An entire function sharing a polynomial with its derivatives
The paper deals with unicity theorems for entire functions. Let \(f\) be an entire function and \(a=a(z)\) a polynomial of degree \(p\). We let \(E(a;f)\) denote the set of zeros of \(f-a\) counted with multiplicities. We also denote by \(\overline{E}(a;f)\) the set of distinct zeros of \(f-a\). In the case where \(f\) is a transcendental function, we set \(\text{deg}\,f=+\infty\). We define the counting function \(n_A(r, f, a)\) as the number of zeros of \(f\) in \(A\cap \left\{|z|<r \right\}\) with multiplicities, and let \[ N_A(r, f, a)=\int_0^r (n_A(t, f, a)-n_A(0, f, a))dt/t. \] Suppose that \(n \geq p+2\) is a positive integer. Let \(A=\overline{E}(a;f)\setminus \overline{E}(a;f')\) and \(B=\overline{E}(a;f')\setminus (\overline{E}(a;f^{(n)})\cap \overline{E}(a;f^{(n+1)}))\). Then the authors prove the following unicity theorem: If \[ N_A(r, a, f)+N_B(r, a, f')=S(r, f) \] and each common zero of \(f-a\) and \(f'-a\) has the same multiplicity, then \(f=\lambda e^z\) for some constant \(\lambda \neq 0\).
scientific article; zbMATH DE number 6802157