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
DOI:
https://doi.org/10.1090/S0002-9939-1976-0427564-2
MathSciNet review:
0427564
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Abstract | References | Similar Articles | Additional Information
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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- A. M. Fink, Mean value inequalities, SIAM J. Math. Anal. 8 (1977), no. 5, 785–791. MR 499039, DOI https://doi.org/10.1137/0508061
- G. H. Hardy, J. E. Littlewood, and G. Pólya, Inequalities, Cambridge, at the University Press, 1952. 2d ed. MR 0046395
- D. S. Mitrinović, Analytic inequalities, Springer-Verlag, New York-Berlin, 1970. In cooperation with P. M. Vasić. Die Grundlehren der mathematischen Wissenschaften, Band 165. MR 0274686
- Carl Ludwig Siegel, The trace of totally positive and real algebraic integers, Ann. of Math. (2) 46 (1945), 302–312. MR 12092, DOI https://doi.org/10.2307/1969025
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Additional Information
Keywords:
Inequalities,
arithemetic-geometric mean inequality,
convex functions
Article copyright:
© Copyright 1976
American Mathematical Society
