On the average value of a function of generalized mean

Abstract

In this short note we find the exact formula for

$$\lim_{n\rightarrow\infty}\idotsint_{[0,1]^n}f(M_{n,p}(x_1,\dots, x_n))\,dx_1\dots dx_n,$$

where $M_{n,p}(x_1,\dots, x_n)$, $p\in [-\infty,\infty]$ is the generalized mean and $f$ is an arbitrary continuous function.