Brück conjecture and linear differential polynomials
The paper deals with a unicity theorem of meromorphic functions. Let \(f\) be a meromorphic function on \(\mathbb{C}\). We let \(\sigma(f)\) denote the order of \(f\). Let \(f\) and \(a\) be meromorphic functions on \(\mathbb{C}\). We assume that \(f\) and \(a\) do not have common poles and \(\sigma(a)<\sigma(f)<+\infty\). Let \(L(f)=a_0f+a_1f'+\cdots+a_kf^{(k)}\), where \(k\) is a positive integer and \(a_0,\ldots,a_k\) (\(\neq 0\)) are constants. The main result in this paper can be stated as follows. Suppose \(f\) and \(L(f)\) share the function \(a\) CM. Then \(L(f)-a=c(f-a)\) for a nonzero constant \(c\), provided that one of the following holds: \[ (1)\quad \sigma(f)\neq 1\qquad \text{or} \qquad (2)\quad \sigma(f)= 1 \text{ and } a_0=\cdots=a_{k-1}. \] The authors also give an example that shows that the above condition (2) is essential,
scientific article; zbMATH DE number 6861518