}}$ convex and 0 at 0 for $ 0 \leqslant k < n$.


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DOI: http://dx.doi.org/10.1090/S0002-9939-1976-0427564-2
Keywords: Inequalities, arithemetic-geometric mean inequality, convex functions
Article copyright: © Copyright 1976 American Mathematical Society