The average value of a function $y=f(x)$, on an interval $[a,b]$, is ${1\over {b-a}}\int_a^b f(t)dt$. This of course relates to the arithmetic average. It is easy to see that a corresponding formula for the geometric average is $\exp\left({1\over {b-a}}\int_a^b \ln(f(t))dt\right)$.
There are many other types of averages. In particular the ones motivated by the elementary symmetric polynomials are interesting as they "mix" the function values. My question is: How can we evaluate those averages?
To be specific, consider a real positive continuous function $y=f(x)$ on $[a,b]$. Create a partition of $n$ sub-intervals of width $\Delta x$. Let $Y=(y_1,y_2,\cdots, y_n)$ be the values of the function $f$ at some point in those intervals. Define the elementary symmetric polynomials $e_k=e_k(Y)$, for $1\le k \le n$, through
$$ \prod_{i=1}^n (t+y_i)= t^n+e_1t^{n-1}+\cdots+e_n. $$ Define the average $$ a_k(Y)={\root k \of {{e_k} \over {\left (n \atop k \right )}}}. $$ Define $A_\alpha(f)$, the $\alpha$-average of $f$ over $[a,b]$, as the limit of $a_k(Y)$ as $n \to \infty$, $\Delta x \to 0$, and $k/n \to \alpha$. Note $\alpha=0$ corresponds to the arithmetic average and $\alpha=1$ is the geometric average.
What do we know about $A_\alpha$ for $0<\alpha <1$? How can we compute it? For example if $f(x)=x$, $[a,b]=[1,2]$, and $\alpha=1/2$ what is $A_\alpha$?
Edit 1:
Some related inequalities are Maclaurin's and Newton's.
Edit 2:
I guess the requirement of continuity can be relaxed to piecewise continuity and still have a unique limit. Finding $A_\alpha$ for the following function, for a given $m>0$, will also be of interest: $$f(x)= \cases { 1 & if $ \quad 0 \le x \le 1/2$ \cr m & if $ \quad 1/2 < x \le 1$ }.$$