Dynamics of inequalities in geometric function theory (Q1347689)

A domain \(\Omega \subset \mathbb C\) is called starlike (with respect to \(0\)) if for any given \(\omega \in \Omega\) there holds \(t\omega \in \Omega\) for all \(t \in (0,1]\). Note that then \(0 \in \Omega\) or \(0 \in \partial\Omega\). Let \(\Delta \subset \mathbb C\) denote the open unit disc. Then a univalent function \(f \colon \Delta \to \mathbb C\) is called starlike if \(f(\Delta)\) is a starlike domain. In addition, if \(0 \in f(\Delta)\), then \(f\) is called starlike with respect to the interior point \(f(a)=0\) for some \(a \in \Delta\). If \(0 \in \partial f(\Delta)\), then \(f\) is called starlike with respect to the boundary point \(0\). A well-known result due to Nevanlinna and Alexander states that a univalent function \(f\) in \(\Delta\) with \(f(0)=0\) is starlike if and only if the inequality \[ \text{Re}{zf'(z) \over f(z)}>0 , \quad z \in \Delta \tag{1} \] holds. A conjecture of \textit{M. S. Robertson} [J. Math. Anal. Appl. 81, 327-345 (1981; Zbl 0472.30015)] states that the inequality \[ \text{Re}{\left(2 {zf'(z) \over f(z)}+{1+z \over 1-z}\right)}>0 , \quad z \in \Delta \tag{2} \] characterizes those univalent functions \(f\) in \(\Delta\) with \(f(0)=1\) such that \(f\) is starlike with respect to the boundary point \(f(1) := \lim_{r \to 1-}{f(r)} = 0\) and \(f(\Delta)\) lies in the right half-plane. This conjecture was proved by \textit{A. Lyzzaik} [Proc. Am. Math. Soc. 91, 108-110 (1984; Zbl 0509.30009)]. The main result of this article is a generalization of the inequalities (1) and (2) which reads as follows. Theorem. Let \(f\) be a univalent function in \(\Delta\). Then \(f\) is starlike if and only if there is some \(\tau \in \overline{\Delta}\) such that the inequality \[ \text{Re}{f'(z)(z-\tau)(1-z\bar{\tau}) \over f(z)} \geq 0 , \quad z \in \Delta \tag{3} \] holds. Thus, if \(\tau \in \Delta\), then \(f\) is starlike with respect to \(f(\tau)=0\), and if \(\tau \in \partial\Delta\), then \(f\) is starlike with respect to the boundary point \(f(\tau) = \lim_{r \to 1-}{f(r\tau)} = 0\). For \(\tau=0\) this is inequality (1), and the author shows that for \(\tau=1\) the inequality (3) is equivalent to a generalized version of (2). The method of proof uses that a domain which is starlike with respect to a boundary point can be approximated by domains which are starlike with respect to interior points. This approximation process can be viewed dynamically as an evolution of the null points of the underlying functions from the interior towards a boundary point. Then the framework of complex dynamical systems and hyperbolic monotonicity is used.