Some asymptotic properties of discrete means

In part 1, given n different ways of averaging n positive numbers, we iterate the resulting map in $(0,\infty {)}^n$. 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 $a_i=a(i/n),1⩽i⩽n$, of a given continuous positive function a on the normalized interval $[0,1]$ and we let $p=f(n)$. When ${\lim }_{n\to \infty }f(n)/n=0$, we prove that it admits a limit as $n\to \infty$, called the f-mean of a, which moreover coincides with ${\int }_0^{1}a(x)\mathrm{d}x$ whenever $f(n)=o(\log n)$. We record similar, quite immediate, results on the geometric side $p=n-f(n)$.