Some results on uniqueness of an entire function with its shift difference

In the paper, we have shown that every entire solution of the differential difference equation ${\mathrm{\Delta }}_{\eta }^{m}f-Q_1=({\mathrm{\Delta }}_{\eta }^{k}f-Q_2)e^P$ satisfy hyper order of $f=$degree of $P$ and using this result we prove differential difference counter part Brück conjecture. Also we have proved differential difference counter part Brück conjecture for the special case of hyper order of $f<\frac{1}{2}$.