It's well-known that $$ \liminf_n\frac{\varphi(n)\log\log n}{n}=e^{-\gamma} $$ and there exists an effective version $$ \varphi(n)>\frac {n}{e^\gamma\log\log n+\frac{3}{\log\log n}} $$ valid for $n\ge3.$ Of course the RHS is increasing and so has an inverse, but I would like to know if there is an explicit formula (giving a tight bound) with $$ \varphi(f(n))>n. $$
Is this too much to ask?