Unicity theorems for meromorphic functions sharing four small functions
Let \(f_1\), \(f_2\) be transcendental meromorphic functions on the complex plane \(\mathbb{C}\). The paper is a voluminous work which discusses the equality of \(f_1\) and \(f_2\) under different sets of conditions. In addition to the standard functions of the Nevanlinna theory, the bulk of the discussions involve the ``smallness w.r.t. \(f_j\)' in terms of the classes (1) \(s(f_j)\) of all real functions \(p\) on \((0,+\infty)\) such that \(p(r)/T(r,f_j)\to 0\) as \(r\to+\infty\) possibly excluding a set of finite linear measure and (2) \(S(f_j)\) of all meromorphic functions `\(a\)' on \(\mathbb{C}\) such that \(T(.,a)\in s(f_j)\) \((j= 1,2)\). Most of the results have the common assumption that \(f_1\) and \(f_2\) ``share'' four distinct functions `\(a\)' of \(S(f_1)\cap S(f_2)\) in the sense that \(\{z\in \mathbb{C}: f_j(z)= a(z)\}\) is the same for \(j= 1,2\). An illustration of the extra conditions (taken from cor. 1) is indicated in the abstract and it is the following: \[ \overline N(.,f_j)\leq u_j T(.,f_j)+ p_j\quad\text{for some }p_j\in s(f_j),\quad u_j\in [0,1/19)\quad\text{for }j= 1,2. \] Assumptions, in which five functions of \(S(f_1)\cap S(f_2)\) are shared and in which four constants of the extended complex plane are shared, (by \(f_1\) and \(f_2\)), are also discussed.
scientific article; zbMATH DE number 1286738