The uniqueness of entire functions that share a small function with its differential polynomials
Let \(f\) and \(g\) be two non-constant entire functions, \(\alpha\) be a small function of \(f\) and \(g\). If \(f-\alpha\) and \(g-\alpha\) have the same zeros with the same multiplicities, then we say that \(f\) and \(g\) share \(\alpha\) \(cM\). Denote \[ L[f]=a_kf^{(k)}+ a_{k-1}f^{(k-1)}+\cdots+ a_0f+\beta,\quad (k>0) \] where \(a_j\) \((j=0,1,\dots,k)\) are constants with \(a_k\neq 0\), \(\beta\) is a small functin of \(f\). In this paper the authors proved that if \(f\) is of finite order and \(\sigma(f)\neq 1\), and \(f\) and \(L[f]\) share a small function \(\alpha\) \(cM\), then \[ L[f]-\alpha=c(f-\alpha), \] where \(c\) is a non-constant.
scientific article; zbMATH DE number 2147391