Mixed-mean inequalities for subsets
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- by Gangsong Leng, Lin Si and Qingsan Zhu PDF
- Proc. Amer. Math. Soc. 132 (2004), 2655-2660 Request permission
Abstract:
For $A\subset X=\{x_1,...,x_n ~|~x_i\geq 0, ~i=1,2,...,n\},$ let $a_A$ and $g_A$ denote the arithmetic mean and geometric mean of elements of $A$, respectively. It is proved that if $k$ is an integer in $(\frac {n}{2}, n]$, then \[ \Big (\prod _{|A|=k\atop A\subset X}a_{A}\Big )^{\frac {1}{C_n^k}} \geq \frac {1}{C_n^k}\Big (\sum _{|A|=k\atop A\subset X}g_{A}\Big ),\] with equality if and only if $x_1=...=x_n$. Furthermore, as a generalization of this inequality, a mixed power-mean inequality for subsets is established.References
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Additional Information
- Gangsong Leng
- Affiliation: Department of Mathematics, Shanghai University, Shanghai, 200436, People’s Republic of China
- MR Author ID: 323352
- Email: gleng@mail.shu.edu.cn
- Lin Si
- Affiliation: Department of Mathematics, Shanghai University, Shanghai, 200436, People’s Republic of China
- Email: silin_mail@sohu.com
- Qingsan Zhu
- Affiliation: School of Mathematical Sciences, Peking University, Beijing 100871, People’s Republic of China
- Received by editor(s): May 15, 2003
- Received by editor(s) in revised form: June 9, 2003
- Published electronically: March 24, 2004
- Additional Notes: This work was supported partly by the National Natural Sciences Foundation of China (10271071)
- Communicated by: Carmen C. Chicone
- © Copyright 2004 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 132 (2004), 2655-2660
- MSC (2000): Primary 26A51; Secondary 26B25, 26D07
- DOI: https://doi.org/10.1090/S0002-9939-04-07384-8
- MathSciNet review: 2054791