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Mixed-mean inequalities for subsets

Author(s): Gangsong Leng; Lin Si; Qingsan Zhu
Journal: Proc. Amer. Math. Soc. 132 (2004), 2655-2660.
MSC (2000): Primary 26A51; Secondary 26B25, 26D07
Posted: March 24, 2004
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Abstract: For $A\subset X=\{x_1,...,x_n ~\vert~x_i\geq 0, ~i=1,2,...,n\},$ 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

\begin{displaymath}\Big(\prod_{\vert A\vert=k\atop A\subset X}a_{A}\Big)^{\frac{... ...{1}{C_n^k}\Big(\sum_{\vert A\vert=k\atop A\subset X}g_{A}\Big),\end{displaymath}

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.

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Additional Information:

Gangsong Leng
Affiliation: Department of Mathematics, Shanghai University, Shanghai, 200436, People's Republic of China
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

DOI: 10.1090/S0002-9939-04-07384-8
PII: S 0002-9939(04)07384-8
Keywords: Mixed mean, power mean, Carlson inequality
Received by editor(s): May 15, 2003
Received by editor(s) in revised form: June 9, 2003
Posted: 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 of article: Copyright 2004, American Mathematical Society

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