Exceptional sets for compositions involving Euler's function,the divisor-sum function and Dedekind's function

Let $\psi (n)$, $\phi (n)$, and $\sigma (n)$ denote Dedekind’s arithmetic function, Euler’s totient function, and the sum-of-divisors function, respectively. We study exceptional sets associated with the compositions $\phi (\psi (n))$, $\phi (\sigma (n))$, $\psi (\psi (n))$, and $\psi (\sigma (n))$. For every fixed $c>0$, we obtain quantitative upper bounds for the numbers of integers $n\le x$ satisfying $\phi (\psi (n))\ge cn$ and $\phi (\sigma (n))\ge cn$, thereby refining density results of Sándor and Dixit and Bhattacharjee, respectively. We further establish quantitative density-zero estimates for the sets of integers $n\le x$ for which $\psi (\psi (n))\le cn$ or $\psi (\sigma (n))\le cn$. More generally, we allow the fixed thresholds in the first two problems to be replaced by thresholds involving non-decreasing functions subject to mild growth conditions.