Meromorphic functions whose derivatives share four small functions
Let \(f\) be a non-constant meromorphic function in the plane. A meromorphic function \(b\) is said to be a small function with respect to \(f\) if \(T(r,b)=o(T(r,f))\), where \(T(r,f)\) denotes the Nevanlinna characteristic of \(f\). Of course, any constant function is a small function with respect to \(f\). Two non-constant meromorphic functions \(f\) and \(g\) are said to share a small function (or value) \(b\) CM (counting multiplicities) if \(f-b\) and \(g-b\) have the same zeros with the same multiplicities. The well-known four-point-theorem of R.~Nevanlinna states that if \(f\) and \(g\) are non-constant meromorphic functions which share four distinct values CM, then \(f\) is a Möbius transformation of \(g\). In this paper the author proves the following theorem. Theorem. Let \(f\) and \(g\) be transcendental meromorphic functions, let \(k\) be a positive integer, and let \(b_1\), \(b_2\), \(b_3\), \(b_4\) be four distinct small functions with respect to \(f\) and \(g\). If \(f^{(k)}\) and \(g^{(k)}\) share \(b_1\), \(b_2\), \(b_3\), \(b_4\) CM, then only one of the following possibilities occur: (1) If \(b_j \not\equiv \infty\) (\(j=1,2,3,4\)), then \(f^{(k)} \equiv g^{(k)}\). (2) If \(a_4 \equiv \infty\), then \((f^{(k)}-a_3)(g^{(k)}-a_3) \equiv (a_2-a_3)^2\), and \(\infty\) and \(a_3\) are exceptional functions of \(f^{(k)}\) such that \(a_1+a_2 \equiv 2a_3\). Here \((a_1,a_2,a_3,a_4)\) is a permutation of \((b_1,b_2,b_3,b_4)\). This theorem generalizes a previous result of the author [Syst. Sci. Math. Sci. 11, 245-248 (1998; Zbl 0933.30021)]. The example \(f(z)=e^z\) and \(g(z)=e^{-z}\) with an even positive integer \(k\) shows that case (2) actually may occur.
scientific article; zbMATH DE number 1791140