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Monotonic averages of convex functions.

Research output: Contribution to journalJournal article

Published

Journal publication date1/12/2000
JournalJournal of Mathematical Analysis and Applications
Journal number1
Volume252
Number of pages21
Pages410-430
Original languageEnglish

Abstract

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.