Note on a sum involving the divisor function

Let $d\left(n\right)$ be the divisor function and denote by $\left[t\right]$ the integral part of the real number $t$. In this paper, we prove that $\sum _{n\le x^{1/c}}d\left(\left(\frac{x}{n^c}\right)\right)=d_c{x}^{1/c}+O_{\varepsilon ,c}\left(x^{max\{(2c+2)/(2c^2+5c+2),5/(5c+6)\}+\varepsilon }\right),$ where $d_c={\sum }_{k\ge 1}d\left(k\right)\left(\frac{1}{k^{1/c}}-\frac{1}{{(k+1)}^{1/c}}\right)$ is a constant. This result constitutes an improvement upon that of Feng.