Some asymptotic properties of discrete means

In part 1, given n different ways of averaging n positive numbers, we iterate the resulting map in (0,∞)n. We prove convergence toward the diagonal, with rate estimates under smoothness assumptions. In part 2, we consider the elementary symmetric means of order p applied to the values ai=a(i/n),1⩽i⩽n, of a given continuous positive function a on the normalized interval [0,1] and we let p=f(n). When limn→∞f(n)/n=0, we prove that it admits a limit as n→∞, called the f-mean of a, which moreover coincides with ∫01a(x)dx whenever f(n)=o(logn). We record similar, quite immediate, results on the geometric side p=n-f(n).