The Schwarzian derivative and the Poincaré metric

Proposition 1. If $\phi$ is holomorphic and univalent in $\Delta$ , then $$|S_{\scriptscriptstyle{\phi}}(z)| \leq 6(1-|z|^{\scriptscriptstyle{2}})^{\scriptscriptstyle{-2}}$$ , $z \in \varDelta$ , and the constant 6 is sharp.

Proposition 2. Let $\Omega$ be a simply connected domain and let $\lambda_{\varrho}(z)$ be its Poincaré metric. If $\phi$ is holomorphic and univalent in $\Omega$ , then $$|S_{\scriptscriptstyle{\phi}}(z)| \leqq 12 \lambda_{\scriptscriptstyle{\varOmega}}^{\scriptscriptstyle{2}}(z)$$ , $z \in \varOmega$ , and the constant 12 is sharp. Proposition 1 is due to Kraus [5] and Proposition 2 is due to Lehto [6]. In this direction Nehari [7] has shown that if $\phi$ is holomorphic with $|S_{\phi}(z)| \leq 2(1-|z|^2)^{-2}$ in $\Delta$ , then $\phi$ is univalent in $\Delta$ with the constant 2 being the best possible. We shall show that the above two propositions can be immediately read off from one single inequality (Corollary 3) which is valid for any plane region. Our result can be easily extended to open Riemann surfaces too but we shall not pursue this point. Our arguments rely heavily on well known classical results of Bergman and Schiffer [2]. In order to be self contained, however, we will attempt to provide proofs to most crucial statements. The final result obtained in this paper (Theorem 2) involves a string of sharp inequalities amongst the Schwarzian derivative, the span or condenser capacity, the analytic capacity, the capacity and the Poincaré metric. In this string of inequalities, the inequality between the span (condenser) capacity and the analytic capacity is a well known result of Ahlfors and Beurling [1]. Here we provide a different proof of this result which is based on our Theorem 1. The representation formula of Theorem 1 was first mentioned in Schiffer [9] in case $\zeta = \infty \in \Omega$ and $\Omega$ has the largest complementary area amongst all regions which are conformally equivalent to $\Omega$ . 2. Capacities and the Poincaré metric. Let $\Omega$ be an open region in the extended plane and let $\zeta \in \Omega$ . Usually, $\zeta \neq \infty$ but the transition to $\zeta = \infty$ is trivial. $H(\Omega)$ stands for the class of holomorphic functions in $\Omega$ and $H_m(\Omega)$ denotes the class of multivalued holomorphic functions f in $\Omega$ such that |f(z)|, $z \in \Omega$ , is single valued. We write $$||f||_{\scriptscriptstyle{\infty}} = \sup_{z \in \varOmega} |f(z)|$$ , $D[f] = \int_{\varrho} |f'(z)|^2 d\sigma(z)$ , where $d\sigma(z) = dxdy$ is the Lebesgue area measure. Consider the following families: $$\begin{aligned} \mathscr{B}_{\zeta}(\varOmega) &= \{f \in H(\varOmega) \colon ||f||_{\infty} \leqq 1, f(\zeta) = 0\} \;, \\ \mathscr{C}_{\zeta}(\varOmega) &= \{f \in H_m(\varOmega) \colon ||f||_{\infty} \leqq 1, f(\zeta) = 0\} \;, \\ \mathscr{D}_{r}(\varOmega) &= \{f \in H(\varOmega) \colon D[f] \leqq \pi, f(\zeta) = 0\} \;. \end{aligned}$$ We now introduce (cf. [1]) the analytic capacity $$C_{\scriptscriptstyle B}(\zeta) = C_{\scriptscriptstyle B}(\zeta\colon\varOmega) = \max\left\{|f'(\zeta)|\colon f\in\mathscr{D}_{\scriptscriptstyle \zeta}(\varOmega)\right\}$$ , the capacity $$C_{\beta}(\zeta) = C_{\beta}(\zeta; \Omega) = \max \{ |f'(\zeta)| : f \in \mathscr{C}_{\zeta}(\Omega) \}$$ and the span or condenser capacity $$C_{\mathbf{D}}(\zeta) = C_{\mathbf{D}}(\zeta; \ \Omega) = \max \left\{ |f'(\zeta)| : f \in \mathscr{D}_{\zeta}(\Omega) \right\}$$ . We note that $C_{\beta}(\zeta)$ is well defined and that by $f(\zeta) = 0$ in $\mathscr{C}_{\zeta}(\Omega)$ we mean that at least one branch of f(z) vanishes at $\zeta$ . Assume now that $\Omega \notin \mathbf{0}_{G}$ (i.e., $\Omega$ has a nontrivial Green's function) and thus $\Omega$ has the unit disk $\Delta$ as its universal covering space. The Poincaré metric $\lambda_{\Omega}(z)$ is defined as follows: For the unit disk $\Delta$ , $\lambda_{\Delta}(\omega) = (1 - |\omega|^{2})^{-1}$ while for $\Omega$ $$\lambda_{\mathcal{Q}}(z) = \lambda_{\mathcal{A}}(\omega) |\pi'(\omega)|^{-1}, \qquad z = \pi(\omega) \in \Omega,$$ where $\pi: \Delta \to \Omega$ is a universal cover map. We denote by $\delta_{\Omega}(z)$ the distance from z to the boundary of $\Omega$ . $\lambda_{\Omega}(z)$ is monotonic decreasing with $\Omega$ and thus $$\lambda_{\varrho}(z)\delta_{\varrho}(z) \leq 1$$ , $z \in \Omega$ . Moreover, if $\Omega$ is simply connected then, in view of Koebe's 1/4 theorem, $$\lambda_{\varOmega}(z)\delta_{\varOmega}(z) \geq 1/4$$ , $z \in \varOmega$ . Clearly, $C_B$ , $C_B$ , and $\lambda_B$ are conformally invariant and therefore $$(2.1) C_{\mathcal{D}}(z;\Omega) = C_{\mathcal{B}}(z;\Omega) = C_{\mathcal{B}}(z;\Omega) = \lambda_{\mathcal{Q}}(z), z \in \Omega,$$ whenever $\Omega$ is simply connected. It is also evident that $C_B(z;\Omega) \leq C_\beta(z;\Omega)$ and it is a theorem of Ahlfors and Beurling [1] (see also Corollary 2) that $C_D(z;\Omega) \leq C_B(z;\Omega)$ . Moreover, $C_\beta(z;\Omega) \leq \lambda_\Omega$ . Indeed, let $\pi; \Delta \to \Omega$ be a universal cover map $z = \pi(\omega)$ . Then $$\begin{aligned} C_{\beta}(z;\varOmega) &= \max\left\{ |f'(z)| \colon f \in \mathscr{C}_z(\varOmega) \right\} \ &= |\pi'(\omega)|^{-1} \max\left\{ |g'(\omega)| \colon g = f \circ \pi \in \mathscr{C}_\omega(\varOmega), \ f \in \mathscr{C}_z(\varOmega) \right\} \ &\leq |\pi'(\omega)|^{-1} \max\left\{ |g'(\omega)| \colon g \in \mathscr{C}_\omega(\varOmega) \right\} \ &= |\pi'(\omega)|^{-1} C_{\beta}(\omega;\varOmega) \ &= \lambda_d(\omega) |\pi'(\omega)|^{-1} = \lambda_g(z) \ , \end{aligned}$$ where (2.1) has been used. Consequently, $$(2.2) C_{\mathcal{D}}(z;\Omega) \leq C_{\mathcal{B}}(z;\Omega) \leq C_{\mathcal{B}}(z;\Omega) \leq \lambda_{\mathcal{L}}(z) \leq \delta_{\mathcal{D}}^{-1}(z),$$ and we note that, if $\Omega \notin 0_{AD}$ , then $C_D(z;\Omega) > 0$ for all $z \in \Omega$ . The condition $\Omega \notin 0_{AD}$ means that there exists a nonconstant holomorphic function f in $\Omega$ with $D[f] < \infty$ . We conclude this section by recalling the following well-known lemma of Ahlfors and Beurling [1]:

Lemma 1. Let E be a measurable set with a finite Lebesgue measure $\sigma(E)$ in C. Then $$\sup_{\zeta \in \mathcal{C}} \left| \int_{E} \frac{d\sigma(z)}{z - \zeta} \right| \leq \sqrt{\pi \sigma(E)}$$ with equality holding, if and only if E is a almost everywhere a disk of radius $\sqrt{\sigma(E)/\pi}$ . 3. The slit mappings. Here we assume that the region $\Omega$ is bounded by n closed analytic curves $C_1, \dots, C_n$ and we denote by C the boundary $\partial \Omega = \bigcup_{k=1}^n C_k$ of $\Omega$ . We assume that $C_1$ is outer and we let $\zeta \in \Omega$ . Let $$p(z) = p(z; \zeta) = \frac{1}{z - \zeta} + a(z - \zeta) + \cdots$$ and $$q(z) = q(z; \zeta) = \frac{1}{z - \zeta} + b(z - \zeta) + \cdots$$ be the horizontal and vertical slit mappings, respectively, of $\Omega$ . We write $$\varPhi(z) = \varPhi(z;\zeta) = \frac{1}{2}(p(z) - q(z))$$ and $$\Psi(z) = \Psi(z; \zeta) = \frac{1}{2} (p(z) + q(z)) .$$ Then $\Phi(z)$ and $(z-\zeta)\Psi(z)$ are holomorphic on $\bar{\Omega}$ with $\Phi(\zeta)=0$ . Further, $\Psi(z)$ is univalent on $\Omega$ with pole at $\zeta$ . It maps $\Omega$ onto $\Omega$ with $E=\hat{C}-\bar{\Omega}$ being bounded. Clearly, $d\Phi=d\bar{\Psi}$ on C and therefore $\Phi=\bar{\Psi}-\bar{\lambda}_k$ on $C_k$ , where $\lambda_k$ is a constant depending on the component $C_k$ , $1\leq k\leq n$ . Also $\Phi(C_k)$ and $\Psi(C_k)$ are closed analytic and convex curves. One easily shows that (3.1) $$D[\phi] = \pi C_{\nu}^{2}(\zeta; \Omega) = \sigma(E) = \frac{\pi}{2}(a - b) .$$ Let $$\psi(z) = \psi(z;\zeta) = \zeta + \frac{1}{\Psi(z;\zeta)}$$ . $\psi$ maps $\Omega$ conformally onto $\Omega' = \psi(\Omega)$ with $\phi(\zeta) = \zeta$ . We write $\Gamma = \bigcup_{k=1}^n \Gamma_k = \partial \Omega'$ with $\Gamma_k = \psi(C_k)$ , $1 \le k \le n$ . We now establish an integral formula representing $\Phi$ in terms of $\Psi$ (compare also Schiffer [9]).

Theorem 1. The integral formula $$\varPhi(z;\zeta) = \frac{1}{\pi} \int_E \frac{d\sigma(t)}{\varPsi(z;\zeta) - t} \, ; \qquad E = \hat{\bm{C}} - \overline{\varPsi}(\varOmega) \,$$ , holds.

Corollary 1. Let $\Omega \notin 0_{AD}$ . Then $||\Phi||_{\infty} \leq C_D(\zeta)$ with equality holding if and only if $\Omega$ is conformally equivalent to the unit disk $\Delta$ less (possibly) a closed null $C_D$ -set. Proof. According to Lemma 1 and (3.1) we have $$||\Phi||_{\scriptscriptstyle\infty} \leq \frac{1}{\pi} \sqrt{\pi \sigma(E)} = C_{\scriptscriptstyle D}(\zeta) \; .$$ The statement about equality follows from Lemma 1 too. The theorem of Ahlfors and Beurling [1] is also a consequence of the theorem as the following corollary shows.

Corollary 2. $C_p(\zeta) \leq C_p(\zeta)$ .

Theorem 2. Let $\Omega \notin \mathbf{0}_{AD}$ . If $\phi$ is holomorphic and univalent in $\Omega$ we have the following sharp string of inequalities $$\begin{align} |S_{\beta}(z,\zeta)| & \leq 6C_{\scriptscriptstyle D}(z)C_{\scriptscriptstyle D}(\zeta)\biggl[ 1+\biggl(1-\frac{\alpha^2(\zeta)}{C_{\scriptscriptstyle D}^2(\zeta)}\biggr)^{\!1/2}\biggr] \ & \leq 6C_{\scriptscriptstyle B}(z)C_{\scriptscriptstyle B}(\zeta)\biggl[ 1+\biggl(1-\frac{\alpha^2(\zeta)}{C_{\scriptscriptstyle B}^2(\zeta)}\biggr)^{\!1/2}\biggr] \ & \leq 6C_{\scriptscriptstyle \beta}(z)C_{\scriptscriptstyle \beta}(\zeta)\biggl[ 1+\biggl(1-\frac{\alpha^2(\zeta)}{C_{\scriptscriptstyle B}^2(\zeta)}\biggr)^{\!1/2}\biggr] \end{split}$$ $$\leq 6\lambda_{\scriptscriptstyle \varOmega}(z)\lambda_{\scriptscriptstyle \varOmega}(\zeta)\Big[1+\Big(1-\frac{\alpha^2(\zeta)}{\lambda_{\scriptscriptstyle \varOmega}^2(\zeta)}\Big)^{\!\scriptscriptstyle 1/2}\Big]$$ ,

Corollary 3. Let $\Omega \notin 0_{\sigma}$ and let $\phi$ be holomorphic and univalent in $\Omega$ . Then $$|S_{\phi}(z,\zeta)| \leq 6\lambda_{\varOmega}(z)\lambda_{\varOmega}(\zeta)\bigg[1+\big(1-\frac{\alpha^2(\zeta)}{\lambda_{\varOmega}^2(\zeta)}\big)^{\!\!\!\!1/2}\bigg]\,; \qquad z,\,\zeta\in\varOmega$$ , and in particular $$|S_{\phi}(z)| \leqq 6\lambda_{\varOmega}^{2}(z)\bigg[1+\bigg(1-\frac{\alpha^{2}(z)}{\lambda_{\varOmega}^{2}(z)}\bigg)^{^{1/2}}\bigg]\;; \qquad z\in\varOmega\;.$$ The inequalities are sharp. The inequality $$|S_{\phi}(z)| \leq 6 \lambda_{\mathcal{Q}}^{\scriptscriptstyle 2}(z)$$ is sharp only when $\Omega$ is a disk less (possibly) a closed subset of innear capacity zero. Otherwise, we have the sharp inequality $$|S_o(z)| \leq 12\lambda_o^2(z)$$ .