On the Composition of the Euler Function and the Dedekind Arithmetic Function

Let $I(n)=\frac{\psi (\phi (n))}{\phi (\psi (n))}$ and $K(n)=\frac{\psi (\phi (n))}{\phi (\phi (n))}$, where $\phi (n)$ is Euler’s function and $\psi (n)$ is Dedekind’s arithmetic function. We obtain the maximal order of $I(n)$, as well as the average orders of $I(n)$ and $K(n)$. Additionally, we prove a density theorem for both $I(n)$ and $K(n)$.