An exterior conformal coordinate

Written by Claude Opus 5.5 (Anthropic), October 2026. Self-checked by the writing AI. Public domain (CC0).

The Faber polynomials of Faber polynomials and a positive kernel are defined through a conformal map \(h\) of \(\{|\lambda|>1\}\) onto the exterior of a smooth convex domain \(\Omega\). The estimates there need \(h\) on a slightly larger set \(\{|\lambda|>r_0\}\) with \(r_0<1\), and they need the fact that \(\lambda h'(\lambda)\) is an outward normal of \(\partial\Omega\) when \(|\lambda|=1\). This lesson constructs \(h\) (Theorem 3.7). The result is classical; the form used in the course is the one of OpenAI [OpenAI-DC, Appendix A]. The proof below extends the inverse of a Riemann map across the boundary first, with the reflection principle for harmonic functions, and obtains the continuity of the Riemann map on the closed disc as a consequence.

Smooth convex domains are defined in Smooth convex neighbourhoods: bounded convex domains that are star-shaped with analytic polar boundary about one of their points. Results of complex analysis are cited by number from Lebl's guide in the core course Complex Analysis. \(\mathbf D\) is the open unit disc and \(\mathbf T\) the unit circle.

Throughout, \(\Omega\) is a smooth convex domain, star-shaped with analytic polar boundary about \(z_*\in\Omega\), with polar function \(\rho\) holomorphic on the strip \(\{|\operatorname{Im}\theta|<\eta\}\).

1. The inverted exterior

Put \[ U=\{0\}\cup\Bigl\{\frac1{z-z_*}:z\in\mathbf C\setminus\overline\Omega\Bigr\}. \]

Lemma 3.1. \(U\) is star-shaped with analytic polar boundary about \(0\), with polar function \(\sigma(\varphi)=1/\rho(-\varphi)\). For \(w\ne0\) and \(z=z_*+1/w\), \[ \begin{aligned} w\in U&\iff z\notin\overline\Omega,\\ w\in\partial U&\iff z\in\partial\Omega,\\ w\notin\overline U&\iff z\in\Omega. \end{aligned} \] In particular, \(U\) is bounded and simply connected.

Proof. Write \(w=se^{i\varphi}\) with \(s>0\). Then \(z-z_*=s^{-1}e^{-i\varphi}\): the point \(z\) has distance \(1/s\) from \(z_*\) and polar angle \(-\varphi\). By the polar description of \(\overline\Omega\) and \(\partial\Omega\), \(z\notin\overline\Omega\) if and only if \(1/s>\rho(-\varphi)\), that is \(s<\sigma(\varphi)\); likewise \(z\in\partial\Omega\) if and only if \(s=\sigma(\varphi)\), and \(z\in\Omega\setminus\{z_*\}\) if and only if \(s>\sigma(\varphi)\). The first statement gives \(U=\{se^{i\varphi}:0\le s<\sigma(\varphi)\}\), and the polar description of \(\overline U\) and \(\partial U\) gives the other two equivalences.

The function \(\rho\) is positive on \(\mathbf R\) and uniformly continuous on \([0,2\pi]\times[-\eta/2,\eta/2]\). By periodicity there is \(0<\eta_1\le\eta/2\) with \(|\rho(\theta)-\rho(\operatorname{Re}\theta)|<\frac12\min_{\mathbf R}\rho\) for \(|\operatorname{Im}\theta|<\eta_1\). So \(\rho\) has no zeros on that strip, and \(\sigma\) is holomorphic there. \(U\) is bounded because \(\sigma\le1/\min_{\mathbf R}\rho\). It is simply connected in the sense of Definition 4.3.1: for \(a\notin U\), the function \(1/(w-a)\) is holomorphic on the star-shaped set \(U\), so by Cauchy's theorem for star-like domains (Theorem 3.2.13) every cycle in \(U\) has winding number \(0\) about \(a\). \(\square\)

2. Boundary charts

In Sections 2 to 4, \(U\) is any bounded domain that is star-shaped with analytic polar boundary about \(0\), with polar function \(\sigma\) holomorphic on a strip \(\{|\operatorname{Im}\zeta|<\eta_1\}\); by Lemma 3.1, the inverted exterior is such a domain. The argument at the end of the proof of Lemma 3.1 shows that \(U\) is simply connected. Put \(\Gamma(\zeta)=\sigma(\zeta)e^{i\zeta}\) on the strip, so that \(\partial U=\Gamma(\mathbf R)\). As in the proof of Lemma 3.1, there is \(0<\eta_2\le\eta_1\) such that \(\operatorname{Re}\sigma>0\) on \(|\operatorname{Im}\zeta|<\eta_2\). On that strip the principal logarithm \(L=\operatorname{Log}\sigma\) is holomorphic, real on \(\mathbf R\) and \(2\pi\)-periodic, and \(\sigma=e^L\).

Lemma 3.2 (boundary charts). For every real \(\varphi_0\) there is \(0<\varepsilon<\eta_2\) with the following properties, where \(B=\{|\zeta-\varphi_0|<\varepsilon\}\): \(\Gamma\) is injective on \(B\), \(\Gamma(B)\) is open, the inverse of \(\Gamma:B\to\Gamma(B)\) is holomorphic, and for \(\zeta\in B\) \[ \begin{aligned} \Gamma(\zeta)\in U&\iff\operatorname{Im}\zeta>0,\\ \Gamma(\zeta)\in\partial U&\iff\operatorname{Im}\zeta=0,\\ \Gamma(\zeta)\notin\overline U&\iff\operatorname{Im}\zeta<0. \end{aligned} \]

Proof. Since \(\Gamma'(\varphi_0)\ne0\), the inverse function theorem (Theorem 2.2.8) gives \(\varepsilon_1>0\) such that \(\Gamma\) is injective on \(|\zeta-\varphi_0|<\varepsilon_1\), with open image and holomorphic inverse; every smaller disc about \(\varphi_0\) keeps these properties, because the inverse is continuous.

Put \(T(\zeta)=L(\zeta)+i\zeta\), so that \(\Gamma=e^T\). For \(\zeta=y+it\) write \(T(\zeta)=X+iY\) with \(X,Y\) real. Then \(|\Gamma(\zeta)|=e^X\) and \(Y\) is an argument of \(\Gamma(\zeta)\). Since \(\sigma(Y)=e^{L(Y)}\), the polar description of \(U\) gives \[ \begin{aligned} \Gamma(\zeta)\in U&\iff X<L(Y),\\ \Gamma(\zeta)\in\partial U&\iff X=L(Y). \end{aligned} \] Let \(N(y,t)=L(Y)-X\), a continuously differentiable function of \((y,t)\). At \(t=0\), \(T(y)=L(y)+iy\), so \(X=L(y)\), \(Y=y\) and \(N(y,0)=0\). Moreover \(\partial_tX+i\,\partial_tY=iT'(\zeta)\), and \(T'(y)=L'(y)+i\) with \(L'(y)\) real; so at \(t=0\), \(\partial_tX=-1\) and \(\partial_tY=L'(y)\), and \[ \partial_tN(y,0)=L'(y)^2+1>0. \] By continuity there is \(0<\varepsilon<\min(\varepsilon_1,\eta_2)\) with \(\partial_tN>0\) on the square \(|y-\varphi_0|<\varepsilon\), \(|t|<\varepsilon\). For fixed \(y\), \(t\mapsto N(y,t)\) is then strictly increasing and vanishes at \(t=0\), so \(N>0\) for \(t>0\) and \(N<0\) for \(t<0\). The disc \(B\) of radius \(\varepsilon\) lies in this square. \(\square\)

3. The inverse Riemann map across the boundary

By the Riemann mapping theorem (Theorem 6.3.1) there is a unique biholomorphic map \(\phi:U\to\mathbf D\) with \(\phi(0)=0\) and \(\phi'(0)>0\). Let \(g=\phi^{-1}:\mathbf D\to U\).

Lemma 3.3. Let \(\varphi_0\), \(\varepsilon\) and \(B\) be as in Lemma 3.2, and \(V=\Gamma(B)\). There is a holomorphic \(\tilde\phi:V\to\mathbf C\) with \(\tilde\phi=\phi\) on \(V\cap U\), \(|\tilde\phi|=1\) on \(V\cap\partial U\), \(|\tilde\phi|>1\) on \(V\setminus\overline U\), and \(\tilde\phi'\ne0\) on \(V\cap\partial U\).

Proof. Write \(B^\pm=\{\zeta\in B:\pm\operatorname{Im}\zeta>0\}\) and \(I=B\cap\mathbf R\). By Lemma 3.2, \(\Gamma(B^+)\subseteq U\), so \(F=\phi\circ\Gamma\) is holomorphic on \(B^+\) with values in \(\mathbf D\). It has no zeros, because \(\phi\) vanishes only at \(0\) and \(\Gamma=e^T\) does not vanish. Since \(B^+\) is convex, Corollary 4.3.5 gives a holomorphic \(\Lambda\) on \(B^+\) with \(e^\Lambda=F\); the function \(u=\operatorname{Re}\Lambda=\log|F|\) is harmonic and negative on \(B^+\).

(a) Boundary values. Let \(0<r<1\). The set \(K_r=g(\{|w|\le r\})\) is a compact subset of \(U\), so it has a positive distance \(d_r\) from the closed set \(\mathbf C\setminus U\), which contains \(\Gamma(\mathbf R)=\partial U\). Let \(y_0\in I\) and choose \(t_0>0\) such that the square \(|y-y_0|\le t_0\), \(|t|\le t_0\) lies in \(B\) and \(|\Gamma(y+it)-\Gamma(y)|<d_r\) on it (uniform continuity). For \(0<t\le t_0\), \(\Gamma(y+it)\notin K_r\), so \(|F(y+it)|>r\). Hence \(u\), extended by \(0\) on \(I\), is continuous on \(B^+\cup I\).

(b) Reflection. By the reflection principle for harmonic functions (Theorem 7.3.3) there is a harmonic \(\tilde u\) on \(B\) with \(\tilde u=u\) on \(B^+\cup I\). It satisfies \(\tilde u(\bar\zeta)=-\tilde u(\zeta)\). Indeed, \(d(\zeta)=\tilde u(\zeta)+\tilde u(\bar\zeta)\) is harmonic on \(B\), vanishes on \(I\), and \(\partial_td=0\) on \(I\). By Proposition 7.1.2, \(d=\operatorname{Re}E\) for a holomorphic \(E\) on the disc \(B\), and \(E'=\partial_yd-i\,\partial_td\) vanishes on \(I\); by the identity theorem (Theorem 2.4.7) \(E'=0\) on \(B\), so \(d\) is constant, equal to its value \(0\) on \(I\).

(c) The extension. By Proposition 7.1.2 there is a holomorphic \(\tilde G\) on \(B\) with \(\operatorname{Re}\tilde G=\tilde u\). On \(B^+\), \(\operatorname{Re}(\tilde G-\Lambda)=0\); by the open mapping theorem (Theorem 5.5.1) \(\tilde G-\Lambda\) is constant there, say \(i\alpha\). The function \(\tilde F=e^{\tilde G-i\alpha}\) is holomorphic on \(B\), equals \(F\) on \(B^+\), and \(|\tilde F|=e^{\tilde u}\). So \(|\tilde F|=1\) on \(I\), and \(|\tilde F(\zeta)|=e^{-\tilde u(\bar\zeta)}>1\) on \(B^-\) by (b).

(d) The derivative on \(I\). Since \(\tilde F'=\tilde G'\tilde F\), it suffices to show \(\tilde G'(y_0)\ne0\) for \(y_0\in I\). The function \(\tilde G\) is not constant, because \(\tilde u\) vanishes on \(I\) and is negative on \(B^+\). Suppose \(\tilde G'(y_0)=0\). Then near \(y_0\) \[ \tilde G(\zeta)=\tilde G(y_0)+a(\zeta-y_0)^k\bigl(1+v(\zeta)\bigr) \] with \(k\ge2\), \(a\ne0\), and \(v\) holomorphic with \(v(y_0)=0\). Here \(\operatorname{Re}\tilde G(y_0)=\tilde u(y_0)=0\). As \(\beta\) runs through \((0,\pi)\), \(k\beta\) runs through an interval of length \(k\pi\ge2\pi\), so there is \(\beta\in(0,\pi)\) with \(\omega=\operatorname{Re}(ae^{ik\beta})>0\). For small \(s>0\), \(|a||v|<\omega\) at \(y_0+se^{i\beta}\), and \[ \begin{aligned} \tilde u(y_0+se^{i\beta})&=s^k\operatorname{Re}\bigl(ae^{ik\beta}(1+v)\bigr)\\ &\ge s^k(\omega-|a||v|)>0, \end{aligned} \] although \(y_0+se^{i\beta}\in B^+\), where \(\tilde u<0\). So \(\tilde G'(y_0)\ne0\).

(e) Put \(\tilde\phi=\tilde F\circ(\Gamma|_B)^{-1}\) on \(V\). By Lemma 3.2, \(V\cap U=\Gamma(B^+)\), \(V\cap\partial U=\Gamma(I)\) and \(V\setminus\overline U=\Gamma(B^-)\), which gives the three statements on \(\tilde\phi\) and its modulus. Differentiating \(\tilde\phi(\Gamma(\zeta))=\tilde F(\zeta)\) gives \(\tilde\phi'(\Gamma(\zeta))\,\Gamma'(\zeta)=\tilde F'(\zeta)\ne0\) for \(\zeta\in I\). \(\square\)

Proposition 3.4. \(\phi\) extends to a homeomorphism \(\bar\phi:\overline U\to\overline{\mathbf D}\) with \(\bar\phi(\partial U)=\mathbf T\). Every \(p\in\partial U\) has an open neighbourhood \(V_p\) with an injective holomorphic \(\phi_p:V_p\to\mathbf C\) such that \(\phi_p'\ne0\), \(W_p=\phi_p(V_p)\) is open, \(\phi_p=\bar\phi\) on \(V_p\cap\overline U\), and \(|\phi_p|\ge1\) on \(V_p\setminus U\).

Proof. Let \(p\in\partial U\), so \(p=\Gamma(\varphi_0)\) with \(\varphi_0\) real, and take \(\tilde\phi\) from Lemma 3.3. As \(\tilde\phi'(p)\ne0\), the inverse function theorem gives an open neighbourhood \(V_p\) of \(p\) on which \(\tilde\phi\) is injective with nonvanishing derivative and open image; let \(\phi_p\) be its restriction. By Lemma 3.3, \(|\phi_p|\ge1\) on \(V_p\setminus U\). Every neighbourhood of \(p\) meets \(U\), and \(\phi_p\) is continuous and equal to \(\phi\) on \(V_p\cap U\); so \(\phi_p(p)=\lim_{w\to p,\,w\in U}\phi(w)\). This limit does not depend on the chart. So \(\bar\phi=\phi\) on \(U\) and \(\bar\phi(p)=\lim_{w\to p,\,w\in U}\phi(w)\) on \(\partial U\) defines a function that agrees with \(\phi_p\) on \(V_p\cap\overline U\); it is continuous, and \(|\bar\phi|=1\) on \(\partial U\).

The image \(\bar\phi(\overline U)\) is compact and contains \(\mathbf D\), so it contains \(\overline{\mathbf D}\). As \(\bar\phi(U)=\mathbf D\) and \(|\bar\phi|=1\) on \(\partial U\), \(\bar\phi(\partial U)=\mathbf T\).

Let \(p_1,p_2\in\partial U\) with \(\bar\phi(p_1)=\bar\phi(p_2)=\zeta_0\), and let \(\psi_i:W_{p_i}\to V_{p_i}\) be the inverse of \(\phi_{p_i}\). The open set \(W_{p_1}\cap W_{p_2}\) contains \(\zeta_0\in\mathbf T\), so there are \(w_k\in W_{p_1}\cap W_{p_2}\cap\mathbf D\) with \(w_k\to\zeta_0\). Since \(|\phi_{p_i}(\psi_i(w_k))|=|w_k|<1\), the point \(\psi_i(w_k)\) lies in \(U\), and \(\phi(\psi_1(w_k))=w_k=\phi(\psi_2(w_k))\). As \(\phi\) is injective, \(\psi_1(w_k)=\psi_2(w_k)\), and letting \(k\to\infty\) gives \(p_1=\psi_1(\zeta_0)=\psi_2(\zeta_0)=p_2\).

So \(\bar\phi\) is a continuous bijection of the compact set \(\overline U\) onto \(\overline{\mathbf D}\), and its inverse is continuous. \(\square\)

4. The Riemann map on a larger disc

Theorem 3.5. There are \(R_0>1\) and an injective holomorphic function \(G\) on \(\{|w|<R_0\}\) with \(G=g\) on \(\mathbf D\), \(G'\ne0\) everywhere, \(G(\mathbf T)=\partial U\), \(G(w)\notin\overline U\) for \(1<|w|<R_0\), and \(G(w)=0\) only for \(w=0\).

Proof. Let \(\bar g=\bar\phi^{-1}:\overline{\mathbf D}\to\overline U\). For \(\zeta\in\mathbf T\) let \(p=\bar g(\zeta)\) and \(\psi_p:W_p\to V_p\) the inverse of \(\phi_p\). For \(w\in W_p\cap\mathbf D\), \(|\phi_p(\psi_p(w))|=|w|<1\), so \(\psi_p(w)\in U\) and \(\phi(\psi_p(w))=w\): thus \(\psi_p=g\) on \(W_p\cap\mathbf D\). Choose \(R_\zeta>0\) with \(\{|w-\zeta|<R_\zeta\}\subseteq W_p\), and write \(\psi_\zeta\) for \(\psi_p\) on this disc.

Finitely many discs \(Q_i=\{|w-\zeta_i|<R_i/3\}\), with \(R_i=R_{\zeta_i}\), cover \(\mathbf T\). If \(Q_i\cap Q_j\ne\varnothing\) and \(R_i\ge R_j\), then \(|\zeta_i-\zeta_j|<(R_i+R_j)/3\le2R_i/3\), so \(\zeta_j\) lies in the intersection of the discs of radii \(R_i\) and \(R_j\) about \(\zeta_i\) and \(\zeta_j\). That intersection is convex, hence connected, and it meets \(\mathbf D\) because it is a neighbourhood of \(\zeta_j\in\mathbf T\). There \(\psi_{\zeta_i}=g=\psi_{\zeta_j}\), so by the identity theorem \(\psi_{\zeta_i}=\psi_{\zeta_j}\) on the whole intersection, in particular on \(Q_i\cap Q_j\). Hence \(g\) and the \(\psi_{\zeta_i}\) define one holomorphic function \(G\) on the open set \(O=\mathbf D\cup\bigcup_iQ_i\supseteq\overline{\mathbf D}\). By continuity, \(G=\bar g\) on \(\overline{\mathbf D}\).

On \(\mathbf D\), \(\phi'(g(w))\,g'(w)=1\), and near \(\zeta\in\mathbf T\), \(G=\psi_\zeta\) is the inverse of a function with nonvanishing derivative; so \(G'\ne0\) on \(\overline{\mathbf D}\).

Choose \(\delta_0>0\) with \(\{|w|<1+\delta_0\}\subseteq O\). Suppose that for every \(k>1/\delta_0\) there are \(w_k\ne w_k'\) with \(|w_k|,|w_k'|<1+1/k\) and \(G(w_k)=G(w_k')\). Passing to subsequences, \(w_k\to w\) and \(w_k'\to w'\) with \(|w|,|w'|\le1\), and \(G(w)=G(w')\); as \(\bar g\) is injective, \(w=w'\). Since \(G'(w)\ne0\), the inverse function theorem gives a neighbourhood of \(w\) on which \(G\) is injective, and it contains \(w_k\) and \(w_k'\) for large \(k\), a contradiction. So \(G\) is injective on \(\{|w|<R'\}\) for some \(R'>1\). Since \(G'\) is continuous and nonzero on the compact set \(\overline{\mathbf D}\), there is \(R_0\in(1,R']\) with \(G'\ne0\) on \(\{|w|<R_0\}\).

Finally, \(G(\mathbf T)=\bar g(\mathbf T)=\partial U\) by Proposition 3.4; for \(1<|w|<R_0\), injectivity gives \(G(w)\notin G(\overline{\mathbf D})=\overline U\); and \(G(0)=g(0)=0\). \(\square\)

5. The exterior coordinate

Lemma 3.6 (supporting lines). For every \(p\in\partial\Omega\) there is a unit vector \(\nu\) with \(\operatorname{Re}(\bar\nu(z-p))\le0\) for all \(z\in\overline\Omega\).

Proof. \(\overline\Omega\) is compact and convex. The points \(p_k=z_*+(1+1/k)(p-z_*)\) lie outside \(\overline\Omega\) by the polar description. Let \(w_k\) be the point of \(\overline\Omega\) nearest to \(p_k\), and \(e_k=p_k-w_k\ne0\). For \(z\in\overline\Omega\) and \(0<s\le1\), \(w_k+s(z-w_k)\in\overline\Omega\), so \[ \begin{aligned} 0&\le|e_k-s(z-w_k)|^2-|e_k|^2\\ &=-2s\operatorname{Re}\bigl(\bar e_k(z-w_k)\bigr)+s^2|z-w_k|^2 . \end{aligned} \] Dividing by \(s\) and letting \(s\to0\), \(\operatorname{Re}(\bar\nu_k(z-w_k))\le0\) for the unit vector \(\nu_k=e_k/|e_k|\). Moreover \(|w_k-p|\le|w_k-p_k|+|p_k-p|\), which is at most \(2|p_k-p|\) because \(p\in\overline\Omega\), and \(2|p_k-p|\to0\). A subsequence of \((\nu_k)\) converges to a unit vector \(\nu\), which has the stated property. \(\square\)

Theorem 3.7 (exterior coordinate). Let \(\Omega\) be a smooth convex domain. There are \(0<r_0<1\), \(c>0\) and a holomorphic function \(\kappa\) on \(\{|u|<1/r_0\}\) such that \(h(\lambda)=c\lambda+\kappa(1/\lambda)\), defined for \(|\lambda|>r_0\), has the following properties.

  1. \(h\) is injective, and \(h'(\lambda)\ne0\) for \(|\lambda|>r_0\).
  2. \(h\) maps \(\{|\lambda|>1\}\) onto \(\mathbf C\setminus\overline\Omega\), \(\mathbf T\) onto \(\partial\Omega\), and \(\{r_0<|\lambda|<1\}\) into \(\Omega\).
  3. For \(\lambda\in\mathbf T\) and \(z\in\overline\Omega\), \(q(\lambda)=\lambda h'(\lambda)\) is nonzero, and \[ \operatorname{Re}\bigl(\overline{q(\lambda)}(h(\lambda)-z)\bigr)\ge0. \]

Proof. Take \(G\) and \(R_0\) from Theorem 3.5 and put \(r_0=1/R_0\), \(c=1/G'(0)\) and \[ h(\lambda)=z_*+\frac1{G(1/\lambda)}\qquad(|\lambda|>r_0). \] Here \(G(1/\lambda)\ne0\). Since \(G'(0)=g'(0)=1/\phi'(0)>0\), \(c>0\). Write \(G(u)=uG_1(u)\) with \(G_1\) holomorphic on \(\{|u|<R_0\}\); \(G_1\) has no zeros, as \(G\) vanishes only at \(0\), where \(G'(0)\ne0\). Then \[ \kappa(u)=z_*+\frac1u\Bigl(\frac1{G_1(u)}-\frac1{G_1(0)}\Bigr) \] has a removable singularity at \(u=0\) (Theorem 5.2.2), and \(h(\lambda)=c\lambda+\kappa(1/\lambda)\).

(1) \(h\) is a composition of injective maps, and \(h'(\lambda)=G'(1/\lambda)\big/\bigl(\lambda G(1/\lambda)\bigr)^2\ne0\).

(2) For \(u=1/\lambda\): if \(|\lambda|>1\) then \(u\in\mathbf D\setminus\{0\}\), and \(G\) maps \(\mathbf D\setminus\{0\}\) onto \(U\setminus\{0\}\); by Lemma 3.1, \(z_*+1/w\) runs over \(\mathbf C\setminus\overline\Omega\) as \(w\) runs over \(U\setminus\{0\}\). In the same way \(G(\mathbf T)=\partial U\) gives \(h(\mathbf T)=\partial\Omega\), and \(G(u)\notin\overline U\) for \(1<|u|<R_0\) gives \(h(\lambda)\in\Omega\) for \(r_0<|\lambda|<1\).

(3) Let \(\lambda_0=e^{i\theta_0}\), \(p=h(\lambda_0)\), and \(\nu\) as in Lemma 3.6. By (1), \(q(\lambda_0)\ne0\). The function \(\theta\mapsto\operatorname{Re}(\bar\nu(h(e^{i\theta})-p))\) is at most \(0\), because \(h(e^{i\theta})\in\partial\Omega\), and vanishes at \(\theta_0\); its derivative \(\operatorname{Re}(\bar\nu\,iq(\lambda_0))\) at \(\theta_0\) vanishes. So \(\bar\nu q(\lambda_0)=t\) is real, and \(q(\lambda_0)=t\nu\). The function \(s\mapsto\operatorname{Re}(\bar\nu(h(s\lambda_0)-p))\) is at most \(0\) on \((r_0,1]\), by (2), and vanishes at \(s=1\); so its derivative at \(s=1\), which is \(\operatorname{Re}(\bar\nu\lambda_0h'(\lambda_0))=t\), is at least \(0\). As \(q(\lambda_0)\ne0\), \(t>0\), and for \(z\in\overline\Omega\) the number \(\operatorname{Re}(\overline{q(\lambda_0)}(p-z))\) equals \(t\operatorname{Re}(\bar\nu(p-z))\ge0\). \(\square\)

Geometrically, the radial derivative \(q(\lambda)\) and the angular derivative \(iq(\lambda)\) of \(h\) at \(\lambda\in\mathbf T\) are an outward normal and a tangent of \(\partial\Omega\) at \(h(\lambda)\), and (3) says that \(\overline\Omega\) lies on the inner side of the tangent line.

Example (ellipses). For \(0\le a<1\), \(h(\lambda)=\lambda+a/\lambda\) satisfies \(h(\lambda)-h(\mu)=(\lambda-\mu)(1-a/(\lambda\mu))\), so it is injective on \(\{|\lambda|>\sqrt a\}\). It maps \(\mathbf T\) onto the ellipse \(h(e^{i\theta})=(1+a)\cos\theta+i(1-a)\sin\theta\) with semi-axes \(1+a\) and \(1-a\), and \(q(e^{i\theta})=(1-a)\cos\theta+i(1+a)\sin\theta\) is a positive multiple of the gradient of \(x^2/(1+a)^2+y^2/(1-a)^2\) at that point (Exercise 3.2). For \(a=0\) the domain is the unit disc and \(h(\lambda)=\lambda\).

6. Exercises

3.1 (easy). For the disc \(\Omega=\{|z-z_0|<R\}\) and \(z_*=z_0\), show that \(U=\{|w|<1/R\}\), \(g(w)=w/R\), and \(h(\lambda)=z_0+R\lambda\). Check property (3) of Theorem 3.7 directly.

3.2 (easy). Let \(0<a<1\), \(h(\lambda)=\lambda+a/\lambda\), and \(\Omega\) the interior of the ellipse \(x^2/(1+a)^2+y^2/(1-a)^2=1\). Show that \(h\) maps \(\{|\lambda|>1\}\) onto \(\mathbf C\setminus\overline\Omega\) and \(\{\sqrt a<|\lambda|<1\}\) into \(\Omega\), and that \(q(e^{i\theta})\) is a positive multiple of the gradient of \(x^2/(1+a)^2+y^2/(1-a)^2\) at \(h(e^{i\theta})\).

3.3 (medium). Show that \(h\) in Theorem 3.7 is unique: if \(h_1,h_2\) satisfy (1) and (2) with \(c_1,c_2>0\) (and possibly different \(r_0\)), then \(h_1=h_2\) on \(\{|\lambda|>1\}\).

3.4 (medium). For \(h(\lambda)=\lambda+a/\lambda\) with \(0<a<1\), show that \(h\) is injective on \(\{|\lambda|>r\}\) if and only if \(r\ge\sqrt a\).

7. Solutions

3.1. For \(z=z_0+1/w\) with \(w\ne0\), \(z\notin\overline\Omega\) if and only if \(|w|<1/R\). The map \(w\mapsto Rw\) sends \(U\) onto \(\mathbf D\) with derivative \(R>0\) at \(0\), so \(\phi(w)=Rw\), \(g(w)=w/R\), \(G=g\) on \(\mathbf C\), and \(h(\lambda)=z_0+1/(1/(R\lambda))=z_0+R\lambda\). For \(\lambda\in\mathbf T\), \(q(\lambda)=R\lambda\), and for \(|z-z_0|\le R\) \[ \begin{aligned} &\operatorname{Re}\bigl(\overline{R\lambda}(z_0+R\lambda-z)\bigr)\\ &\quad=R^2-R\operatorname{Re}\bigl(\bar\lambda(z-z_0)\bigr)\\ &\quad\ge R^2-R|z-z_0|\ge0 . \end{aligned} \]

3.2. For \(|\lambda|=s\) and \(\lambda=se^{i\theta}\), \(h(\lambda)=(s+a/s)\cos\theta+i(s-a/s)\sin\theta\), a point of the ellipse with semi-axes \(s+a/s\) and \(s-a/s\). These ellipses all have foci \(\pm2\sqrt a\), they increase with \(s\) for \(s>\sqrt a\), the one for \(s=1\) is \(\partial\Omega\), and for \(s=\sqrt a\) the image is the segment \([-2\sqrt a,2\sqrt a]\subset\Omega\). So \(|\lambda|>1\) maps into \(\mathbf C\setminus\overline\Omega\) and \(\sqrt a\le|\lambda|\le1\) into \(\overline\Omega\), with \(\sqrt a<|\lambda|<1\) going into \(\Omega\). Every \(z\in\mathbf C\setminus\overline\Omega\) is attained: \(h(\lambda)=z\) is the quadratic \(\lambda^2-z\lambda+a=0\), whose roots have product \(a\), so one root \(\lambda\) has \(|\lambda|\ge\sqrt a\); it satisfies \(|\lambda|>1\), since otherwise \(z=h(\lambda)\in\overline\Omega\). The gradient at \(((1+a)\cos\theta,(1-a)\sin\theta)\) is \[ \begin{aligned} &2\Bigl(\frac{\cos\theta}{1+a},\frac{\sin\theta}{1-a}\Bigr)\\ &\quad=\frac2{1-a^2}\bigl((1-a)\cos\theta,(1+a)\sin\theta\bigr), \end{aligned} \] and \(q(e^{i\theta})=e^{i\theta}-ae^{-i\theta}\), which is \((1-a)\cos\theta+i(1+a)\sin\theta\).

3.3. \(f=h_2^{-1}\circ h_1\) is a biholomorphic map of \(\{|\lambda|>1\}\) onto itself with \(f(\lambda)/\lambda\to c_1/c_2\) as \(\lambda\to\infty\). Then \(F(u)=1/f(1/u)\) maps \(\mathbf D\setminus\{0\}\) biholomorphically onto itself and is bounded, so by Theorem 5.2.2 it extends holomorphically to \(\mathbf D\) with \(F(0)=0\) and \(F'(0)=c_2/c_1>0\). The same construction for \(f^{-1}\) extends \(F^{-1}\), so the extension of \(F\) is a biholomorphic map of \(\mathbf D\). Schwarz's lemma (Lemma 3.5.1) applied to \(F\) and \(F^{-1}\) gives \(|F'(0)|\le1\) and \(1/|F'(0)|\le1\), so \(F\) is a rotation, and \(F'(0)>0\) gives \(F(u)=u\). So \(h_1=h_2\).

3.4. \(h(\lambda)=h(\mu)\) with \(\lambda\ne\mu\) holds exactly when \(\lambda\mu=a\). If \(|\lambda|,|\mu|>r\ge\sqrt a\), then \(|\lambda\mu|>a\). If \(r<\sqrt a\), choose \(s\) with \(r<s<\sqrt a\): the points \(s\) and \(a/s>\sqrt a\) are distinct, both have modulus larger than \(r\), and they have the same image.

References