Certain meromorphic functions sharing a nonconstant polynomial with their linear polynomials

Abstract

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 of $f.$ 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 $L[f]=f^{(k)}+a_{k-1}f^{(k-1)}+\cdots+a_1f'+a_0f,$ where $k$ is a positive integer and $a_0,$ $a_1,$ $\cdots,$ $a_{k-1}$ are complex numbers, $k$ is a positive integer, $P$ is a nonconstant polynomial. Then, the hyper-order of $f$ is nonnegative integer or $\infty.$ 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 Brück [5]. An example is provided to show that the results in this paper are best possible.