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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,  
 
  Abstract:

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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