Let's call $K$ the set of all possible functions $f_n(x)$ bounded by $o(1)$ as $x \rightarrow \infty$. Here, o is the little-o-notation.
Now integrate all of these functions and let's say new set is $L$. I would like to ask that, what is the minimum function $g(x)$ that can bound all of the elements of $L$ with notation $o(g(x))$? I mean there are infinitely many function that can bound all of the elements of $L$, but I want the minimum of these.
For instance you claim that $g_0$ is the minimum function that can bound $L$. If someone can find $g_1$ also bounds $L$, we will compare these two functions like follows:
- if we can write $g_0 = o(g_1)$ so $g_0$ is the minimum.
- if we can write $g_1 = o(g_0)$ so $g_1$ is the minimum
Is the answer trivially $o(x)$ or is it more complicated than I thought?