A class of small multiplicative functions in Piatetski-Shapiro sequences

For any real number $x,[x]$ denotes the integer part of $x$. ${\mathcal{F}}_1$ denotes a class of multiplicative functions that are small in a numerical sense. In this paper, we study the distribution of the functions from ${\mathcal{F}}_1$ in Piatetski-Shapiro sequences. In particular, we prove that for $1<c<\frac{4}{3},$

$$\sum _{\stackrel{n\le x}{f\in {\mathcal{F}}_1}}f([n^c])={\int }_1^{x^c}\gamma u^{\gamma -1}d\left(\sum _{1\le n\le u}f(n)\right)+O(x^{1-\epsilon }),$$

where $\gamma =\frac{1}{c}$ and $[n^c]$ is the Piatetski-Shapiro sequence.