Uniqueness of entire functions sharing polynomials with their derivatives
The authors use the theory of normal families to prove the following. Let \(Q_{1}(z)=a_{1}z^{p}+a_{1,p - 1}z^{p - 1}+ \dots +a_{1,0}\) and \(Q_{2}(z)=a_{2}z^{p}+a_{2,p - 1}z^{p - 1}+ \dots +a_{2,0}\) be two polynomials such that \(\text{deg }Q_{1}=\text{deg } Q_{2}=p\) (where \(p\) is a nonnegative integer), and let \(a_{1},a_{2}\) \((a_{2}\neq 0)\) be two distinct complex numbers. Let \(f(z)\) be a transcendental entire function. If \(f(z)\) and \(f^{\prime }(z)\) share the polynomial \(Q_{1}(z)\) CM and if \(f(z)=Q_{2}(z)\) whenever \(f^{\prime }(z)=Q_{2}(z)\), then \(f\equiv f^{\prime }\). This result improves a result due to \textit{X. M. Li} and \textit{H. X. Yi} [Arch. Math. 89, No. 3, 216--225 (2007; Zbl 1131.30013)].
scientific article; zbMATH DE number 5574391
scientific article