Assume there is a function $f(x)$ which oscillates over some mean value in an interval $x \in [a..b]$. The mean formula is: $$m(a,b)=\frac{1}{b-a} \int_a^b f(x)dx$$
Assume the function is positive in the inteval $[a,b]$.
Now for the same interval lets consider: $$l(a,b)= \int_a^b w(a,b,x)\ln f(x)dx$$
I would like to understand if there is any relation between the mean of function and logarithmic mean. I am not sure how to determine the $w(a,b)$ in order to keep the $l(a,b)$ to have the meaning of mean. Not sure if the obvious $w(a,b,x)=\frac{1}{b-a}$ will bring us to proper mean definition of log case.
And also is there any relation between the $l(a,b)$ and $m(a,b)$.
I tried to find any definition of mean based on logarithm but was unable to find some. After long searches found the only article (that more or less correspond to this topic) is https://en.wikipedia.org/wiki/Logarithmic_mean
If we look at the below sample images we will see that the topologically they have the same structure, so the averages also should be somehow related.
This is the base image:
This is the log image:
