On a generalisation sum involving the Euler function

Let $j\ge 1$, $k\ge 0$ be real numbers and $\phi (n)$ be the Euler function. In this paper, we study the asymptotical behaviour of the summation function $$S_{j,k}(x):=\sum _{n\le x}\frac{\phi \left({\left[\frac{x}{n}\right]}^j\right)}{{\left[\frac{x}{n}\right]}^k}$$ as $x\to \mathrm{\infty }$, where $[\cdot ]$ is the integral part function. Our results combine and generalize the recent work of Zhai, Wu and Ma.