Univalent analytic functions and the Poincaré metric
The author investigates two types of problems on a hyperbolic region \(\Omega\) in the complex plane \({\mathbb{C}}\). The density \(\rho_{\Omega}\) of the Poincaré, or hyperbolic, metric plays a role in both types of problems. First, the author relates the geometry of \(\Omega\) to various differential inequalities involving the quantity \(1/\rho_{\Omega}\). Second, the author studies the family S(\(\Omega\)) of holomorphic, univalent functions defined on \(\Omega\). In particular, he extends a result of \textit{P. L. Duren}, \textit{H. S. Shapiro} and \textit{A. L. Shields} [Duke Math. J. 33, 247-254 (1966; Zbl 0174.375)] to hyperbolic regions of finite type. This result gives an upper bound for a norm of the Schwarzian derivative of \(f\in S(\Omega)\) in terms of a second norm related to \(f''/f'\). Also, the author shows that if a function f in S(\(\Omega\)) is ``isolated'' in a sense related to the second norm, then \({\mathbb{C}}\setminus \Omega\) has zero area. Reviewer's comments. Note that the regions of finite type in this paper are sometimes called uniformly perfect by other authors. Also, \textit{R. Harmelin} [Isr. J. Math. 70, 111-128 (1990; Zbl 0708.30045)] independently obtained certain results relating the geometry of a hyperbolic region to differential inequalities involving \(\rho_{\Omega}\). There is some material common to the two papers; anyone interested in the subject matter of one of these papers should certainly read the other.
scientific article; zbMATH DE number 4198423
scientific article