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Monotonic averages of convex functions.

Research output: Contribution to journalJournal articlepeer-review

<mark>Journal publication date</mark>1/12/2000
<mark>Journal</mark>Journal of Mathematical Analysis and Applications
Issue number1
Number of pages21
Pages (from-to)410-430
Publication StatusPublished
<mark>Original language</mark>English


We investigate the monotonicity of various averages of the values of a convex (or concave) function at n equally spaced points. For a convex function, averages without end points increase with n, while averages with end points decrease. Averages including one end point are treated as a special case of upper and lower Riemann sums, which are shown to decrease and increase, respectively. Corresponding results for mid-point Riemann sums and the trapezium estimate require convexity or concavity of the derivative as well as the function. Special cases include some known results and some new ones, unifying them in a more systematic theory. Further applications include results on series and power majorization.