I am trying to bound a function that includes $\sum\limits_{\substack{d < n^{1/3} \\ d \mid n}} 1$.

Is there an upper bound known for this sum, either in general or in terms of $\sum\limits_{\substack{d \mid n}} 1$? Or in general is there a bound for $\sum\limits_{\substack{d < n^{1/k} \\ d \mid n}} 1$? Any help is appreciated.

Edit: I am realizing that a lower bound for this sum in terms of $\sum\limits_{\substack{d \mid n}} 1$ would also be useful if anyone can help with that.