Meromorphic functions share one value with their derivatives (Q2721547)

For a non-constant meromorphic function \(f\) in \(\mathbb C\) and \(a \in \mathbb C\) let \(E(a,f)\) denote the set of \(a\)-points of \(f\), and for \(z_0 \in E(a,f)\) let \(\nu(z_0,a,f)\) denote the multiplicity of the \(a\)-point \(z_0\). We say that two non-constant meromorphic functions \(f\) and \(g\) share the value \(a \in \mathbb C\) if \(E(a,f)=E(a,g)\). If in this case \(\nu(z_0,a,f)=\nu(z_0,a,g)\) for all \(z_0 \in E(a,f)\), then \(f\) and \(g\) share the value \(a\) CM (counting multiplicities). \textit{R.~Brück} [Result. Math. 30, No. 1-2, 21-24 (1996; Zbl 0861.30032)] proved that if \(f\) is a non-constant entire function such that \(f\) and \(f'\) share the value \(1\) CM and \(N(r,{1 \over f'})=S(r,f)\) then \(f-1=c(f'-1)\) for some constant \(c \in \mathbb C \setminus \{0\}\). In this article, the author generalizes this result in several ways. NEWLINENEWLINENEWLINEIn order to state some special cases of the main results we need some further notation. For \(k \in \mathbb N \cup \{\infty\}\) let \(E_k(a,f) := \{ z_0 \in E(a,f) : \nu(z_0,a,f) \leq k \}\) (so that \(E_\infty(a,f)=E(a,f)\)). Furthermore, let \(N_1(r,{1 \over f-a})\) denote the counting function of the simple \(a\)-points of \(f\). Then the following results hold. NEWLINENEWLINENEWLINETheorem 1. Let \(n \in \mathbb N \cup \{\infty\}\), \(k \in \mathbb N\), and let \(f\) be a non-constant meromorphic function such that \(E_n(1,f) = E_n(1,f^{(k)})\). If \(N(r,{1 \over f'}) + \overline{N}(r,f) = S(r,f)\), then \(f-1=c(f^{(k)}-1)\) for some constant \(c \in \mathbb C \setminus \{0\}\). NEWLINENEWLINENEWLINETheorem 2. Let \(f\) be a non-constant entire function such that \(f\) and \(f'\) share the value \(1\) CM. If \(N_1(r,{1 \over f'}) < (\lambda+o(1))T(r,f')\) for some \(\lambda \in (0,{1 \over 2})\), then \(f-1=c(f'-1)\) for some constant \(c \in \mathbb C \setminus \{0\}\).