(Apology in advance for the broadness of this question) I recently came across a relatively simple application where I needed to "balance" the "spreaded-out-ness" of a function with the "peaked-ness" of a function, and my approach was to average over a few norms of the function, and for the purposes of my application seems to be working well enough, though I'm interested in any possible existing theory of what I've done. I apologize in advance for not being very technical; my explanation is basically entirely intuitive.
Specifically, suppose a bounded $f\colon[0,1]\rightarrow\mathbb{R}$$f\colon[0,1]\rightarrow\mathbb{R}_{\geq0}$ is given, and then sup-norm-normalize $f\leftarrow f/\|f\|_\infty$ so that $0\leq f\leq1$. Then as I understand, fractional powers $0<p<1$ will "lift the function away from zero", and so fractional "$p-$norms" will give a better idea of how much the function is spread out, whereas standard $p-$norms, $p\geq1$ will more accentuate the peaks as these powers "push the low spots closer to zero".
My painfully-simple solution was to just average the $\frac{1}{2}-$ and $2-$norms to "balance" the "spreaded-out-ness" and "peaked-ness" of the function in question, but I am very curious if there is any existing research on averaging norms to detect this type of structure of a function.
A little more precisely, I'm looking for literature related to quantities like $$\frac{1}{n}\sum_{k=1}^{n}\|f\|_{p_k}, \ \ 0<p_k<\infty$$ or even $$\frac{1}{p_2-p_1}\int_{p_1}^{p_2}\|f\|_pdp, \ \ 0<p_1<p_2<\infty.$$
Any ideas or direction will be greatly appreciated.