It is well known that matrices with a $UV$-displacement structure possess generalized inverse with a $VU$-displacement structure. Estimation for the displacement rank of $A_{T,S}^{(1,2)}U-VA_{T,S}^{(1,2)}$ are presented, where $A_{T,S}^{(1,2)}$ is the $(1,2)$-inverse of $A$ with prescribed range $T$ and null space $S$. We extend the results due to G. Heinig and F. Hellinger, Wei and Ng, Cai and Wei for the Moore-Penorse inverse, group inverse and weighted Moore-Penrose inverse, respectively.