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  Volume 6, Issue 2, Article 43
Note on Certain Inequalities for Means in Two Variables

    Authors: Tiberiu Trif,  
    Keywords: Arithmetic mean, Geometric mean, Identric mean.  
    Date Received: 13/10/04  
    Date Accepted: 16/03/05  
    Subject Codes:

Primary: 26E60, 26D07.

    Editors: Sever S. Dragomir,  

Given the positive real numbers $ x$ and $ y$, let $ A(x,y)$, $ G(x,y)$, and $ I(x,y)$ denote their arithmetic mean, geometric mean, and identric mean, respectively. It is proved that for $ pgeq 2$, the double inequality

$displaystyle alpha A^p(x,y)+(1-alpha)G^p(x,y)I^p(x,y)beta A^p(x,y)+(1-beta)G^p(x,y) $

holds true for all positive real numbers $ xneq y$ if and only if $ alphaleq left(frac{2}{e}right)^p$ and $ betageqfrac{2}{3}$. This result complements a similar one established by H. Alzer and S.-L. Qiu [Inequalities for means in two variables, Arch. Math. (Basel) 80 (2003), 201-215].

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