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)) + ... + b(n)(z) f (z + c(n)) be a linear difference polynomial, where b(1)(z), b(2)(z), ..., b(n)(z) are nonzero small functions relative to f(z), and c(1), c(2), ..., 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 Bruck conjecture. Some sufficient conditions to show that f(z) and L(z, f) cannot share some small functions are also presented.