Note on a paper by Bordellès, Dai, Heyman, Pan and Shparlinski, 3

Denote by $[t]$ the integral part of $t$. Under some simple hypothesis on the growth of arithmetic function $f$, we prove asymptotic formulas for $$ S_f(x):= \sum_{n\leqslant x} f\Big(\Big[\frac{x}{n}\Big]\Big) $$ as $x\to\infty$ and give somme applications. These improve or generalize some recent results of Zhao and Wu.