Entire functions that share pair of values with their derivatives (Q2744262)

Let \(f\) and \(g\) be non-constant entire functions, and let \(a\), \(b \in \mathbb C\). We say that \(f\) and \(g\) share the pair of values \((a,b)\) provided that \(f(z_0)=a\) if and only if \(g(z_0)=b\). If \(z_0\) is an \(a\)-point of \(f\) of multiplicity \(p\) and a \(b\)-point of \(g\) of multiplicity \(q\), then we say that \(f\) and \(g\) share the pair of values \((a,b)\) CM (counting multiplicities), if \(p=q\) for all such points \(z_0\). With these notations one of the main results of this paper reads as follows. NEWLINENEWLINENEWLINETheorem. Let \(k \in \mathbb N\), \(a\), \(b \in \mathbb C \setminus \{0\}\), and let \(f\) be a non-constant entire function such that \(f\) and \(f^{(k)}\) share the pair of values \((a,b)\) CM. If \(N(r,{1 \over f^{(k+1)}}) = S(r,f)\), then \(f-a=c(f^{(k)}-b)\) for some constant \(c \in \mathbb C \setminus \{0\}\). NEWLINENEWLINENEWLINEThe special case \(k=1\) and \(a=b\) gives a result of \textit{R. Brück} [Result. Math. 30, No. 1, 21-24 (1996; Zbl 0861.30032)]. Furthermore, the author also generalizes related results of \textit{G. Jank, E. Mues} and \textit{L. Volkmann} [Complex Variables, Theory Appl. 6, 51-71 (1986; Zbl 0603.30037)] as well as of \textit{L.-Z. Yang} [Bull. Aust. Math. Soc. 41, No. 3, 337-342 (1990; Zbl 0691.30022)].