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  Volume 7, Issue 4, Article 148
New Inequalities About Convex Functions

    Authors: Lazhar Bougoffa,  
    Keywords: Jensen's inequality, Convex functions.  
    Date Received: 11/06/06  
    Date Accepted: 15/10/06  
    Subject Codes:


    Editors: Bicheng Yang,  

If $ f$ is a convex function and $ x_{1},dots ,x_{n}$ or $ a_{1},dots,a_{n}$ lie in its domain the following inequalities are proved

begin{multline*} sum_{i=1}^{n}f(x_{i})-fleft(frac{x_{1}+cdots +x_{n}}{n}ri... ...1}+x_{n}}{2}right)+fleft(frac{x_{n}+x_{1} }{2}right) right] end{multline*}

$displaystyle (n-1)left [f(b_{1})+ cdots +f(b_{n})right ]leq nleft[ f(a_{1})+cdots +f(a_{n}) - f(a)right], $

where $ a=frac{a_{1}+cdots+a_{n}}{n}$ and $ b_{i}=frac{n a-a_{i}}{n-1},  i=1,dots,n $.

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