Monotonic averages of convex functions

We investigate the monotonicity of various averages of the values of a conv ex (or concave) function at n equally spaced points. For a convex function, averages without end paints increase with n, while averages with end point s 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 trap ezium estimate require convexity or concavity of the derivative as well as the function. Special cases include some known results and some new ones, u nifying them in a more systematic theory. Further applications include resu lts on series and power majorization. (C) 2000 Academic Press.