Superquadratic functions and refinements of inequalities between averages

Abstract

In this paper upper bounds are given for the successive differences $A_{n+1}-A_n$ and B${}_n-B_{n-1}$ where $A_{n}=1/(n-1) \tsum_{r=1}^{n-1}f(r/n)$, $B_{n}=1/(n+1) \tsum_{r=0}^{n}f(r/n)$ and $f$ is superquadratic function. We obtain bounds for the successive differences of the more general sequence $1/c_{n}\tsum_{r=1}^{n}f(a_{r}/b_{n})$ when $f$ is superquadratic, which refine known results for convex functions. We also obtain bounds for various successive differences when $f$ is an increasing subquadratic function.