Estimates of invariant distances on “convex” domains

Abstract

Estimates for invariant distances of convexifiable, $\mathbb{C }$-convexifiable and planar domains are given.

1 Introduction and results

Diederich and Ohsawa [6, p. 182] asked if $D$ is a smooth bounded pseudoconvex domain in ${\mathbb{C }}^{n}$, then the following lower bound for the Bergman distance $b_{D}$ holds: For fixed $z$ and $w$ close to $\partial D$, one has that

$$\begin{aligned} b_D(z,w)\ge -c\log d_D(w), \end{aligned}$$

where $d_{D}(w)=\text{ dist }(w,\partial D)$ and $c>0$ is a constant depending only on $D.$ Błocki [4, Theorem 1.3] mentioned this fact for bounded convexifiable domains (not necessarily smooth).

We shall prove the estimate in the case of bounded $\mathbb{C }$-convex domains (or, more generally, $\mathbb{C }$-convexifiable). Recall that a set in $\mathbb{C }^{n}$ is called $\mathbb{C }$-convex if all its intersections with complex lines are contractible (cf. [2, p. 25]). Note that a $C^{1}$-smooth bounded domain is $\mathbb{C }$-convex if and only the complex tangent hyperplane through any boundary point does not intersect the domain (cf. [2, Theorem 2.5.2]).

Let $D$ be a domain in $\mathbb{C }^{n}$. Denote by $c_{D}$ and $l_{D}$ the Carathéodory distance and the Lempert function of $D,$ respectively:

$$\begin{aligned} c_D(z,w)&= \sup \{\tanh ^{-1}|f(w)|:f\in \mathcal{O }(D,\mathbb{D }) \text{ with } f(z)=0\},\\ l_D(z,w)&= \inf \{\tanh ^{-1}|\alpha |:\exists \varphi \in \mathcal{O }(\mathbb{D },D) \text{ with } \varphi (0)=z,\varphi (\alpha )=w\}, \end{aligned}$$

where $\mathbb{D }$ is the unit disk (we refer to [10] for basic properties of the objects under consideration). The Kobayashi distance $k_{D}$ is the largest pseudodistance not exceeding $l_{D}$. We have that

$$\begin{aligned} c_D\le k_D,\quad c_D\le b_D \end{aligned}$$

(if $b_{D}$ is well-defined). Note also that $k_{D}=l_{D}$ for any planar domain $D$ (cf. [10, Remark 3.3.8(e)]). By Lempert’s theorem [11, Theorem 1], combining with a result by Jacquet [9, Theorem 5], $c_{D}=l_{D}$ on any $C^{2}$-smooth bounded $\mathbb{C }$-convex domain $D$ and hence on any convex domain. On the other hand, it follows by [14, Theorem 12] that there exists a constant $c_{n}>0,$ depending only on $n$, such that

$$\begin{aligned} k_D\le 4b_D\le c_nk_D \end{aligned}$$

(1)

for any $\mathbb{C }$-convex domain $D$ in $\mathbb{C }^{n}$, containing no complex lines (then $b_{D}$ is well-defined). In other words, to estimate $b_{D}$, it is enough to find lower bounds for $c_{D}$ and upper bounds for $l_{D}$.

Recall that $b_{D}$ is the integrated form of Bergman metric

$$\begin{aligned} \beta _D(z;X)=\frac{M_D(z;X)}{\sqrt{K_D(z)}}, \quad z\in D, X\in \mathbb{C }^n, \end{aligned}$$

where

$$\begin{aligned} M_D(z;X)=\sup \{|f^{\prime }(z)X|:f\in L_h^2(D)\,||f||_D\le 1,\;f(z)=0\} \end{aligned}$$

and

$$\begin{aligned} K_D(z)=\sup \{|f(z)|^2:f\in L_h^2(D),\;||f||_D\le 1\} \end{aligned}$$

is the Bergman kernel on the diagonal ($K_{D}(z)>0$ is assumed). So,

$$\begin{aligned} b_D(z,w)=\inf _\gamma \int \limits _0^1\beta _D(\gamma (t);\gamma ^{\prime }(t)), \end{aligned}$$

where the infimum is taken over all smooth curves $\gamma :[0,1]\rightarrow D$ with $\gamma (0)=z$ and $\gamma (1)=w$.

Estimates for invariant distances of strictly pseudoconvex domains in $\mathbb{C }^{n}$ and pseudoconvex domains of finite type in $\mathbb{C }^{2}$ can be found in [3] (see also [1, 10]) and [8], respectively.

Recall now in details two estimates. The proof of [4, Theorem 5.4] (cf. also [12, Proposition 2.4]) implies that if $D$ is a proper convex domain in $\mathbb{C }^{n}$, then

$$\begin{aligned} c_D(z,w)\ge \frac{1}{2}\log \frac{d_D(z)}{d_D(w)} \end{aligned}$$

(2)

(this proof uses only the existence of an appropriate supporting (real) hyperplane and the formula for the Poincaré distance of the upper half-plane). On the other hand, by [13, Theorem 1], for any $C^{1+\varepsilon }$-smooth bounded domain, there exists a constant $c>0$ such that

$$\begin{aligned} l_D(z,w)\le -\frac{1}{2}\log (d_D(z)d_D(w))+c \end{aligned}$$

(3)

(see [7, Proposition 2.5] for a stronger estimate for $k_{D}$).

The smoothness is essential as an example of a $C^{1}$-smooth bounded $\mathbb{C }$-convex planar domain shows (see [13, Example 2]). Moreover, using [16, p. 146, Theorem 7], one may find a bounded $\mathbb{C }$-convex planar domain for which there is no similar estimate with any constant instead of $-1/2.$

So, it natural to find an upper bound for $l_{D}$ in the convex case and a lower bound for $c_{D}$ in the $\mathbb{C }$-convex case.

Proposition 1

Let $D$ be a proper convex domain in $\mathbb{C }^{n}$. Then

$$\begin{aligned} l_D(z,w)\le \frac{||z-w||}{d(z)-d(w)}\log \frac{d(z)}{d(w)} \le \frac{||z-w||}{\min (d(z),d(w))}. \end{aligned}$$

¹ In particular, if, in addition, $D$ is bounded, then for any compact subset $K$ of $D$, there is a constant $c_{K}>0$ such that

$$\begin{aligned} b_D\le -c_K\log d_D(w)+1/c_K,\quad z\in K,w\in D. \end{aligned}$$

The last estimate for $k_{D}$ instead of $b_{D}$ (and $K$ a singleton) is the content of [12, Proposition 2.3]. Similar estimates for the Kobayashi distance of pseudoconvex Reinhardt domains can be found in [19].

Proposition 2

Let $D$ be a proper $\mathbb{C }$-convex domain in $\mathbb{C }^{n}$. Then

$$\begin{aligned} c_D(z,w)\ge \frac{1}{4}\log \frac{d_D(z)}{4d_D(w)}. \end{aligned}$$

Hence, if, in addition, $D$ is bounded, then for any compact subset $K$ of $D$, there is a constant $c_{K}>0$ such that

$$\begin{aligned} b_D(z,w)\ge -\frac{1}{4}\log d_D(w)-c_K,\quad z\in K,w\in D. \end{aligned}$$

Note that by [5, p. 2381], the first estimate in Proposition 2 implies the following

Corollary 3

The Bergman and Szegö kernels (on the diagonal) are comparable on any $C^{2}$-smooth bounded $\mathbb{C }$-convex domain.

We point out that [5, Theorem 1.3] deals with the convex case.

Remark

• (a) The estimate for $l_{D}$ is sharp when $z\rightarrow w$. Moreover, it is sharp up to a constant when $z$ is fixed and $w\rightarrow \partial D$. Indeed, denote by $R_{D}(z,w)$, the right-hand side of the first inequality in Proposition 1. If $\theta \in (0,\pi )$ and $D_{\theta }=\{z\in \mathbb{C }_{*}:|\arg z|<\theta \}$, then

  $$\begin{aligned} \lim _{\theta \rightarrow 0}\lim _{x\rightarrow 0+}\frac{l_{D_\theta }(1,x)}{R_{D_\theta }(1,x)}=\frac{\pi }{4}. \end{aligned}$$

• (b) The factor $1/4$ in the bound for $c_{D}$ is optimal as $D=\mathbb{C }_{*}\setminus {\mathbb{R }}^{+}$ shows.

• (c) Estimates for the infinitesimal forms of the distances under consideration, namely, the Carathéodory, Kobayashi and Bergman metrics, of convex and $\mathbb{C }$-convex domains can be found in [14]. The bounds there depend only on the distance to the boundary from the respective point in the respective direction.

Our main result is in the spirit of [4, Theorem 1.3], where a lower bound for the Bergman metric is mentioned in the locally convexifiable case (and a hint for a proof is given).

Proposition 4

Let $D$ be a bounded domain in $\mathbb{C }^{n}$ which is locally $\mathbb{C }$-convexifiable, i.e., for any point $a\in \partial D$, there exist a neighborhood $U_{a}$ of $a$, an open set $V_{a}$ in $\mathbb{C }^{n}$ and a biholomorphism $F_{a}:U_{a}\rightarrow V_{a}$ such that $F_{a}(D\cap U_{a})$ is $\mathbb{C }$-convex. Then, there exists a constant $c>0$ such that for any compact subset $K$ of $D$ one can find a constant $c_{K}>0$ with

$$\begin{aligned} s_D(z,w)\ge -c\log d_D(w)-c_K,\quad z\in K,w\in D, \end{aligned}$$

where $s_{D}=k_{D}$ or $s_{D}=b_{D}$.

Moreover, if $D$ is locally convexifiable or $C^{1+\varepsilon }$-smooth and locally $\mathbb{C }$-confexifiable, then for any compact subset $K$ of $D$, one can find a constant $c^{\prime }_{K}>0$ with

$$\begin{aligned} s_D(z,w)\le -c_K^{\prime }\log d_D(w)+1/c^{\prime }_K,\quad z\in K,w\in D. \end{aligned}$$

Finally, we consider the planar case. We shall say that a boundary point $p$ of a planar domain $D$ is Dini-smooth if $\partial D$ near $p$ is a Dini-smooth curve $\gamma :[0,1]\rightarrow \mathbb{C }$.² Call a planar domain Dini-smooth if it is Dini-smooth near any boundary point.

Proposition 5

Let $p$ be a Dini-smooth boundary point of a planar domain $D$. Then, for any neighborhood $U$ of $p$ and any compact subset $K$ of $D$, there exist a neighborhood $V$ of $p$ and a constant $c>0$ such that

$$\begin{aligned}&s_D(z,w)\ge -\frac{1}{2}\log d_D(w)-c,\quad z\in D\setminus U, w\in D\cap V,\\&|s_D(z,w)+\frac{1}{2}\log d_D(w)|\le c,\quad z\in K, w\in D\cap V, \end{aligned}$$

where $s_{D}=c_{D},\, s_{D}=l_{D}(=k_{D})$ or $s_{D}=b_{D}/\sqrt{2}$.

Since $k_{D}$ and $b_{D}$ are the integrated forms of $\kappa _{D}$ and $\beta _{D}$, we get the following

Corollary 6

Let $p$ and $q$ be different Dini-smooth boundary points of a planar domain $D$. If $s_{D}=l_{D}(=k_{D})$ or $s_{D}=b_{D}/\sqrt{2}$, then the function

$$\begin{aligned} 2s_D(z,w)+\log d_D(z)+\log d_D(w) \end{aligned}$$

is bounded for $z$ near $q$ and $w$ near $p$.

In general, $c_{D}$ is not an inner distance (even in the plane). So, the next proposition is not a direct consequence of Proposition 5.

Proposition 7

Let $p$ and $q$ be different Dini-smooth boundary points of a planar domain $D$. Then, the function

$$\begin{aligned} 2c_D(z,w)+\log d_D(z)+\log d_D(w) \end{aligned}$$

is bounded for $z$ near $q$ and $w$ near $p$.

The next result is optimal for the boundary behavior of $c_{D}$ and $l_{D}(=k_{D})$ in the planar case. It is more general than the last results, but its proof uses these results. Similar (and slightly weaker) result for $k_{D}$ on $C^{2}$-smooth strictly pseudoconvex bounded follows by [3, Theorem 1, Proposition 1.2].

Proposition 8

Let $D$ be a Dini-smooth bounded planar domain.³ Then, there exists a constant $c\ge 1$ such that

$$\begin{aligned} \log \left( 1+\frac{|z-w|}{c\sqrt{d_D(z)d_D(w)}}+\frac{|z-w|^2}{cd_D(z)d_D(w)} \right)&\!\le \! 2c_D(z,w) \le 2l_D(z,w)\\&\!\le \! \log \left( 1\!+\!\frac{c|z-w|}{\sqrt{d_D(z)d_D(w)}}\!+\!\frac{c|z-w|^2}{d_D(z)d_D(w)}\right) . \end{aligned}$$

In particular, the function $l_{D}-c_{D}$ is bounded on $D\times D$.

It is shown in [18, Theorem 1] that if $D$ is strongly pseudoconvex domain in $\mathbb{C }^{n}$, then

$$\begin{aligned} \lim _{\begin{array}{c} w\rightarrow \partial D\\ z\ne w \end{array}}\frac{c_D(z,w)}{k_D(z,w)}=1\quad \text{ uniformly } \text{ in } z\in D. \end{aligned}$$

We have the following planar extension of this result.

Proposition 9

If $D$ is finitely connected bounded planar domain without isolated boundary points, then

$$\begin{aligned} \lim _{\begin{array}{c} w\rightarrow \partial D\\ z\ne w \end{array}}\frac{c_D(z,w)}{l_D(z,w)}=1\quad \mathrm{uniformly\; in}\quad z\in D. \end{aligned}$$

2 Proofs

Proof of Proposition 1

Denote by $C_{z,w}$ the convex hull of the union of the disks $\mathbb{D }(z,d_{D}(z))$ and $\mathbb{D }(w,d_{D}(w)),$ lying in the complex line through $z$ and $w$. Let $\gamma (t)=z+t(w-z)$. Since $C_{z,w}\subset D$ and $l_{C_{z,w}}=k_{C_{z,w}}$ is the integrated form of the Kobayashi metric $\kappa _{C_{z,w}}$, ⁴ then

$$\begin{aligned} l_D(z,w)&\le l_{C_{z,w}}(z,w)\le \int \limits _0^1\kappa _{C_{z,w}}(\gamma (t);\gamma ^{\prime }(t))dt\\&\le \int \limits _0^1\frac{|\gamma ^{\prime }(t)|}{d_{C_{z,w}}(\gamma (t))}dt= \frac{||z-w||}{d(z)-d(w)}\log \frac{d(z)}{d(w)}. \end{aligned}$$

This inequality and (1) lead to the wanted result for $b_{D}$.

Proof of Proposition 2

Let $p(w)\in \partial D$ be such that $||w-p(w)||=d_{D}(w)$. Since $E$ is ${\mathbb{C }}$-convex, there exists a hyperplane $H_{p(w)}$ through $p(w)$ and disjoint from $D$ (cf. [2, Theorem 2.3.9(ii)]). Denote by $D_{w}$ and $z_{w}$ the projections of $D$ and $z$ onto the complex line through $w$ and $p(w)$ in direction $H_{(p(w)},$ respectively. By [2, Theorem 2.3.6], $D_{w}$ is a simply connected domain and $p(w)\in \partial D_{w}$. Denote by $\psi _{w}\in \mathcal{O }(\mathbb{D },D_{w})$ a Riemann map such that $\psi _{w}(0)=z_{w}$. If $\psi _{w}(\alpha _{w})=w$, then

$$\begin{aligned} c_D(z,w)\ge c_{D_w}(z_w,w)=\tanh ^{-1}|\alpha _w|. \end{aligned}$$

By [16, p. 139, Corollary 6] (which is a consequence of the Köbe 1/4 and the Köbe distortion theorems),

$$\begin{aligned} \tanh ^{-1}|\alpha _w|\ge \frac{1}{4}\log \frac{|\psi {^{\prime }}_{w}(0)|}{4d_{D_w}(w)}. \end{aligned}$$

Since $d_{D_{w}}(w)=d_{D}(w)$ and $|\psi {^{\prime }}_{w}(0)|\ge d_{D_{w}}(z_{w})\ge d_{D}(z),$ it follows that

$$\begin{aligned} c_D(z,w)\ge \frac{1}{4}\log \frac{d_D(z)}{4d_D(w)}. \end{aligned}$$

This inequality and $b_{D}\ge c_{D}$ imply the desired result for $b_{D}$.

Proof of Proposition 4

⁵ First, we shall prove the lower bound.

Note that

$$\begin{aligned} 0<c_a\le \frac{d_{F_a(D\cap U_a)}(F_a(w))}{d_D(w)}\le \frac{1}{c_a}\quad \text{ near } \text{ any } a\in \partial D. \end{aligned}$$

(4)

Then, by Proposition 2, we may find a finite set $M\subset \partial D$ and a constant $c_{1}>0$ such that

$$\begin{aligned} s_{D\cap U_a}(z,w)\ge \frac{1}{4}\log \frac{d_D(z)}{d_D(w)}-c_1,\quad z,w\in D\cap V_a, a\in M, \end{aligned}$$

where $V_{a}\subset U_{a}$ is a neighborhood of $a$ such that $\partial D\subset \cup _{a\in M} V_{a}$.

Denote now by $S_{D}$ the Kobayashi or Bergman metrics of $D$. By localization principles (cf. [10, Proposition 7.2.9 and Proposition 6.3.5], since $D$ is pseudoconvex), there exists a constant $c_{2}>0$ such that

$$\begin{aligned} S_D\ge 4c_2 S_{D\cap U_a}\quad \text{ on } (D\cap V_a)\times \mathbb{C }^n. \end{aligned}$$

Let $W_{a}\Subset V_{a}$ be such that $W=\cup _{a\in M} W_{a}$ does not intersect $K$ and contains $\partial D$. Set $r=\min _{a\in M}\text{ dist }(\partial W_{a},\partial V_{a})$.

Let $\varepsilon >0$. Since $s_{D}$ is the integrated form of $S_{D}$, for any $z\in K$ and $w\in D\cap W$, there exists a smooth curve $\gamma :[0,1]\rightarrow D$ with $\gamma (0)=z,\,\gamma (1)=w$ and

$$\begin{aligned} s_D(z,w)+\varepsilon >\int \limits _0^1S_D(\gamma (t);\gamma ^{\prime }(t))dt. \end{aligned}$$

Let $t_{1}=\max \{t\in (0,1):\gamma (t)\in G=D\setminus W\}$. Choose a point $a_{1}\in M$ such that $\mathbb{B }_{n}(\gamma (t_{1}),r)\subset V_{a_{1}}$. Let $t_{2}=\sup \{t\in (t_{1},1]:\gamma ([t_{1},t))\in V_{a_{1}}\}$ and etc. In this way, we may find numbers $0<t_{1}<\dots <t_{N+1}=1$ and points $a_{1},\dots ,a_{N+1}\in M$ such that $\gamma [t_{j},t_{j+1})\subset D\cap V_{a_{j}}$ and $||\gamma (t_{j+1})-\gamma (t_{j})||\ge r,\,1\le j\le N$. Then

$$\begin{aligned} s_D(z,w)+\varepsilon&> c_2\sum _{j=1}^N s_{D\cap U_{a_j}}(\gamma (t_j),\gamma (t_{j+1}))\\&\ge c_2\sum _{j=1}^N\log \frac{d_D(\gamma (t_j))}{d_D(\gamma (t_{j+1}))} -c_3N\\&\ge c_2\log \frac{\text{ dist }(G,\partial D)}{d_D(w)}-c_3N, \end{aligned}$$

where $c_{3}=4c_{1}c_{2}$.

On the other hand, since $D$ is a bounded domain, there exists a constant $c_{4}>0$ such that $s_{D}(z_{1},z_{2})\ge c_{4}||z_{1}-z_{2}||$. Then

$$\begin{aligned} s_D(z,w)+\varepsilon >\sum _{j=1}^Ns_D(\gamma (t_j),\gamma (t_{j+1}))\ge c_4rN. \end{aligned}$$

So,

$$\begin{aligned} \left( 1+\frac{c_3}{c_4r}\right) (s_D(z,w)+\varepsilon )\ge c_2\log \frac{\text{ dist } (G,\partial D)}{d_D(w)}. \end{aligned}$$

The case when $w\in G$ is trivial which completes the proof of the lower bound.

The proof of the upper bound is easier. Fix a point $a\in \partial D$. It is enough to find a constant $c^{\prime }_{a,K}>0$ such that the estimate holds for $w$ near $a$. Take a point $u\in U_{a}$ and a neighborhood $V_{a}\Subset U_{a}$ of $a$ and a point $u\in D\cap U_{a}$. Proposition 1, (3) and (4) imply that

$$\begin{aligned} k_D(z,w)&\le k_D(z,u)+k_D(u,w)\le k_D(z,u)+k_{D\cap U}(u,w)\\&\le 1/c^{\prime }_{a,K}-c^{\prime }_{a,K}\log d_D(w),\ z\in K,w\in D\cap V_a. \end{aligned}$$

The upper bound for $b_{D}$ follows similarly. It suffices to use that

$$\begin{aligned} b_D\le \widetilde{c}_ab_{D\cap U_a}\le \widetilde{c}_ac_n k_{D\cap U_a} \end{aligned}$$

in view of [10, Proposition 6.3.5] and (1).

Proof of Proposition 5 for

$c_{D}$ and $l_{D}$ We may find a Dini-smooth Jordan curve $\zeta$ such that $\zeta =\partial D$ near $p$ and $D\subset G:=\zeta _{\text{ ext }}$. Take a point $a\not \in \overline{G}$ and consider the union $G_{e}$ of $0$ and the image of $G$ under the map $\varphi :z\rightarrow (z-a)^{-1}$. There exists a conformal map $\psi :G_{e}\rightarrow \mathbb{D }$. It extends to a $C^{1}$-diffeomorphism from $\overline{G_{e}}$ to $\overline{\mathbb{D }}$ (cf. [20, Theorems 3.5]). Setting $\eta =\psi \circ \varphi$, then

$$\begin{aligned} c_D(z,w)\ge c_{\mathbb{D }}(\eta (z),\eta (w)). \end{aligned}$$

Now the lower bound for $c_{D}$ follows by the same bound for $c_{\mathbb{D }}$ and an inequality of type (4).

The estimate

$$\begin{aligned} l_D(z,w)\le -\frac{1}{2}\log d_D(w)-c,\quad z\in K, w\in D\cap V \end{aligned}$$

follows by (3). It can be also obtained in the following way. There exist a Dini-smooth domain simply connected domain $G_{i}\subset D$ and a neighborhood $V$ of $p$ such that $\partial G\cap V=\partial D\cap V$. Take a point $u\in V$. Since $l_{D}=k_{D}$, then

$$\begin{aligned} k_D(z,w)\le k_D(z,u)+k_{G_i}(u,w). \end{aligned}$$

It remains to repeat the final arguments from the first paragraph.

Proof of Proposition 5 for

$b_{D}$ ⁶ Choosing $G$ as above, then

$$\begin{aligned} b_D(z,w)=b_{\eta (D)}(\eta (z),\eta (w)). \end{aligned}$$

By the Dini-smoothness,

$$\begin{aligned} \lim _{z\rightarrow p}\frac{d_{\eta (D)}(\eta (z))}{d_D(z)}=|\eta ^{\prime }(p)|. \end{aligned}$$

We may assume that $\eta (p)=1$. So, it is enough to get the estimates for $D\subset \mathbb{D }$ such that $F=\mathbb{D }\cap \mathbb{D }(1,r)\subset D$ for some $r\in (0,1)$.

First, we shall prove that if $0<r^{\prime }<r$, then

$$\begin{aligned} \sqrt{2}b_D(z,w)\le -\log d_D(w)+c^{\prime },\quad z\in K, w\in F^{\prime }=\mathbb{D }\cap \mathbb{D }(1,r^{\prime }) \end{aligned}$$

for some constant $c^{\prime }>0$.

For a domain ${\Omega }\subset \mathbb{C }$ set $\beta _{{\Omega }}(z)=B_{{\Omega }}(z;1)$ and $\kappa _{{\Omega }}(z)=\kappa _{{\Omega }}(z;1)$. Let ${\check{F}}=\mathbb{D }\setminus F$ and

$$\begin{aligned} l_{\mathbb{D }}(u,{\check{F}})=\inf _{w\in {\check{F}}}l_{\mathbb{D }}(u,w). \end{aligned}$$

Then, for any $r^{\prime \prime }\in (r^{\prime },r)$, we may find a constant ${\tilde{c}}>0$ such that

$$\begin{aligned} \beta _D(u)&\le \beta _F(u)\sqrt{\frac{K_F(u)}{K_{\mathbb{D }}(u)}} =\frac{\sqrt{2}\kappa _{F}^{2}(u)}{\kappa _{\mathbb{D }}(u)} \\&\le \sqrt{2}\coth ^2l_{\mathbb{D }}(u,{\check{F}})\kappa _{\mathbb{D }}(u) \le \frac{\sqrt{2}}{1-|u|^2}+{\tilde{c}}, \quad u\in F^{\prime \prime }=\mathbb{D }\cap \mathbb{D }(1,r^{\prime \prime }). \end{aligned}$$

(for the equality use that $F$ is biholomorphic to $\mathbb{D }$ and for the inequality “between the lines” cf. [10, Proposition 7.2.9]).

Let $z\in K, w\in F^{\prime }$ and $w^{\prime }=[0,w]\cap \partial D(1,r^{\prime \prime })$. Then

$$\begin{aligned} b_D(z,w)&\le b_D(z,w^{\prime })+|w-w^{\prime }|\left( {\tilde{c}}+\sqrt{2}\int \limits _0^1\frac{dt}{1-|w^{\prime }+t(w-w^{\prime })|^2}\right) \\&\le (-\log d_D(w)+c^{\prime })/\sqrt{2} \end{aligned}$$

for some constant $c^{\prime }>0$.

Now, shrinking $r$ such that $\mathbb{D }(1,r)\subset U$, it remains to prove that

$$\begin{aligned} \sqrt{2}b_D(z,w)\ge -\log d_D(w)-c^{\prime \prime },\quad z\in {\check{F}}, w\in F^{\prime } \end{aligned}$$

for some constant $c^{\prime \prime }>0$.

We have that

$$\begin{aligned} \beta _D(u)&\ge \beta _{\mathbb{D }}(u)\sqrt{\frac{K_{\mathbb{D }}(u)}{K_F(u)}} =\frac{\sqrt{2}\kappa ^2_{\mathbb{D }}(u)}{\kappa _F(u)}\\&\ge \sqrt{2}\tanh l_{\mathbb{D }}(u,{\check{F}})\kappa _{\mathbb{D }}(u)\ge \frac{\sqrt{2}}{1-|u|^2}-\hat{c}, \quad u\in F^{\prime \prime }. \end{aligned}$$

For $z\in {\check{F}}, w\in F^{\prime }$, and $\varepsilon >0$, there exists a smooth curve $\gamma :[0,1]\rightarrow D$ with

$$\begin{aligned} b_D(z,w)+\varepsilon >\int \limits _0^1\beta _D(\gamma (t))|\gamma ^{\prime }(t)|dt. \end{aligned}$$

Let $t_{0}=\sup \{t\in (0,1):\gamma (t)\not \in F^{\prime \prime }\}$. Then,

$$\begin{aligned} b_D(z,w)+\varepsilon >\int \limits _{t_0}^1b_D(\gamma (t))|\gamma ^{\prime }(t)|dt\ge \hat{b}_{\mathbb{D }}(w,{\check{F}}), \end{aligned}$$

where $\hat{b}_{\mathbb{D }}$ is the integrated form of the Finsler pseudometric

$$\begin{aligned} \hat{\beta }_{\mathbb{D }}(u;X)=|X|\left( \frac{\sqrt{2}}{1-|u|^2}-\hat{c}\right) ^+. \end{aligned}$$

It remains to use that, shrinking $r^{\prime }$ (if necessary),

$$\begin{aligned} \hat{b}_{\mathbb{D }}(w,{\check{F}})\ge (-\log d_D(w)-c^{\prime \prime })/\sqrt{2} \end{aligned}$$

for some constant $c^{\prime \prime }>0$ (cf. [3, Theorem 1.1]).

Proof of Corollary 6

Since $k_{D}$ and $b_{D}$ are the integrated forms of $\kappa _{D}$ and $\beta _{D}$, the boundedness from below follows by the first inequality in Proposition 5 (cf. the proof of [10, Proposition 10.2.6]). Choosing a point $a\in D$, the boundedness from above is a consequence of the inequality $s_{D}(z,w)\le s_{D}(z,a)+s_{D}(a,w)$ and the second inequality in Proposition 5.

Proof of Proposition 7

In virtue of the inequality $c_{D}\le k_{D}$ and Corollary 6, we have to prove only the boundedness from below. For this, take disjoint Dini-smooth Jordan curves $\zeta ^{\prime }$ and $\zeta ^{\prime \prime }$ such that $\zeta ^{\prime }=\partial D$ near $p, \zeta ^{\prime \prime }=\partial D$ near $q$ and $D\subset G:=\zeta ^{\prime }_{\text{ ext }}\cap \zeta ^{\prime \prime }_{\text{ ext }}$. Note that any Dini-smooth bounded double-connected planar ${\tilde{G}}$ domain can be conformally map to some annulus $A_{r}=\{z\in \mathbb{C }:1/r<|z|<r\} (r>1)$, and the respective mapping extends to a $C^{1}$-diffeomorphism from $\overline{\tilde{G}}$ to $\overline{A_{r}}$.⁷

Then, proceeding similarly to the proof of Proposition 5 for $c_{D}$, it is enough to show that

$$\begin{aligned} 2c_{A_r}(z,w)+\log d_{A_r}(z)+\log d_{A_r}(w) \end{aligned}$$

is bounded from below for $z\in \mathbb{R }$ near $r$ and $w$ near $p$, where $|p|=1/r$; this is equivalent to

$$\begin{aligned} m_{A_r}(z,w):=\tanh c_{A_r}(z,w)\ge 1-c d_{A_r}(z)d_{A_r}(w) \end{aligned}$$

for some constant $c>0$.

Recall that (cf. [10, Proposition 5.5])

$$\begin{aligned} m_{A_r}(z,w)=\frac{f(z,w)f(1/z,-|w|)}{r|w|}, \end{aligned}$$

where $f$ is a holomorphic function on $\overline{A_{r}\times A_{r}}\setminus \{u=v\in \partial A_{r}\}$ and $|f(u,v)|=1$ if $|u|=r, v\in \overline{A_{r}}$ or $u\in \overline{A_{r}}, |v|=1/r (u\ne v)$.

In particular,

$$\begin{aligned} \frac{\partial ^n f}{\partial u^n}=\frac{\partial ^n f}{\partial v^n}=0,\quad n\in {\mathbb{N }}, \end{aligned}$$

at any point $(u,v)$ with $|u|=r$ and $|v|=1/r$. Then, by the Taylor expansion,

$$\begin{aligned} |f(z,w)-f(r,w(r|w|)^{-1})|\le c_1d_{A_r}(z)d_{A_r}(w). \end{aligned}$$

This implies that

$$\begin{aligned} |f(z,|w|)-f(r,1/r)|\le c_1d_{A_r}(z)d_{A_r}(|w|) \end{aligned}$$

(the constant can be chosen the same for $z$ near $r$ and $w$ away from $r$). Since $f(r,\cdot )$ is a unimodular constant and $d_{A_{r}}(w)=d_{A_{r}}(|w|)$, it follows that

$$\begin{aligned} |m_{A_r}(z,w)-m_{A_r}(z,|w|)|\le c_2d_{A_r}(z)d_{A_r}(|w|). \end{aligned}$$

Further, $c_{A_{r}}(z,|w|)=c_{A_{r}}(z,t)+c_{A_{r}}(t,|w|)$ for $t\in [|w|,z]$ (cf. [10, Lemma 5.11(b)]). Then, Proposition 5 implies that

$$\begin{aligned} m_{A_r}(z,|w|)\ge 1-c_3d_{A_r}(z)d_{A_r}(|w|). \end{aligned}$$

Hence we may choose $c=c_{2}+c_{3}$ which completes the proof.

Proof of Proposition 8

Using Corollary 6 and Proposition 7, it is enough to prove the inequalities for $z$ and $w$ near a fixed point $p\in \partial D$. Moreover, it is easy to see that these inequalities are equivalent to

$$\begin{aligned} \frac{|z-w|}{\sqrt{cd_D(z)d_D(w)+|z-w|^2}}&\le \tanh c_D(z,w) \le \tanh l_D(z,w)\\&\le \frac{|z-w|}{\sqrt{c^{-1}d_D(z)d_D(w)+|z-w|^2}} \end{aligned}$$

for some constant $c\ge 1$.⁸

To prove the lower bound for $\tanh c_{D}(z,w)$, let $\eta$ be as in the proof of Proposition 5 for $c_{D}$ and $l_{D}$. Then, it is not difficult to find a constant $c_{1}>0$ such that

$$\begin{aligned} \tanh c_D(z,w)\ge \tanh c_{\mathbb{D }}(z_1,w_1)\ge \frac{|z_1-w_1|}{\sqrt{c_1d_{\mathbb{D }}(z_1)d_{\mathbb{D }}(w_1)+|z_1-w_1|^2}}, \end{aligned}$$

where $z_{1}=\eta (z)$ and $w_{1}=\eta (w)$. It remains to use that, similarly to (4), $d_{D}\ge c_{2} d_{\mathbb{D }}$ and $|z_{1}-w_{1}|\ge c_{2}|z-w|$ for some constant $c_{2}>0$.

The proof of the upper bound for $\tanh l_{D}(z,w)$ is similar (by using $G_{i}$ from the second part of the proof mentioned above) and we skip it.

Proof of Proposition 9

By the Köbe uniformization theorem, we may assume that $\partial D$ consists of disjoint circles. Using Proposition 8 and compactness, it is enough to prove that for any point $p\in \partial D$,

$$\begin{aligned} \lim _{z\ne w\rightarrow p}\frac{c_D(z,w)}{l_D(z,w)}=1. \end{aligned}$$

Applying an inversion, we may suppose that the outer boundary of $D$ is the unit circle $\Gamma$ and $p\in \Gamma$. Let $U$ be a disk centered at $p$ such that $\mathbb{D }\cap U\subset D$. Then,

$$\begin{aligned} 1\ge \frac{c_D(z,w)}{l_D(z,w)}\ge \frac{c_{\mathbb{D }}(z,w)}{l_{\mathbb{D }\cap U}(z,w)}=\frac{k_{\mathbb{D }}(z,w)}{k_{\mathbb{D }\cap U}(z,w)}. \end{aligned}$$

Considering $\mathbb{D }$ as a part of the unit ball in $\mathbb{C }^{2}$, it follows that the last ratio tends to 1 as a particular case of the same result for strongly pseudoconvex domains (see [18, Proposition 3]).