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  Volume 5, Issue 4, Article 91
Inequalities for Averages of Convex and Superquadratic Functions

    Authors: Shoshana Abramovich, Graham Jameson, Gord Sinnamon,  
    Keywords: Inequality, Averages, Convex, Superquadratic, Monotonic  
    Date Received: 26/07/04  
    Date Accepted: 03/08/04  
    Subject Codes:

26A51, 26D15

    Editors: Constantin P. Niculescu,  

We consider the averages $ A_n(f)= 1/(n-1) sum_{r=1}^{n-1} f(r/n)$ and $ B_n(f)=1/(n+1) sum_{r=0}^n f(r/n)$. If $ f$ is convex, then $ A_n(f)$ increases with $ n$ and $ B_n(f)$ decreases. For the class of functions called superquadratic, a lower bound is given for the successive differences in these sequences, in the form of a convex combination of functional values, in all cases at least $ f(1/3n)$. Generalizations are formulated in which $ r/n $ is replaced by $ a_r/a_n$ and $ 1/n$ by $ 1/c_n$. Inequalities are derived involving the sum $ sum_{r=1}^n (2r-1)^p$.

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