Superquadratic functions and refinements of inequalities between averages

In this paper upper bounds are given for the successive differences $A_{n+1}-A_n$ and B${}_n-B_{n-1}$ where $A_n=1\mathrm{/}(n-1){\text{tsum}}_{r=1}^{n-1}f(r\mathrm{/}n)$, $B_n=1\mathrm{/}(n+1){\text{tsum}}_{r=0}^{n}f(r\mathrm{/}n)$ and $f$ is superquadratic function. We obtain bounds for the successive differences of the more general sequence $1\mathrm{/}{c}_n{\text{tsum}}_{r=1}^{n}f(a_r\mathrm{/}{b}_n)$ 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.