Uniqueness on linear difference polynomials of meromorphic functions

Suppose that $ f(z) $ is a meromorphic function with hyper order $ \sigma_{2}(f) < 1 $. Let $ L(z, f) = b_1(z)f(z+c_1)+b_2(z)f(z+c_2)+\cdots+b_n(z)f(z+c_n) $ be a linear difference polynomial, where $ b_1(z), b_2(z), \cdots, b_n(z) $ are nonzero small functions relative to $ f(z) $, and $ c_1, c_2, \cdots, c_n $ are distinct complex numbers. We investigate the uniqueness results about $ f(z) $ and $ L(z, f) $ sharing small functions. These results promote the existing results on differential cases and difference cases of Brück conjecture. Some sufficient conditions to show that $ f(z) $ and $ L(z, f) $ cannot share some small functions are also presented.