In my home I am trying to get the thesis for a specialization for my example* from a statement published related to functions that satisfy certain properties.

I've considered the function* $$f(x)=-\sum_{n=2}^\infty\frac{\mu(n)}{n}x^n,\tag{1}$$ where $\mu(n)$ denotes the Möbius function and our function $(1)$ is defined as a formal series over the real numbers of the unit interval. Using a CAS I think that my example satisfy (each of those) some of such properties, and with this purpose I wondered next question about one of main properties that I need.

Question.Is it possible to prove that $$f(x)=-\sum_{n=2}^\infty\frac{\mu(n)}{n}x^n$$ is bijective over $[0,1]$? Can you provide a good and explicit estimation (in terms of algebraic functions or elementary transcendental functions, see this Wikipedia $u(x)$ and $l(x)$) of $$l(x)\leq f^{-1}(x)\leq u(x)\tag{2}$$ over the unit interval?Many thanks.

*I know that my example of function $(1)$ was in the literature. If there was literature about if previous function is bijective and how to estimate its inverse $f^{-1}(x)$ over the unit interval, then answer this question as a reference request for such literature and I try to find and study such statements.