The Sharp Distortion Estimate Concerning Julia’s Lemma

For \(\alpha > 0\), let \(J_\alpha \) be the class of all analytic functions f in the unit disk \({\mathbb {D}}: = \{ z \in {\mathbb {C}}: |z| < 1 \}\) satisfying \(f({\mathbb {D}}) \subset {\mathbb {D}}\) with the the angular derivative

$$\begin{aligned} \angle \lim _{z \rightarrow 1} \frac{f(z)-1}{z-1} = \alpha . \end{aligned}$$

For \(a,z\in \mathbb {D}\), let

$$\begin{aligned} k(z) = \frac{|1-z|^2}{1-|z|^2}\quad \text {and}\quad \sigma _a(z) = \frac{1-\overline{a}}{1-a} \frac{z-a}{1-\overline{a}z}. \end{aligned}$$

Let \(z_0 \in {\mathbb {D}}\) be fixed. For \(f \in J_\alpha \), we obtain the sharp estimate

$$\begin{aligned} |f'(z_0)| \le \frac{4 \alpha k(z_0)^2}{(\alpha k(z_0)+1)^2 |1-z_0|^2} \qquad \text {when }\alpha k(z_0) \le 1, \end{aligned}$$

with equality if and only if \(f = \sigma _{w_0}^{-1} \circ \sigma _{z_0}\). Here \(w_0 = (1-\alpha k(z_0))/(\alpha k(z_0) +1)\). In case of \(\alpha k(z_0) > 1\) we derive the estimate \(|f'(z_0)| \le k(z_0)/|1-z_0|^2\). It is also sharp, however in contrast to the former case, there are no extremal functions in \(J_\alpha \). The lack of extremal functions is caused by the fact that \(J_\alpha \) is not closed in the topology of local uniform convergence in \({\mathbb {D}}\). Thus we consider the closure \(\bar{J}_\alpha \) of \(J_\alpha \) and study \(\bar{V}_1(z_0, \alpha ):= \{ f'(z_0): f \in \bar{J}_\alpha \}\) which is the variability region of \(f'(z_0)\) when f ranges over \(\bar{J}_\alpha \). We shall show that \(\partial \bar{V}_1(z_0, \alpha )\) is a simple closed curve and \(\bar{V}_1(z_0, \alpha )\) is a convex and closed Jordan domain enclosed by \(\partial \bar{V}_1(z_0, \alpha )\). Moreover, we shall give a parametric representation of \(\partial \bar{V}_1(z_0, \alpha )\) and determine all extremal functions.