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  Volume 9, Issue 3, Article 65
Mixed Arithmetic and Geometric Means and Related Inequalities

    Authors: Takashi Ito,  
    Keywords: Inequalities, Arithmetic means, Geometric means.  
    Date Received: 21/04/08  
    Date Accepted: 30/07/08  
    Subject Codes:

26D20 , 26D99.

    Editors: Sever S. Dragomir,  

Mixed arithmetic and geometric means, with and without weights, are both considered. Related to mixed arithmetic and geometric means, the following three types of inequalities and their generalizations, from three variables to a general $ n$ variables, are studied. For arbitrary $ x,y,zgeq $ 0 we have

$displaystyle left[ {frac{x+y+z}{3}left( {xyz}ight) ^{	ext{$1$}/	ext{$3$}}}ight] ^{	ext{$1$}/	ext{$2$}}$ $displaystyle leq left( {frac{x+y}{2}cdot frac{y+z}{2}cdot frac{z+x}{2}}ight) ^{	ext{$1$}/	ext{$3$}},$ (A)
$displaystyle frac{1}{3}left( {sqrt{xy}+sqrt{yz}+sqrt{zx}}ight)$ $displaystyle leq frac{1}{2} left[ {frac{x+y+z}{3}+left( {xyz}ight) ^{	ext{$1$}/	ext{$3$}}}ight] ,$ (B)
$displaystyle left[ {frac{1}{3}left( {xy+yz+zx}ight) }ight] ^{	ext{$1$}/	ext{$2$} }$ $displaystyle leq left( {frac{x+y}{z}cdot frac{y+z}{2}cdot frac{z+x}{2}}ight) ^{	ext{$1$}/	ext{$3$}}.$ (D)

The main results include generalizations of J.C. Burkill's inequalities (J.C. Burkill; The concavity of discrepancies in inequalities of means and of Hölder, J. London Math. Soc. (2), 7 (1974), 617-626), and a positive solution for the conjecture considered by B.C. Carlson, R.K. Meany and S.A. Nelson (B.C. Carlson, R.K. Meany, S.A. Nelson; Mixed arithmetic and geometric means, Pacific J. of Math., 38 (1971), 343-347).

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