Certain meromorphic functions sharing a nonconstant polynomial with their linear polynomials

Let B be the class of meromorphic functions f such that the set sing (f(-1)) is bounded, where sing (f(-1)) is the set of critical and asymptotic values off. Suppose that f has at most finitely many poles in the complex plane, and that L(f) - P and f - P share 0 CM, where L[f] = f((k)) + a(k-1)f((k-1)) + . . . + a(1)f' + a(0)f, where k is a positive-integer and a(0), a(1), . . . , a(k-1) are complex numbers, P is a nonconstant polynomial. Then, the hyper-order of f is nonnegative integer or infinity. Applying this result, we obtain some uniqueness results for transcendental meromorphic functions having the same fixed points with their linear differential polynomials, where the meromorphic functions belong to B and have at most finitely many poles in the complex plane. The results in this paper are concerning a conjecture of Briick [5]. An example is provided to show that the results in this paper are best possible.