I know the linked articles from Wikipedia for the Euler's totient function and the Dedekind psi function. I know that are in the literature famous, interesting and difficult problems involving the Euler's totient function and that were studied by mathematicians. Feel free to ask about it in comments, since I know from an informative point of view the literature and I can to provide you references.

Question. I would like to know if does make sense similar problems for the Dedekind psi function $\psi(n)$, in particular the exploration of inequalities of the form $$\psi(an+b)<\psi(\hat{a}n+\hat{b})\tag{1}$$ for some suitable choice of integers or the density of sequences $$\{\psi(An+B)/(n^e \cdot\psi(Cn+D))\}_{n=1}^{\infty}\tag{2}$$ for some suitable choice of integers (compare with the section Ratio of consecutive values of Wikipedia). If it is in the literaute refer it and I try to read it from the literature, and other case propose what similar problems should be interesting. Many thanks.

Thus I am asking as a reference request if the problems that I evoke as $(1)$ and $(2)$ for some suitable choice of integers are in the literaute, then refer the literature (I think that it is useful as a reference for all people interesting if it is in this kind of problems). And alternatively in the other case I'm asking about yourself proposals for problems $(1)$ and $(2)$.

After some proposal/reference for $(1)$ and $(2)$ I am going to accept an answer.