Rogosinski's lemma for univalent functions, hyperbolic Archimedean spirals and the Loewner equation (Q2922874)

The authors explicitly describe the region of values NEWLINE\[NEWLINE\mathcal V(z_0):=\{f(z_0):f\in\mathcal H_0(\mathbb D)\;\text{is univalent}\},\;\;z_0\in\mathbb D,NEWLINE\]NEWLINE for the class \(\mathcal H_0(\mathbb D)\) of all holomorphic functions \(f\) which map the unit disk \(\mathbb D\) into itself and are normalized by \(f(0)=0\) and \(f'(0)\geq0\). It is proved that, for \(z_0\in\mathbb D\setminus\{0\}\), NEWLINENEWLINE\[NEWLINE\mathcal V(z_0)\cup\{0\}=\Big\{z=|z|e^{i\varphi}\in\mathbb D:d_{\mathbb D}(0,z)-d_{\mathbb D}(0,z_0)\leq-|\varphi-\arg z_0|,\;\varphi\in\mathbb R\Big\}NEWLINE\]NEWLINE with the hyperbolic distance \(d_{\mathbb D}(z,w)\) between \(z,w\in\mathbb D\) given by NEWLINE\[NEWLINEd_{\mathbb D}=\log\frac{1+|(z-w)/(1-\overline wz)|}{1-|(z-w)/(1-\overline wz)|}.NEWLINE\]NEWLINENEWLINENEWLINEThe boundary curves \(\gamma^{\pm}(z_0)\), \(\partial\mathcal V(z_0)=\gamma^+(z_0)\cup\gamma^-(z_0)\cup\{0\}\), are the arcs of the hyperbolic Archimedean spirals. In particular, the origin is an isolated boundary point of \(\mathcal V(z_0)\) if and only if \(|z_0|>\tanh(\pi/2)\), and \(\mathcal V(z_0)\) is convex if and only if \(|z_0|\leq\tanh(\pi/4)\). Both \(\gamma^{\pm}(z_0)\) are parameterized by the two optimal trajectories of the classical Loewner differential equation NEWLINE\[NEWLINE\dot w(t)=-w(t)\frac{\kappa(t)+w(t)}{\kappa(t)-w(t)},\;\;t\geq0,\;\;w(0)=z_0\in\mathbb D,NEWLINE\]NEWLINE with continuous driving functions \(\kappa:[0,\infty)\to\partial\mathbb D\). So, \(\mathcal V(z_0)\) is a reachable set for the Loewner equation.NEWLINENEWLINEThe authors present an analogous result for the upper half-plane \(\mathbb H\) describing the region of values NEWLINE\[NEWLINE\{g(z_0):g\in\mathcal H_{\infty}(\mathbb H)\;\text{is univalent}\}=\{z\in\mathbb C:\Im z>\Im z_0\}\cup\{z_0\}NEWLINE\]NEWLINE for the class \(\mathcal H_{\infty}(\mathbb H)\) of all holomorphic functions \(g:\mathbb H\to\mathbb H\) such that the normalization NEWLINE\[NEWLINEg(z)-z\to0,\;S_{\beta}:=\{z:|\arg z-\pi/2|<\beta\}\ni z\to\infty,\;0<\beta<\pi/2,NEWLINE\]NEWLINE is satisfied. This value region coincides with a reachable set \(\mathcal R(z_0)\) for the chordal Loewner differential equation NEWLINE\[NEWLINE\dot w=\frac{-2}{w(t)-U(t)},\;\;t\geq0,\;\;w(0)=z_0\in\mathbb H,NEWLINE\]NEWLINE with continuous driving functions \(U:[0,\infty)\to\mathbb R\).