On a sum involving certain arithmetic functions on Piatetski–Shapiro and Beatty sequences

Let $c$, $\alpha$, $\beta \in \mathbb{R}$ be such that $1<c<2$, $\alpha>1$ is irrational and with bounded partial quotients, $\beta\in [0, \alpha)$. In this paper, we study asymptotic behaviour of the summations of the form $\displaystyle \sum\limits_{n\leq N}\frac{f(\lfloor n^c \rfloor)}{ \lfloor n^c \rfloor}$ and $\displaystyle \sum\limits_{n\leq N}\frac{f(\lfloor \alpha n+\beta \rfloor)}{\lfloor \alpha n+\beta \rfloor}$, where $f$ is the Euler totient function $\phi$, Dedekind function $\Psi$, sum-of-divisors function $\sigma$, or the alternating sum-of-divisors function $\sigma_{alt}$.