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.
Categories: math.NA