On a sum involving the Euler totient function

In this short note, we prove that $\frac{4}{{\pi }^2}xlogx+O\left(x\right)?\sum _{n?x}\phi \left(\frac{x}{n}]\right)?\left(\frac{1}{3}+\frac{4}{{\pi }^2}\right)xlogx+O\left(x\right),$ for $x\to \infty$, where $\phi \left(n\right)$ is the Euler totient function and $t]$ is the integral part of real $t$. This improves recent results of Bordellès–Heyman–Shparlinski and of Dai–Pan.