Transcendental entire functions of finite order sharing two sets of small functions with their shift differential operators

Dealing with a question initiated by Liu [Meromorphic functions sharing set applications to difference equations, J. Math. Anal. Appl., 2009], we have investigated the situation when finite order entire function and its shift differential operator share two sets of small functions. Our result has improved extended results Chen-Chen [Entire their operators or shifts, Slovaca, 2013] Cui-Chen [The conjecture on unity meromorphic concerning differences, Difference Equ. 2016]. We exhibited several examples relevant content paper.