The asymptotic behavior for a heriditary recursion

$$x_1>a\text{and}{x}_{n+1}=\frac{1}{n^s}\sum _{k=1}^{n}f\left(\frac{x_k}{k}\right)\text{for every}n\ge 1$$

is studied, where $f$ is decreasing, continuous on $(a,\mathrm{\infty })$ ( $a<0$), and twice differentiable at $0$. The result has been known for the case $s=1$. This paper analyzes the case $s>1$. We obtain an asymptotic sequence that is quite different from the case $s=1$. Some examples and applications are provided.