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  Volume 8, Issue 2, Article 51
$S$-Geometric Convexity of a Function Involving Maclaurin's Elementary Symmetric Mean

    Authors: Xiao-Ming Zhang,  
    Keywords: Geometrically convex function, $S$-geometrically convex function, Inequality, Maclaurin-Inequality, Logarithm majorization.  
    Date Received: 23/02/07  
    Date Accepted: 27/04/07  
    Subject Codes:

Primary 26D15.

    Editors: Peter S. Bullen,  

Let $ x_{i}>0,i=1,2,dots ,n$, $ x=left( {x_{1},x_{2},dots ,x_{n}}right) $, the $ k$th elementary symmetric function of $ x$ is defined as $ P_{n}left( {x,k}right) =left( {{binom{n}{k}} ^{-1}E_{n}left( {x,k}right) }right) ^{frac{1}{k}}$, and the function $ f$ is defined as $ fleft( xright) =P_{n}left( {x,k-1}right) -P_{n}left( {x,k}right) $. The paper proves that $ f$ is a S-geometrically convex function. The result generalizes the well-known Maclaurin-Inequality.

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