What is the relationship between the complexity class $L^n$ and $NL^m$?
It is not known whether $\mathsf{L} = \mathsf{NL}$, and this is a famous open question. More generally, I believe that no better bound than $\phi(m) \leq 2m$ is known for any value of $m$, nor is this bound known to be optimal for any value of $m$.