TIME(n) versus TIME(nlogn)
For all $a≥0$ and $ε&gt;0$, $\mathrm{DTIME}(O(n \log^{a} n))$ is strictly contained in $\mathrm{DTIME}(O(n \log^{a+ε} n))$; see my answer here.

To recap, the logarithmic overhead comes from tape reduction (and is not needed if the number of tapes is fixed and is at least two).  There is a nonconstructive argument that converts the logarithmic overhead into time hierarchy at $\log^{ε} \mathrm{time}$ multiplicative precision for well-behaved (time-constructible and in a certain sense continuous) time bounds.

A constructive proof of the strict inclusion of $\mathrm{DTIME}(O(n))$ in $\mathrm{DTIME}(O(n \log n))$ remains (to my knowledge) an open problem.