On the meromorphic functions in the punctured plane without multiple values
Let \(f\) be a meromorphic function of finite order \(\rho(f) \geq 1\) in the punctured plane \(\mathbb{C}\setminus\{0\}\), and let \(\delta(a,f)\) denote the Nevanlinna defect of \(f\) for \(a\in\mathbb{C}\cup\{\infty\}\). In the paper under review the author shows that if \(f\) does not have any multiple values, then \(f\) is of regular growth at zero and infinity, \(2\rho(f)\in\mathbb{N}\), \[ \sum_{a\in\mathbb{C}\cup\{\infty\}} \delta(a,f) = 2, \] and every deficient value is an asymptotic value, either at zero or at infinity. In addition, for every deficient value \(a\) of \(f\), \[ \delta(a,f) = c\cdot \frac{k_1}{\rho(f)} + (1-c)\cdot \frac{k_2}{\rho(f)}, \] where \(c\) is a constant such that \(0\leq c\leq 1\) and \(k_1,k_2\in\mathbb{N}\cup\{0\}\). The paper is concluded with a discussion on differential equations in \(\mathbb{C}\setminus\{0\}\). It is shown, for instance, that if \(p\) is holomorphic in \(\mathbb{C}\setminus\{0\}\) and if \(f\) is a locally injective meromorphic solution of the Schwarzian differential equation \[ \frac{f'''}{f''} - \frac32 \left(\frac{f''}{f'}\right)^2 = p(z) \] in \(\mathbb{C}\setminus\{0\}\), then there exist two linearly independent solutions \(u_1\) and \(u_2\) of the differential equation \[ u'' + \frac12 p(z)u =0 \] such that \[ f(z)=\frac{u_1(z)}{u_2(z)} \] for all \(z\in\mathbb{C}\setminus\{0\}\).
scientific article; zbMATH DE number 6814833