Construction of nearly hyperbolic distance on punctured spheres

Abstract

We define a distance function on the bordered punctured disk $0<|z|\le 1/e$ in the complex plane, which is comparable with the hyperbolic distance of the punctured unit disk $0<|z|<1.$ As an application, we will construct a distance function on an n-times punctured sphere which is comparable with the hyperbolic distance. We also propose a comparable quantity which is not necessarily a distance function on the punctured sphere but easier to compute.

1 Introduction

A domain $\Omega$ in the Riemann sphere ${\widehat{\mathbb {C}}}={\mathbb {C}}\cup \{\infty \}$ has the upper half-plane ${\mathbb {H}}=\{\zeta \in {\mathbb {C}}: \,{{\text {Im}}\,}\zeta >0\}$ as its holomorphic universal covering space precisely if the complement ${\widehat{\mathbb {C}}}{\setminus }\Omega$ contains at least three points. Such a domain is called hyperbolic. Since the Poincaré metric $|d\zeta |/(2\,{{\text {Im}}\,}\zeta )$ is invariant under the pullback by analytic automorphisms of ${\mathbb {H}}$ [in other words, under the action of ${{\text {PSL}}}(2,{\mathbb {R}})$], it descends to a metric on $\Omega ,$ called the hyperbolic metric of $\Omega$ and denoted by $\lambda _\Omega (w)|dw|.$ More explicitly, they are related by the formula $\lambda _\Omega (p(\zeta ))|p'(\zeta )|=1/(2\,{{\text {Im}}\,}\zeta ),$ where $p:{\mathbb {H}}\rightarrow \Omega$ is a holomorphic universal covering projection of ${\mathbb {H}}$ onto $\Omega .$ The quantity $\lambda _\Omega (w)$ is sometimes called the hyperbolic density of $\Omega$ and it is independent of the particular choice of p and $\zeta \in p^{-1}(w).$ Note that $\lambda _\Omega$ has constant Gaussian curvature $-4$ on $\Omega .$ We denote by $h_\Omega (w_1,w_2)$ the distance function induced by $\lambda _\Omega ,$ called the hyperbolic distance on $\Omega$ and the distance is known to be complete. That is, $h_\Omega (w_1,w_2)=\inf _\alpha \ell _\Omega (\alpha ),$ where the infimum is taken over all rectifiable curves $\alpha$ joining $w_1$ and $w_2$ in $\Omega$ and

$$\begin{aligned} \ell _\Omega (\alpha )=\int _\alpha \lambda _\Omega (w)|dw|. \end{aligned}$$

One of the most important properties of the hyperbolic metric is the principle of hyperbolic metric, which asserts the monotonicity $\lambda _\Omega (w)\ge \lambda _{\Omega _0}(w)$ and thus $h_\Omega (w_1,w_2)\ge h_{\Omega _0}(w_1,w_2)$ for $w, w_1, w_2\in \Omega \subset \Omega _0$ (cf. [15, III.3.6]). For basic facts about hyperbolic metrics, we refer to recent textbooks [11] by Keen and Lakic or a survey paper [6] by Beardon and Minda as well as classical book [3] by Ahlfors. For instance,

$$\begin{aligned} h_{\mathbb {H}}(\zeta _1, \zeta _2)={{\text {arth}}\,}\left| \frac{\zeta _1-\zeta _2}{\zeta _1-\overline{\zeta _2}}\right| , \quad \zeta _1, \zeta _2\in {\mathbb {H}}, \end{aligned}$$

(1.1)

where ${{\text {arth}}\,}x=\frac{1}{2}\log \frac{1+x}{1-x}$ for $0\le x<1$ (see [5, Chap. 7] for instance). Therefore, $h_\Omega (w_1,w_2)$ can be expressed via the universal covering projection $w=p(\zeta )$ as follows (see [11, Theorem 7.1.3]):

$$\begin{aligned} h_\Omega (w_1,w_2)=\min _{\zeta _2\in p^{-1}(w_2)} h_{\mathbb {H}}(\zeta _1,\zeta _2) =\min _{\gamma \in \Gamma } h_{\mathbb {H}}(\zeta _1,\gamma (\zeta _2)), \end{aligned}$$

where $\zeta _1\in p^{-1}(w_1), \zeta _2\in p^{-1}(w_2)$ and $\Gamma =\{\gamma \in {{\text {PSL}}}(2,{\mathbb {R}}): p\circ \gamma =p\}$ is the covering transformation group of $p:{\mathbb {H}}\rightarrow \Omega$ (also called a Fuchsian model of $\Omega$).

It is, however, difficult to obtain an explicit expression of $\lambda _\Omega (w)$ or $h_\Omega (w_1,w_2)$ for a general hyperbolic domain $\Omega$ because a concrete form of its universal covering projection is not known except for several special domains. (It is not easy even for a simply connected domain because it is hard to find its Riemann mapping function in general.) Therefore, as a second choice, estimates of $\lambda _\Omega (w)$ are useful. Indeed, Beardon and Pommerenke [7] supplied a general but concrete bound for $\lambda _\Omega (w).$ However, it is still difficult to estimate the induced hyperbolic distance $h_\Omega (w_1,w_2)$ due to complexity of the fundamental group of $\Omega .$

On the other hand, an explicit bound for the hyperbolic distance may be of importance. For instance, if $f:\Omega \rightarrow X$ is a holomorphic map between hyperbolic domains, then the principle of hyperbolic metric yields the inequality

$$\begin{aligned} h_X(f(z),f(z_0))\le h_\Omega (z,z_0),\quad z_0,z\in \Omega , \end{aligned}$$

(1.2)

which contains some important information about the function f. As a maximal hyperbolic plane domain, the thrice-punctured sphere (the twice-punctured plane) ${\mathbb {C}}_{0,1}:={\mathbb {C}}{\setminus }\{0,1\}$ is particularly important. Letting $X={\mathbb {C}}_{0,1},$ the inequality (1.2) leads to Schottky’s theorem when $\Omega$ is the unit disk (cf. [9, 10]), and it leads to the big Picard theorem when $\Omega$ is a punctured disk (cf. [3, §1–9]). Though the hyperbolic density of ${\mathbb {C}}_{0,1}$ was essentially computed by Agard [2] (see also [18]) and the holomorphic universal covering projection of ${\mathbb {H}}$ onto ${\mathbb {C}}_{0,1}$ is known as an elliptic modular function (see Sect. 2 below and, e.g., [4, p. 279] or [14, Chap. VI]), we do not have any convenient expression of the hyperbolic distance $h_{{\mathbb {C}}_{0,1}}(w_1,w_2)$ except for special configurations of the points $w_1,w_2$ (see, for instance, [17, Lemma 3.10], [18, Lemma 5.1]).

In this paper, we consider punctured spheres $X={\widehat{\mathbb {C}}}{\setminus }\{a_1,\dots ,a_n\}.$ This is still general enough in the sense that for any hyperbolic domain $\Omega \subset {\widehat{\mathbb {C}}},$ there exists a sequence of punctured spheres $X_k\supset \Omega ~(k=1,2,\dots )$ such that $\lambda _{X_k}\rightarrow \lambda _\Omega$ locally uniformly on $\Omega$ as $k\rightarrow \infty$ (see [8, §5]). We also note that Rickman [16] constructed a conformal metric on a punctured sphere of higher dimensions to show a Picard–Schottky type result for quasiregular mappings. Our main purpose of this paper is to give a distance function $d_X(w_1,w_2)$ on the punctured sphere X which can be computed (or estimated) more easily than the hyperbolic distance $h_X(w_1,w_2)$ but still comparable with it by concrete bounds. To this end, we first propose a distance function $D(z_1,z_2)$ on $0<|z|\le e^{-1}$ given by the formula

$$\begin{aligned} D(z_1, z_2)=\frac{2\sin (\theta /2)}{\max \{\log (1/|z_1|),\, \log (1/|z_2|)\}} +\left| \log \log \frac{1}{|z_2|}-\log \log \frac{1}{|z_1|}\right| , \end{aligned}$$

(1.3)

where $\theta =|{{\text {arg}}\,}(z_2/z_1)| \in [0,\pi ]$. In Sect. 3, we will show that $D(z_1,z_2)$ is indeed a distance function and comparable with $h_{{\mathbb {D}}^*}(z_1,z_2)$ on the set $0<|z|\le e^{-1}.$ Note also that $D(z_1, z_2)=e|z_1-z_2|$ for $|z_1|=|z_2|=e^{-1}.$

As another extremal case, the punctured disk ${\mathbb {D}}^*$ is also important. Here and hereafter, we set ${\mathbb {D}}(a,r)=\{z\in {\mathbb {C}}: |z-a|<r\}, {\overline{\mathbb {D}}}(a,r)=\{z\in {\mathbb {C}}: |z-a|\le r\}, ~{\mathbb {D}}^*(a,r)={\mathbb {D}}(a,r){\setminus }\{a\}$ for $a\in {\mathbb {C}}$ and $0<r<+\,\infty$ and ${\mathbb {D}}={\mathbb {D}}(0,1)$ and ${\mathbb {D}}^*={\mathbb {D}}^*(0,1).$ In this case, a holomorphic universal covering projection $p:{\mathbb {H}}\rightarrow {\mathbb {D}}^*$ is given by $z=p(\zeta )=e^{\pi i\zeta }$ and the hyperbolic density is expressed by

$$\begin{aligned} \lambda _{\mathbb {D}^*}(z)=\frac{1}{2|z| \log (1/|z|)}. \end{aligned}$$

A concrete formula of $h_{{\mathbb {D}}^*}(z_1,z_2)$ can also be given but its form is not so convenient [cf. (3.3) below].

In order to understand the hyperbolic distance $h_X(w_1,w_2)$ when one of $w_1,w_2$ is close to a puncture, we should take a careful look at the hyperbolic geodesic nearby the puncture. In Sect. 2, we investigate it by making use of an elliptic modular function as well as the punctured unit disk model. Section 3 is devoted to the study of the function $D(z_1,z_2).$ In particular, we show that D gives a distance on $0<|z|\le e^{-1}$ and compare with the hyperbolic distances $h_{{\mathbb {D}}^*}(z_1,z_2)$ of ${\mathbb {D}}^*$ and $h_{{\mathbb {C}}_{0,1}}(z_1,z_2)$ of ${\mathbb {C}}_{0,1}.$ As an application of the function $D(z_1,z_2),$ in Sect. 4, we will construct a distance function $d_X(w_1,w_2)$ on n-times punctured spheres X which are comparable with the hyperbolic distance $h_X(w_1,w_2).$ We will summarise our main results in Theorem 4.2. Unfortunately, $d_X(w_1,w_2)$ is not very easy to compute because we have to take an infimum in the definition. In Sect. 5, we introduce yet another quantity $e_X(w_1,w_2),$ which can be computed without taking an infimum though it is no longer a distance function on X. We will show our main result that $d_X(w_1,w_2)$ and $e_X(w_1,w_2)$ are both comparable with the hyperbolic distance $h_X(w_1,w_2)$ in a quantitative way in Sect. 5.

We would finally thank Matti Vuorinen for posing, more than ten years ago, the problem of finding a quantity comparable with the hyperbolic distance on ${\mathbb {C}}_{0,1}$ and for helpful suggestions.

2 Hyperbolic geodesics near the puncture

In order to estimate the hyperbolic distance $h_X(w_1,w_2)$ of a punctured sphere $X={\widehat{\mathbb {C}}}{\setminus }\{a_1,\dots ,a_n\},$ we have to investigate the behaviour of a hyperbolic geodesic joining two points near a puncture. Here and in what follows, a curve $\alpha$ joining $w_1$ and $w_2$ in a hyperbolic domain $\Omega$ is called a hyperbolic geodesic if $\ell _\Omega (\alpha )\le \ell _\Omega (\beta )$ whenever $\beta$ is a curve joining $w_1$ and $w_2$ which is homotopic to $\alpha$ in $\Omega .$ In particular, $\alpha$ is called shortest if $\ell _\Omega (\alpha ) =h_\Omega (w_1,w_2).$ Note that the shortest hyperbolic geodesic is not unique in general. Our basic model for that is the punctured disk ${\mathbb {D}}^*.$ In this case, we have precise information about the hyperbolic geodesic.

Lemma 2.1

Let $z_1, z_2\in {\mathbb {D}}^*$ with $\eta _1=-(\log |z_1|)/\pi \ge \eta _2=-(\log |z_2|)/\pi .$ Then a shortest hyperbolic geodesic $\beta$ joining $z_1$ and $z_2$ in ${\mathbb {D}}^*$ is contained in the set $e^{-\pi \delta }|z_1|\le |z|\le |z_2|,$ where

$$\begin{aligned} \delta =\sqrt{\eta _1^2+\frac{1}{4}}-\eta _1 =\frac{1}{4\big (\eta _1+\sqrt{\eta _1^2+1/4}\big )}. \end{aligned}$$

Proof

First note that the function $p(\zeta )=e^{\pi i\zeta }$ is a universal covering projection of the upper half-plane ${\mathbb {H}}$ onto ${\mathbb {D}}^*$ with period 2. We may assume that $z_1\in (0,1)$ and $\theta =({{\text {arg}}\,}z_2)/\pi \in (0,1].$ Then $p(i\eta _1)=z_1,~ p(i\eta _2+\theta )=z_2,$ and $h_{\mathbb {H}}(i\eta _1,i\eta _2+\theta )=h_{{\mathbb {D}}^*}(z_1,z_2).$ Let $\tilde{\beta }$ be the hyperbolic geodesic joining $\zeta _1=i\eta _1$ and $\zeta _2=i\eta _2+\theta$ in ${\mathbb {H}}.$ Recall that $\tilde{\beta }$ is part of the circle orthogonal to the real axis. If we fix $\eta _1,$ the possible largest imaginary part of $\tilde{\beta }$ is attained when $\eta _2=\eta _1$ and $\theta =1.$ Therefore, $\,{{\text {Im}}\,}\zeta \le \eta _1+\delta$ for $\zeta \in \tilde{\beta },$ where $\delta =\sqrt{\eta _1^2+1/4}-\eta _1.$ Hence, we conclude that $\beta =p(\tilde{\beta })$ is contained in the closed annulus $e^{-\pi (\eta _1+\delta )}=e^{-\pi \delta }|z_1|\le |z|\le |z_2|$ as required. $\square$

By the proof, we observe that the above constant $\delta$ is sharp. Note here that $\delta$ is decreasing in $\eta _1$ and that $0<\delta <\frac{1}{8\eta _1}.$

In the above theorem, we see that the subdomain ${\mathbb {D}}^*(0,\rho )$ of ${\mathbb {D}}^*$ with $0<\rho <1$ is hyperbolically convex. This is also true in general. Indeed, the following result is a special case of Minda’s reflection principle [13] (apply his Theorem 6 to the case when $\overline{R}=\overline{\Delta }$).

Lemma 2.2

Let $\Omega$ be a hyperbolic subdomain of ${\mathbb {C}}$ and let $\Delta$ be an open disk centered at a point $a\in {\mathbb {C}}{\setminus }\Omega .$ Suppose that $I(\Omega {\setminus }\Delta )\subset \Omega ,$ where I denotes the reflection in the circle $\partial \Delta .$ Then $\Delta \cap \Omega$ is hyperbolically convex in $\Omega .$

In particular, we have

Corollary 2.3

Let $\Delta$ be an open disk centered at a puncture a of a hyperbolic punctured sphere $X\subset {\mathbb {C}}$ with $\Delta ^*=\Delta {\setminus }\{a\}\subset X.$ Then $\Delta ^*$ is hyperbolically convex in X.

As another extremal case, we now consider the thrice-punctured sphere ${\mathbb {C}}_{0,1}.$ It is well known that the elliptic modular function, which is denoted by $J(\zeta ),$ on the upper half-plane ${\mathbb {H}}=\{\zeta :\,{{\text {Im}}\,}\zeta >0\}$ serves as a holomorphic universal covering projection onto ${\mathbb {C}}_{0,1}.$ The reader can consult [14, Chap. VI] for general facts about the function J and related functions. The covering transformation group is the modular group $\Gamma (2)$ of level 2; namely, $\Gamma (2)=\{A\in {{\text {SL}}}(2,{\mathbb {Z}}): A\equiv I~{{\text {mod}}\,}2\}/\{\pm I\}.$ Since J is periodic with period 2, it factors into $J=Q\circ p,$ where $p(\zeta )=e^{\pi i\zeta }$ as before and Q is an intermediate covering projection of ${\mathbb {D}}^*$ onto ${\mathbb {C}}_{0,1}.$ Since $J(\zeta )\rightarrow 0$ as $\eta =\,{{\text {Im}}\,}\zeta \rightarrow +\infty ,$ the origin is a removable singularity of Q(z) and indeed the function Q(z) has the following representations (see [14, VI.6] or [11, Theorem 14.2.2]):

$$\begin{aligned} Q(z)&=16z\prod _{n=1}^\infty \left( \frac{1+z^{2n}}{1+z^{2n-1}}\right) ^8 =16z\left[ \frac{\sum _{n=0}^\infty z^{n(n+1)}}{1+2\sum _{n=1}^\infty z^{n^2}}\right] ^4\\&=16(z-8z^2+44z^3-192z^4+718z^5-\cdots ). \end{aligned}$$

We also remark that the function Q(z) has been recently used to improve coefficient estimates of univalent harmonic mappings on the unit disk in Abu Muhanna, Ali and Ponnusamy [1]. By its form, Q(z) is locally univalent at $z=0.$ In fact, we can show the following.

Lemma 2.4

The function Q(z) is univalent on the disk $|z|<e^{-\pi /2}\approx 0.20788.$ The radius $e^{-\pi /2}$ is best possible.

Proof

Suppose that $Q(z_1)=Q(z_2)$ for a pair of points $z_1,z_2$ in the disk $|z|<e^{-\pi /2}.$ Take the unique point $\zeta _l=\xi _l+i\eta _l\in p^{-1}(z_l)\subset {\mathbb {H}}$ such that $-1<\xi _l\le 1$ and $\eta _l>1/2$ for $l=1,2.$ We now recall the well-known fact that $\omega =\{\zeta \in {\mathbb {H}}: -\,1<{{\text {Re}}\,}\zeta \le 1, |\zeta +1/2|<1/2, |\zeta -1/2|\le 1/2\}$ is a fundamental set for the modular group $\Gamma (2)$ of level 2 (see [4, Chap.7, §3.4]). In other words, $J(\omega )={\mathbb {C}}_{0,1}$ and no pairs of distinct points in $\omega$ have the same image under the mapping J. We now note that $\zeta _l\in \omega$ for $l=1.2.$ Since $J(\zeta _1)=Q(z_1)=Q(z_2)=J(\zeta _2),$ we conclude that $\zeta _1=\zeta _2.$ Hence, $z_1=z_2,$ which implies that Q(z) is univalent on $|z|<e^{-\pi /2}.$ To see sharpness, we consider the pair of points $\zeta _1=(1+i)/2$ and $\zeta _2=(-1+i)/2.$ Since $\zeta _2=T(\zeta _1)$ for the modular transformation $T(\zeta )=\zeta /(-2\zeta +1)$ in $\Gamma (2),$ we have $J(\zeta _1)=J(\zeta _2).$ Thus $Q(ie^{-\pi /2})=Q(-ie^{-\pi /2}).$ $\square$

We remark that the formula $Q(ie^{-\pi /2})=2$ is valid. Indeed, by recalling the functional identity $Q(-z)=Q(z)/(Q(z)-1)$ (see [14, (92) in p. 328]), we have $Q(ie^{-\pi /2})=Q(-ie^{-\pi /2}) =Q(ie^{-\pi /2})/(Q(ie^{-\pi /2})-1)$, which implies $Q(ie^{-\pi /2})=2.$

We now recall the growth theorem for a normalized univalent analytic function $f(z)=z+a_2z^2+\dots$ on $|z|<1$ (see [3, §5–1] for instance):

$$\begin{aligned} \frac{r}{(1+r)^2}\le |f(z)|\le \frac{r}{(1-r)^2},\quad |z|<1. \end{aligned}$$

Applying this result to $f(z)=e^{\pi /2}Q(ze^{-\pi /2})/16,$ we obtain the following estimates:

$$\begin{aligned} \frac{16r}{(1+re^{\pi /2})^2}\le |Q(z)|\le \frac{16r}{(1-re^{\pi /2})^2}, \quad r=|z|<e^{-\pi /2}. \end{aligned}$$

(2.1)

Observe that the lower bound in (2.1) tends to $4e^{-\pi /2}$ as $r\rightarrow e^{-\pi /2}.$ Hence ${\mathbb {D}}(0,4e^{-\pi /2})\subset Q({\mathbb {D}}(0,e^{-\pi /2})).$ Note that $4e^{-\pi /2}\approx 0.8315.$

Lemma 2.5

Let ${\mathbb {C}}_{0,1}.$ For $w_1, w_2\in {\mathbb {C}}_{0,1}$ with $|w_1|, |w_2|\le \rho \le 4e^{-\pi /2},$ a shortest hyperbolic geodesic $\alpha$ joining $w_1,w_2$ in ${\mathbb {C}}_{0,1}$ is contained in the closed annulus

$$\begin{aligned} \min \{|w_1|,|w_2|\}e^{-K}\le |w|\le \max \{|w_1|,|w_2|\}, \end{aligned}$$

where $K=K(\rho )>0$ is a constant depending only on $\rho .$

Proof

The right-hand inequality follows from Corollary 2.3. We now show the left-hand inequality. Take $0<r\le e^{-\pi /2}$ so that $16r/(1+re^{\pi /2})^2=\rho$ and put $\mu =re^{\pi /2}\le 1.$ Note that r and $\mu$ can be computed by the formula

$$\begin{aligned} \mu =re^{\pi /2} =\rho ^{-1}e^{-\pi /2}\big (8-\rho e^{\pi /2}-4\sqrt{4-\rho e^{\pi /2}}\big ). \end{aligned}$$

By (2.1), we can choose $z_j\in {\mathbb {D}}^*$ with $|z_j|\le r$ and $Q(z_j)=w_j$ for $j=1,2$ in such a way that a lift $\beta$ of the curve $\alpha$ joins $z_1$ and $z_2$ in ${\mathbb {D}}^*$ via the covering map Q. We may assume that $|z_1|\le |z_2|.$ It follows from Lemma 2.1 that $\beta$ is contained in the annulus $|z_1|e^{-\pi \delta }\le |z|\le |z_2|,$ where $\delta$ is given in the lemma with $\eta _1=-(\log r)/\pi \le -(\log |z_1|)/\pi .$ Hence, by (2.1), the curve $\alpha =Q(\beta )$ is contained in the annulus

$$\begin{aligned} \frac{16|z_1|e^{-\pi \delta }}{(1+|z_1|e^{\pi /2-\pi \delta })^2}\le |w|\le \frac{16|z_2|}{(1-|z_2|e^{\pi /2})^2}. \end{aligned}$$

By (2.1) and $|z_1|e^{\pi /2}\le \mu ,$ we see that

$$\begin{aligned} |w_1|\le & {} \frac{16|z_1|}{(1-|z_1|e^{\pi /2})^2} =\frac{16|z_1|e^{-\pi \delta }}{(1+|z_1|e^{\pi /2-\pi \delta })^2} \cdot \frac{e^{\pi \delta }(1+|z_1|e^{\pi /2-\pi \delta })^2}{(1-|z_1|e^{\pi /2})^2}\\\le & {} \frac{16|z_1|e^{-\pi \delta }}{(1+|z_1|e^{\pi /2-\pi \delta })^2} e^K, \end{aligned}$$

where

$$\begin{aligned} K=K(\rho )=\pi \delta +2\log \frac{1+\mu e^{-\pi \delta }}{1-\mu }. \end{aligned}$$

(2.2)

Thus we have seen that $\alpha$ is contained in the set $|w_1|e^{-K}\le |w|.$ $\square$

As an immediate consequence, we obtain the following result.

Corollary 2.6

Let $0<\sigma \le \rho \le 4e^{-\pi /2}.$ A shortest hyperbolic geodesic $\alpha$ joining $z_1,z_2$ in ${\mathbb {C}}_{0,1}$ with $|z_1|,|z_2|\ge \sigma$ does not intersect the disk $\Delta ={\mathbb {D}}(0,\sigma e^{-K}),$ where $K=K(\rho )>0$ is the constant in Lemma 2.5.

Proof

Suppose that $\alpha$ intersects the disk $\Delta .$ Then we can choose a subarc $\alpha _0$ of $\alpha$ such that $\alpha _0$ intersects $\Delta$ and that both endpoints $w_1,w_2$ of $\alpha _0$ have modulus $\sigma .$ Applying Lemma 2.5 to $\alpha _0$ yields a contradiction.

When $\rho =\rho _0=e^{-1},$ we compute $r_0=e^{1-\pi }\big (8-e^{\pi /2-1}-4\sqrt{4-e^{\pi /2-1}}\big ) \approx 0.0301441$ and $\mu _0=r_0 e^{\pi /2}\approx 0.145007.$ Also, we have $\delta _0=\sqrt{\eta _0^2+1/4}-\eta _0\approx 0.107007$ with $\eta _0=-(\log r_0)/\pi \approx 1.11465.$ Then $K_0=K(e^{-1})=\pi \delta _0+2\log [(1+\mu _0e^{-\pi \delta _0})/(1-\mu _0)]\approx 0.846666$ and $e^{K_0}\approx 2.33186.$ Thus we obtain the following statement as a special case of Corollary 2.6.

Corollary 2.7

Let $z_1, z_2$ be two points in ${\mathbb {C}}_{0,1}$ with $|z_1|, |z_2|\ge \sigma$ for some number $\sigma \in (0, e^{-1}].$ Then, a shortest hyperbolic geodesic $\alpha$ joining $z_1,z_2$ in ${\mathbb {C}}_{0,1}$ does not intersect the disk $|z|<\sigma e^{-K_0},$ where $K_0=K(e^{-1})\approx 0.846666.$

3 Basic properties of the distance function $D(w_1,w_2)$

First we show the following result in the present section. Let $E^*=\{z: 0<|z|\le e^{-1}\}.$

Lemma 3.1

The function $D(z_1, z_2)$ given by (1.3) is a distance function on the set $E^*.$

Proof

First we note that $D(z_1, z_2)$ can also be described by

$$\begin{aligned} D(z_1,z_2)=\frac{|\zeta _1-\zeta _2|}{\max \{\tau _1, \tau _2\}} +|\log \tau _1-\log \tau _2|, \end{aligned}$$

where $\tau _j=\log (1/|z_j|)$ and $\zeta _j=z_j/|z_j|$ for $j=1,2.$ It is easy to see that $D(z_1,z_2)=D(z_2,z_1)$ and that $D(z_1,z_2) \ge 0,$ where equality holds if and only if $z_1=z_2$. It remains to verify the triangle inequality. Our task is to show that the inequality $\Delta :=D(z_1,z)+D(z,z_2)-D(z_1, z_2)\ge 0$ for $z\in E^*.$ Set $\tau =\log (1/|z|)$ and $\zeta =z/|z|.$ We may assume that $\tau _2\le \tau _1.$ Then

$$\begin{aligned} D(z_1, z_2) =\frac{|\zeta _1-\zeta _2|}{\tau _1} +\log \tau _1-\log \tau _2. \end{aligned}$$

First we assume that $\tau \le \tau _1.$ Since $\tau _1\ge \max \{\tau , \tau _2\},$ we have

$$\begin{aligned} D(z_1,z_2)&\le \frac{|\zeta _1-\zeta |+|\zeta -\zeta _2|}{\tau _1} +|\log \tau _1-\log \tau |+|\log \tau -\log \tau _2| \\&= D(z_1,z)+D(z,z_2). \end{aligned}$$

Secondly, we assume that $\tau >\tau _1.$ Then, in a similar manner, we have

$$\begin{aligned} \Delta&\ge \left( \frac{1}{\tau }-\frac{1}{\tau _1}\right) |\zeta _1-\zeta _2|+2\log \tau -2\log \tau _1 \\&\ge \frac{2}{\tau }-\frac{2}{\tau _1}+2\log \tau -2\log \tau _1. \end{aligned}$$

Since the function $f(x)=2/x+2\log x$ is increasing in $1\le x<+\infty ,$ we have $\Delta \ge 0$ as required.

Next we compare $D(z_1,z_2)$ with the hyperbolic distance $h_{{\mathbb {D}}^*}(z_1,z_2)$ of ${\mathbb {D}}^*$ on the set $E^*.$

Theorem 3.2

The distance function $D(z_1, z_2)$ given by (1.3) satisfies

$$\begin{aligned} \frac{4}{\pi }\,h_{{\mathbb {D}}^*}(z_1,z_2) \le D(z_1,z_2) \le M_0\, h_{{\mathbb {C}}_{0,1}}(z_1,z_2) \end{aligned}$$

(3.1)

for $0<|z_1|, |z_2|\le e^{-1}.$ Here, ${\mathbb {C}}_{0,1}={\mathbb {C}}{\setminus }\{0,1\},$ $M_0$ is a positive constant with $M_0<24$ and the constant $4/\pi$ is sharp.

We remark that $4/\pi \approx 1.27324.$

Proof

We consider the quantity

$$\begin{aligned} D'(z_1, z_2)=\frac{\theta }{\max \{\log (1/|z_1|),\, \log (1/|z_2|)\}} +\left| \log \log \frac{1}{|z_2|}-\log \log \frac{1}{|z_1|}\right| \end{aligned}$$

for $z_1, z_2\in {\mathbb {D}}^*,$ where $\theta =|{{\text {arg}}\,}(z_2/z_1)|\in [0,\pi ].$ Since $2x/\pi \le \sin x\le x$ for $0\le x\le \pi /2,$ we can easily obtain

$$\begin{aligned} \frac{2}{\pi } D'(z_1,z_2) \le D(z_1,z_2) \le D'(z_1,z_2) \end{aligned}$$

for $z_1, z_2 \in E^*$. We show now the inequality

$$\begin{aligned} 2h_{{\mathbb {D}}^*}(z_1,z_2) \le D'(z_1,z_2) \end{aligned}$$

(3.2)

for $z_1, z_2\in E^*.$ Combining these two inequalities, we obtain the first inequality in (3.1).

Without loss of generality, we may assume that ${{\text {arg}}\,}z_1 =0$, ${{\text {arg}}\,}z_2 = \theta \in [0, \pi ]$, and $\tau _2 \le \tau _1,$ where $\tau _j=\log (1/|z_j|).$ Recall that $p(\zeta )=e^{\pi i\zeta }$ is a holomorphic universal covering projection of the upper half-plane ${\mathbb {H}}$ onto ${\mathbb {D}}^*.$ Let $\zeta _1=i\tau _1/\pi$ and $\zeta _2=(\theta +i\tau _2)/\pi .$ Then $p(\zeta _j)=z_j$ for $j=1,2$ and, by (1.1),

$$\begin{aligned} h_{{\mathbb {D}}^*}(z_1,z_2)=h_{\mathbb {H}}(\zeta _1,\zeta _2) ={{\text {arth}}\,}\left| \frac{\zeta _1-\zeta _2}{\zeta _1-\overline{\zeta _2}}\right| ={{\text {arth}}\,}\sqrt{\frac{\theta ^2+(\tau _1-\tau _2)^2}{\theta ^2+(\tau _1+\tau _2)^2}}. \end{aligned}$$

(3.3)

We consider the function

$$\begin{aligned} G(\theta , \tau _1,\tau _2)&= 2h_{{\mathbb {D}}^*}(z_1,z_2)-D'(z_1,z_2) \\&=2{{\text {arth}}\,}\sqrt{\frac{\theta ^2+(\tau _1-\tau _2)^2}{\theta ^2+(\tau _1+\tau _2)^2}} -\frac{\theta }{\tau _1}-\log \tau _1+\log \tau _2 \end{aligned}$$

on $0\le \theta \le \pi , 1\le \tau _2\le \tau _1.$ A straightforward computation yields

$$\begin{aligned} \frac{\partial G}{\partial \tau _2}=\frac{1}{\tau _2} -\frac{\tau _1^2-\tau _2^2+\theta ^2}{\tau _2\sqrt{(\theta ^2+\tau _2^2+\tau _1^2)^2-4\tau _1^2 \tau _2^2}}. \end{aligned}$$

Since

$$\begin{aligned} \left[ (\theta ^2+\tau _2^2+\tau _1^2)^2-4\tau _1^2 \tau _2^2\right] -(\tau _1^2-\tau _2^2+\theta ^2)^2 =4\tau _2^2\theta ^2>0 \end{aligned}$$

for $0<\theta <\pi ,$ we see that $\partial G/\partial \tau _2>0$ and therefore $G(\theta ,\tau _1,\tau _2)$ is increasing in $1\le \tau _2\le \tau _1$ so that

$$\begin{aligned} G(\theta ,\tau _1,1)<G(\theta ,\tau _1,\tau _2)<G(\theta ,\tau _1,\tau _1) \end{aligned}$$

for $1<\tau _2<\tau _1.$ We observe that $G(\theta ,\tau _1,\tau _1)=2f(2\tau _1/\theta ),$ where

$$\begin{aligned} f(x)={{\text {arth}}\,}\frac{1}{\sqrt{1+x^2}}-\frac{1}{x} =\frac{g(x)-1}{x}, \end{aligned}$$

and $g(x)=x\,{{\text {arth}}\,}(1/\sqrt{1+x^2}).$ Since $g'(x)={{\text {arth}}\,}(1/\sqrt{1+x^2})-1/\sqrt{1+x^2}>0$ for $x>0,$ we have $g(x)<g(+\infty )=1.$ Hence $G(\theta ,\tau _1,\tau _1)<0$ for $0<\theta <\pi , \tau _1>1.$ We have thus proved the inequality $2h_{{\mathbb {D}}^*}(z_1,z_2)\le D'(z_1,z_2).$ Since $h_{{\mathbb {D}}^*}(e^{-\tau },-e^{-\tau })=h_{\mathbb {H}}(i\tau ,i\tau +\pi ) ={\,{\text {arth}}\,}(\pi /\sqrt{\pi ^2+4\tau ^2}),$

$$\begin{aligned} \frac{h_{{\mathbb {D}}^*}(e^{-\tau },-e^{-\tau })}{D(e^{-\tau },-e^{-\tau })} =\frac{\tau }{2}\,{{\text {arth}}\,}\frac{\pi }{\sqrt{\pi ^2+4\tau ^2}}\rightarrow \frac{\pi }{4} \end{aligned}$$

as $\tau \rightarrow +\infty .$ Hence the constant $4/\pi$ is sharp in (3.1).

Finally, we show the second inequality in (3.1). Assume that $0<|z_1|\le |z_2|\le e^{-1}, \theta _0={{\text {arg}}\,}(z_2/z_1)\in [0,\pi ]$ and $\tau _j=-\log |z_j|\ge 1$ for $j=1,2.$ We now show the following two inequalities to complete the proof:

$$\begin{aligned} \frac{\theta _0}{\tau _1}\le M_1H {\quad \text {and}\quad }\log \frac{\tau _1}{\tau _2}\le M_2H \end{aligned}$$

(3.4)

for some constants $M_1$ and $M_2,$ where $H=h_{{\mathbb {C}}_{0,1}}(z_1,z_2).$ Let $\alpha$ be a shortest hyperbolic geodesic joining $z_1$ and $z_2$ in ${\mathbb {C}}_{0,1}.$ Then, by Corollary 2.7, $\alpha$ is contained in the annulus $|z_1|e^{-K_0}\le |z|\le |z_2|.$ Thus $-\log |z|\le K_0+\tau _1$ for $z\in \alpha .$ We recall the following lower estimate of $\lambda (z)=\lambda _{{\mathbb {C}}_{0,1}}(z)$ (see [9]):

$$\begin{aligned} \frac{1}{2|z|(C_0+|\log |z||)}\le \lambda (z),\quad z\in {\mathbb {C}}_{0,1}, \end{aligned}$$

(3.5)

where $C_0=1/2\lambda (-1)=\Gamma (1/4)^4/4\pi ^2\approx 4.37688.$ We remark that the bound in (3.5) is monotone decreasing in |z|. Noting the inequality $|dz|\ge rd\theta$ for $z=re^{i\theta },$ we have

$$\begin{aligned} H=\int _\alpha \lambda (z)|dz| \ge \int _\alpha \frac{rd\theta }{2r(C_0-\log r)} \ge \int _\alpha \frac{d\theta }{2(C_0+\tau _1+K_0)} =\frac{\theta _0}{2(C_0+K_0+\tau _1)}. \end{aligned}$$

Since $C_0+K_0+\tau _1\le (C_0+K_0+1)\tau _1,$ we obtain the first inequality in (3.4) with $M_1=2(C_0+K_0+1).$ Similarly, by using $|dz|\ge |dr|$ for $z=re^{i\theta }=e^{-t+i\theta },$ we obtain

$$\begin{aligned} H\ge \int _\alpha \frac{|dr|}{2r(C_0-\log r)} \ge \int _{\tau _2}^{\tau _1}\frac{dt}{2(C_0+t)} \ge \frac{1}{2(C_0+1)}\log \frac{\tau _1}{\tau _2}. \end{aligned}$$

Hence we have the second inequality in (3.4) with $M_2=2(C_0+1).$ Combining the two inequalities in (3.4), we get

$$\begin{aligned} D(z_1,z_2)\le D'(z_1,z_2)\le (M_1+M_2)H. \end{aligned}$$

Hence, $M_0=M_1+M_2=4C_0+4+2K_0\approx 23.2008$ works. $\square$

4 Construction of a distance function on n-times punctured sphere

In this section, we construct a distance function on an n-times punctured sphere $X={\widehat{\mathbb {C}}}{\setminus }\{a_1,\dots , a_n\}.$ As we noted, X is hyperbolic if and only if $n\ge 3.$ Thus we will assume that $n\ge 3$ in the sequel. After a suitable Möbius transformation, without much loss of generality, we may assume that $0,1,\infty \in {\widehat{\mathbb {C}}}{\setminus } X$ and $a_n =\infty$. Let

$$\begin{aligned} \tilde{\rho }_j={\left\{ \begin{array}{ll} \displaystyle \min _{1\le k<n, k\ne j}|a_k-a_j|&{}\quad \text {for}~j=1,2,\dots , n-1, \\ \displaystyle \max _{1<k<n}|a_k| &{}\quad \text {for}~j=n \end{array}\right. } \end{aligned}$$

and $\rho _j=\tilde{\rho }_j/e$ for $j=1,2,\dots , n-1$ and $\rho _n=e\tilde{\rho }_n.$ Since $a_1=0,$ we have $e\rho _j\le |a_j|$ for $1<j<n$ and $e\rho _j\le \max |a_k|=\rho _n/e$ for $j<n.$ Set $E_{j}={\overline{\mathbb {D}}}(a_j,\rho _j), E^*_j=E_j{\setminus }\{a_j\}$ and $\Delta _{j}={\mathbb {D}}^*(a_j,\tilde{\rho }_j)$ for $j=1,\dots , n-1,$ and set $E_{n}=\{w\in {\widehat{\mathbb {C}}}: |w|\ge \rho _n\},~E^*_n=E_n{\setminus }\{\infty \}$ and $\Delta _{n}=\{w\in {\mathbb {C}}: |w|>\tilde{\rho }_n\}.$ It is easy to see that the Euclidean distance between $E_j$’s are computed and estimated by

$$\begin{aligned} {{\text {dist}}}(E_j,E_k)=|a_j-a_k|-\rho _j-\rho _k \ge (e-2)\max \{\rho _j,\rho _k\} \end{aligned}$$

for $j,k<n,$ and

$$\begin{aligned} {{\text {dist}}}(E_j,E_n)=\rho _n-|a_j|-\rho _j\ge (1-e^{-1}-e^{-2})\rho _n\ge (e^2-e-1)\rho _j>(e-2)\rho _j \end{aligned}$$

for $1\le j<n.$ In particular, $E^*_j$’s are mutually disjoint. Noting the inequality $1-e^{-1}-e^{-2}\approx 0.49678<e-2,$ we also have the estimate

$$\begin{aligned} {{\text {dist}}}(E_j,E_k)\ge (1-e^{-1}-e^{-2})\rho _j \end{aligned}$$

(4.1)

for any pair of distinct j, k. Note also that $\Delta _j\subset X$ for $j=1,\dots , n.$ Finally, let $W=X{\setminus } (E^*_1\cup \dots \cup E^*_n)$.

We are now ready to construct a distance function on X. Set

$$\begin{aligned} D_j(w_1, w_2)= {\left\{ \begin{array}{ll} \rho _jD((w_1-a_j)/\tilde{\rho }_j,(w_2-a_j)/\tilde{\rho }_j) &{} \text {if}\quad j=1,\dots , n-1, \\ \rho _n D(\tilde{\rho }_n/w_1,\tilde{\rho }_n/w_2) &{} \text {if}\quad j=n \end{array}\right. } \end{aligned}$$

for $w_1, w_2\in E^*_j,$ where $D(z_1,z_2)$ is given in (1.3). By definition, we have $D_j(w_1,w_2)=|w_1-w_2|$ for $w_1$, $w_2 \in \partial E_j$. By Lemma 3.1, we know that $D_j(w_1, w_2)$ is a distance function on $E^*_{j}.$ We further define

$$\begin{aligned} d_X(w_1, w_2)= {\left\{ \begin{array}{ll} \displaystyle D_j(w_1, w_2) &{} \text{ if }\quad \ w_1,\, w_2 \in E^*_{j}, \\ \displaystyle \inf _{\zeta \in \partial E_j} (D_j(w_1, \zeta ) +|\zeta -w_2|) &{}\text{ if }\quad \ w_1 \in E^*_{j}, \, w_2 \in W, \\ \displaystyle \inf _{\zeta \in \partial E_j} (|w_1-\zeta | +D_j(\zeta ,w_2)) &{}\text{ if }\quad \ w_1 \in W,\, w_2\in E^*_{j}, \\ \displaystyle \inf _{\begin{array}{c} \zeta _1 \in \partial E_j\\ \zeta _2 \in \partial E_k \end{array}} (D_j(w_1,\zeta _1) +|\zeta _1-\zeta _2|+D_k(\zeta _2, w_2) ) &{}\text{ if }\quad \ w_1 \in E^*_{j}, \, w_2 \in E^*_{k},\, j\ne k,\\ \displaystyle |w_1-w_2| &{} \text{ if }\quad \ w_1,\, w_2 \in W \end{array}\right. } \end{aligned}$$

(4.2)

for $w_1, w_2\in X.$ Note that the infima in the above definition can be replaced by minima. Then we have the following result.

Lemma 4.1

$d_X$ is a distance function on X.

Proof

It is easy to see that $d_X(w_1,w_2)=d_X(w_2,w_1)$, and $d_X(w_1,w_2) \ge 0,$ where equality holds if and only if $w_1=w_2$, for $w_1, w_2 \in X$. It remains to verify the triangle inequality: $d_X(w_1,w_2)\le d_X(w_1,w_3)+d_X(w_3,w_2).$ According to the location of these points, we need to consider several cases. For instance, we consider the case when $w_1 \in E^*_{j}$, $w_2 \in E^*_{k}$ and $w_3\in E^*_l$ for distinct j, k, l. Then,

$$\begin{aligned}&d_X(w_1,w_3)+d_X(w_3,w_2)\\= & {} \inf _{\begin{array}{c} \zeta _1\in \partial E_j,\zeta _2 \in \partial E_k\\ \zeta _3,\zeta _4\in \partial E_l \end{array}} (D_j(w_1,\zeta _1)+|\zeta _1-\zeta _3|+D_l(\zeta _3,w_3)+D_l(w_3,\zeta _4)\\&+\,|\zeta _4-\zeta _2|+D_k(\zeta _2, w_2))\\\ge & {} \inf (D_j(w_1,\zeta _1)+|\zeta _1-\zeta _3|+D_l(\zeta _3,\zeta _4) +|\zeta _4-\zeta _2|+D_k(\zeta _2,w_2))\\= & {} \inf (D_j(w_1,\zeta _1)+|\zeta _1-\zeta _3|+|\zeta _3-\zeta _4| +|\zeta _4-\zeta _2|+D_k(\zeta _2,w_2))\\\ge & {} \inf _{\zeta _1 \in \partial E_j,\zeta _2 \in \partial E_k} (D_j(w_1,\zeta _1)+|\zeta _1-\zeta _2|+D_k(\zeta _2,w_2))= d_X(w_1,w_2). \end{aligned}$$

The other cases can be handled similarly and therefore will be omitted. $\square$

We remark that we can construct a similar distance when $n=2.$ Let $a_1=0$ and $a_2=\infty$ and consider $X={\widehat{\mathbb {C}}}{\setminus }\{a_1,a_2\} ={\mathbb {C}}^*.$ Then, we set

$$\begin{aligned} d_{{\mathbb {C}}^*}(w_1,w_2)= {\left\{ \begin{array}{ll} D(w_1/e, w_2/e) &{}\text {if}\quad 0<|w_1|\le 1, 0<|w_2|\le 1, \\ D(1/(ew_1), 1/(ew_2)) &{} \text {if}\quad 1\le |w_1|, 1\le |w_2|, \\ \displaystyle \inf _{|\zeta |=1} (D(w_1/e, \zeta /e)+D(1/(e\zeta ),1/(ew_2)) &{}\text {if}\quad 0<|w_1|\le 1, 1\le |w_2|, \\ \displaystyle \inf _{|\zeta |=1} (D(1/(ew_1), 1/(e\zeta ))+D(\zeta /e,w_2/e) &{}\text {if}\quad 1\le |w_1|, 0<|w_2|\le 1, \end{array}\right. } \end{aligned}$$

where D is defined by (1.3). Then we can see that $d_{{\mathbb {C}}^*}$ is a distance function on ${\mathbb {C}}^*.$ The asymptotic behaviour of $d_{{\mathbb {C}}^*}$ near the punctures are rather different from that of the quasi-hyperbolic distance q on ${\mathbb {C}}^*$ since $q(w_1,w_2)=|\log |w_1|-\log |w_2||+O(1)$ as $w_1,w_2\rightarrow 0$ (see [12] for instance).

Our main result in the present paper is the following. In the next section, we will prove it in a stronger form (Theorem 5.1).

Theorem 4.2

The distance function $d_X(w_1, w_2)$ given in (4.2) on the n-times punctured sphere $X={\widehat{\mathbb {C}}}{\setminus }\{a_1,\dots ,a_n\} \subset {\mathbb {C}}_{0,1}$ is comparable with the hyperbolic distance $h_X(w_1,w_2)$ on X.

5 Proof of Theorem 4.2

We recall that $X={\widehat{\mathbb {C}}}{\setminus } \{a_1,\dots , a_n\}\subset {\mathbb {C}}_{0,1}$ with $a_n=\infty .$ The function $d_X$ defined in the previous section has the merit that it gives a distance on X. On the other hand, it is not easy to compute the exact value of $d_X(w_1,w_2)$ for a given pair of points $w_1, w_2\in X.$ The following quantity can be a good substitute of $d_X(w_1,w_2)$ because it is computed easily, though it is not necessarily a distance function:

$$\begin{aligned} e_X(w_1, w_2)= {\left\{ \begin{array}{ll} \displaystyle D_j(w_1, w_2) &{} \text{ if }\quad \ w_1,\, w_2 \in E^*_{j}, \\ \displaystyle D_j(w_1, \zeta ) +|\zeta -w_2| &{}\text{ if }\quad \ w_1 \in E^*_{j}, \, w_2 \in W \\ \displaystyle |w_1-\zeta | +D_j(\zeta ,w_2) &{}\text{ if }\quad \ w_1 \in W,\, w_2\in E^*_{j}, \\ \displaystyle D_j(w_1,\zeta _1) +|\zeta _1-\zeta _2|+D_k(\zeta _2, w_2) &{}\text{ if }\quad \ w_1 \in E^*_{j}, \, w_2 \in E^*_{k},\, j\ne k,\\ \displaystyle |w_1-w_2| &{} \text{ if }\quad \ w_1,\, w_2 \in W \end{array}\right. } \end{aligned}$$

for $w_1, w_2\in X,$ where $\zeta$ is the intersection point of the line segment $[w_1,w_2]$ with the circle $\partial E_j$ in the second case, and $\zeta _1$ and $\zeta _2$ are the intersection points of the line segments $[w_1,w_2]$ with the circles $\partial E_j$ and $\partial E_k,$ respectively, in the third case. By definition, the inequality $d_X(w_1,w_2)\le e_X(w_1,w_2)$ holds obviously. Theorem 4.2 now follows from the next result.

Theorem 5.1

There exist positive constants $N_1$ and $N_2$ such that the following inequalities hold:

$$\begin{aligned} N_1h_X(w_1,w_2)\le d_X(w_1,w_2)\le e_X(w_1,w_2)\le N_2h_X(w_1,w_2) \end{aligned}$$

for $w_1,w_2\in X={\widehat{\mathbb {C}}}{\setminus }\{a_1,a_2,\dots ,a_n\}\subset {\mathbb {C}}_{0,1}.$

Proof

Assume that $a_1=0$ and $a_n=\infty$ as before. We recall that $W=X{\setminus }(E^*_1\cup \dots \cup E^*_n).$ Since $\lambda _X(w)\delta _X(w)\le 1,$ we obtain

$$\begin{aligned} \lambda _X(w)\le \frac{1}{m},\quad w\in \overline{W}, \end{aligned}$$

(5.1)

where $m=\min _{1\le j<n}\rho _j.$ We show the first inequality. Fix $w_1, w_2\in X$ and assume that $w_1\in E^*_j$ and $w_2\in W$ for some j. We can deal with the other cases similarly and thus we will omit it. We further assume, for a moment, that $j\ne n.$ By definition,

$$\begin{aligned} d_X(w_1,w_2)=D_j(w_1,\zeta _0)+|\zeta _0-w_2| \end{aligned}$$

for some $\zeta _0\in \partial E_j.$ If the line segment $L=[\zeta _0,w_2]$ intersects $E^*_k$ for some $k\ne j,$ we replace the part $L\cap {\mathbb {D}}(a_k,\rho _k)$ of L by the shorter component of $\partial E_k{\setminus } L$ for each such k. The resulting curve will be denoted by $L'.$ It is obvious from construction that the Euclidean length of $L'$ is bounded by $\pi |\zeta _0-w_2|/2.$ Therefore, by (5.1),

$$\begin{aligned} |\zeta _0-w_2|\ge \frac{2}{\pi }\int _{L'}|dw| \ge \frac{2m}{\pi }\int _{L'}\lambda _X(w)|dw| \ge \frac{2m}{\pi }h_X(\zeta _0,w_2). \end{aligned}$$

Let $z_1=g(w_1)$ and $z_2=g(\zeta _0),$ where $g(w)=(w-a_j)/\tilde{\rho }_j.$ Then, by the definition of $D_j$ and Theorem 3.2,

$$\begin{aligned} D_j(w_1,\zeta _0)=\rho _j D(z_1,z_2)\ge \frac{4\rho _j}{\pi } h_{{\mathbb {D}}^*}(z_1,z_2). \end{aligned}$$

Since g maps $\Delta _j$ conformally onto ${\mathbb {D}}^*,$ the principle of the hyperbolic metric leads to the following:

$$\begin{aligned} h_{{\mathbb {D}}^*}(z_1,z_2)=h_{\Delta _j}(w_1,\zeta _0)\ge h_X(w_1,\zeta _0). \end{aligned}$$

When $j=n,$ with $g(w)=\tilde{\rho }_n/w,$ we have the estimate $D_n(w_1,\zeta _0)\ge (4\rho _n/\pi )h_X(w_1,\zeta _0)$ in a similar way. Since $4\rho _j/\pi \ge 2m/\pi =2\min \rho _k/\pi ,$ we obtain

$$\begin{aligned} d_X(w_1,w_2)\ge N_1\big [ h_X(w_1,\zeta _0)+h_X(\zeta _0,w_2)\big ] \ge N_1h_X(w_1,w_2), \end{aligned}$$

where $N_1=2m/\pi .$ Similarly, we get the first inequality in the other cases with the same constant $N_1.$ Thus the first inequality has been shown.

We next show the inequality $e_X(w_1,w_2)\le N_2h_X(w_1,w_2).$ We consider several cases according to the location of $w_1,w_2.$

Case (i) $w_1, w_2\in E^*_j:$ We first assume that $j\ne n$ and choose $k\ne j$ so that $\tilde{\rho }_j=|a_k-a_j|.$ Let $X_1={\mathbb {C}}{\setminus }\{a_j,a_k\}.$ Then $X_1\supset X$ and $g(w)=(w-a_j)/(a_k-a_j)$ maps $X_1$ conformally onto ${\mathbb {C}}_{0,1}.$ Set $z_1=g(w_1)$ and $z_2=g(w_2).$ By Theorem 3.2, we obtain

$$\begin{aligned} D_j(w_1,w_2)= & {} \rho _j D(z_1,z_2)\le M_0\rho _jh_{{\mathbb {C}}_{0,1}}(z_1,z_2) =M_0\rho _jh_{X_1}(w_1,w_2)\\\le & {} M_0\rho _jh_X(w_1,w_2), \end{aligned}$$

and thus $e_X(w_1,w_2)\le M_0 \rho _j h_X(w_1,w_2).$ When $j=n,$ we set $g(w)=a_k/w,$ where $a_k$ is chosen so that $\tilde{\rho }_n=|a_k|.$ Then, we also have the estimate $D_n(w_1,w_2)\le M_0\rho _n h_X(w_1,w_2).$ In summary, we have $D_j(w_1,w_2)=e_X(w_1,w_2)\le M_0\rho _{n} h_X(w_1,w_2)$ for $j=1,\dots ,n,$ because $\rho _j\le e^{-2}\rho _n<\rho _n.$

Case (ii) $w_1\in E^*_j$ and $w_2\in W:$ Let $\zeta$ be the intersection point of the line segment $[w_1,w_2]$ with the boundary circle $\partial E_j.$ It is thus enough to show the inequalities

$$\begin{aligned} D_j(w_1,\zeta )\le B_1h_{X}(w_1,w_2) {\quad \text {and}\quad }|\zeta -w_2|\le B_2h_{X}(w_1,w_2) \end{aligned}$$

(5.2)

for some constants $B_1$ and $B_2.$

We start with the second one. Assume that $j\ne n$ for a while. Then the function $g(w)=a_j/w$ maps X conformally into ${\mathbb {C}}_{0,1}.$ (When $j=1,$ we set $g(w)=1/w.$) Put $z_l=g(w_l)$ for $l=1,2$ and set $X_1 =g^{-1}({\mathbb {C}}_{0,1})$ which contains X. We consider a shortest hyperbolic geodesic $\alpha$ joining $w_1$ and $w_2$ in $X_1.$ Note that $\ell _{X_1}(\alpha )=h_{X_1}(w_1,w_2)\le h_X(w_1,w_2)$ by the principle of hyperbolic metric. In order to complete the proof, we need to analyse the location of the geodesic $\alpha .$ Since $|w_l|\le \rho _n,~l=1,2,$ the points $z_l$ satisfy $|z_l|=|a_j/w_l|\ge \sigma ,$ where $\sigma =|a_j|/\rho _n\le \tilde{\rho }_n/\rho _n\le e^{-1}.$ (When $j=1,$ $\sigma :=1/\rho _n\le e^{-1}$ because $1\notin X.$) We now apply Corollary 2.7 to see that $g(\alpha )$ is contained in the set $|z|\ge \sigma e^{-K_0},$ where we recall that $K_0\approx 0.85.$ Thus, $\alpha$ lies in the set $|w|\le |a_j|e^{K_0}/\sigma =\rho _ne^{K_0}.$ We note also that $\sigma =|a_j|/\rho _n\ge \tilde{\rho }_1/\rho _n=e\rho _1/\rho _n.$ Hence, by (3.5), we have the lower estimate

$$\begin{aligned} \lambda _{X_1}(w)=\lambda (g(w))|g'(w)|&\ge \frac{1}{2|w|(C_0+|\log |a_j/w||)}\ge \frac{e^{-K_0}}{2\rho _n(C_0-K_0-\log \sigma )} \\&\ge \frac{e^{-K_0}}{2\rho _n(C_0-K_0-1+\log (\rho _n/\rho _1))}=:\frac{1}{U_1} \end{aligned}$$

for $w\in \alpha , 1<j<n.$ Since $e\rho _1\le 1,$ this estimate is valid also for $j=1.$

We are now ready to show the second inequality in (5.2) for $j\ne n.$ We denote by $\beta$ the line passing through $\zeta$ and orthogonal to the line segment $[w_1,w_2].$ Take an intersection point $\zeta _1$ of the line $\beta$ with the geodesic $\alpha$ and denote by $\alpha _1$ the part of $\alpha$ joining $\zeta _1$ and $w_2.$ From the above inequality, we derive

$$\begin{aligned} h_X(\zeta _1,w_2) \ge h_{X_1}(\zeta _1,w_2) =\int _{\alpha _1}\lambda _{X_1}(w)|dw|\ge U_1^{-1}\int _{\alpha _1}|dw| \ge U_1^{-1}|\zeta _1-w_2|. \end{aligned}$$

Now we have

$$\begin{aligned} |\zeta -w_2| \le |\zeta _1-w_2| \le U_1 h_{X_1}(\zeta _1,w_2) \le U_1 h_{X_1}(w_1,w_2) \le U_1 h_{X}(w_1,w_2). \end{aligned}$$

We now turn to the case when $j=n.$ Then we need to modify the above argument a bit. In this case, we set $g(w)=w$ and $X_1={\mathbb {C}}_{0,1}.$ If $\alpha$ is contained in the disk $|w|\le 3\rho _n,$ (3.5) yields

$$\begin{aligned} \lambda _{X_1}(w)\ge \frac{1}{6\rho _n(C_0+\log 3+\log \rho _n)}=:\frac{1}{U_2} \end{aligned}$$

(5.3)

for $w\in \alpha .$ Then the same argument as above yields the inequality $|\zeta -w_2| \le U_2 h_X(w_1,w_2).$ Otherwise, we define $\zeta _2$ to be the first hitting point of the geodesic $\alpha$ to the circle $\Gamma =\{w:|w-w_2|=|\zeta -w_2|\}$ from $w_2.$ Let $\alpha _2$ be the part of $\alpha$ joining $\zeta _2$ and $w_2$ as before. Since the inside of $\Gamma$ is contained in the disk $|w|\le 3\rho _n,$ the inequality (5.3) holds for $w\in \alpha _2.$ Thus, we have

$$\begin{aligned} |\zeta -w_2|=|\zeta _2-w_2|\le U_2 h_{X_1}(\zeta _2,w_2) \le U_2 h_{X_1}(w_1,w_2) \le U_2 h_{X}(w_1,w_2). \end{aligned}$$

In this way, we saw that the second inequality in (5.2) with $B_2=\max \{U_1,U_2\}$ holds at any event.

Next we show the first inequality in (5.2). By case (i), we have

$$\begin{aligned} D_j(w_1,\zeta )\le M_0\rho _n h_X(w_1,\zeta ). \end{aligned}$$

On the other hand, by making use of the first part of the theorem and the second inequality in (5.2), we have

$$\begin{aligned} h_X(\zeta ,w_2)\le N_1^{-1}d_X(\zeta ,w_2)=N_1^{-1}|\zeta -w_2| \le N_1^{-1}B_2h_X(w_1,w_2). \end{aligned}$$

Combining these inequalities, we get

$$\begin{aligned} D_j(w_1,\zeta )&\le M_0\rho _n h_X(w_1,\zeta ) \le M_0\rho _n \big \{h_X(w_1,w_2)+h_X(w_2,\zeta )\big \} \\&\le M_0\rho _n(1+N_1^{-1}B_2)h_X(w_1,w_2). \end{aligned}$$

Thus $D_j(w_1,\zeta )\le B_1h_X(w_1,w_2)$ for $j=1,\dots ,n$ with $B_1=M_0\rho _n(1+N_1^{-1}B_2).$

We have now shown the inequality $e_X(w_1,w_2)\le N_2' h_X(w_1,w_2)$ in this case, where $N_2'=B_1+B_2.$

Case (iii) $w_1\in W, w_2\in E_j^*$: This is essentially same as case (ii).

Case (iv) $w_1,w_2\in W$: Similarly, we obtain $|w_1-w_2|\le B_2h_X(w_1,w_2)$ with the same constant $B_2$ as in case (ii).

Case (v) $w_1\in E_j^*$ and $w_2\in E_k^*$ with $j\ne k$: Then, by definition,

$$\begin{aligned} e_X(w_1,w_2)=D_j(w_1,\zeta _1) +|\zeta _1-\zeta _2|+D_k(\zeta _2, w_2), \end{aligned}$$

where $\zeta _1$ and $\zeta _2$ are the intersection points of the line segment $[w_1,w_2]$ with $\partial E_j$ and $\partial E_k,$ respectively. By using the auxiliary lines $\beta _l$ orthogonally intersecting $[w_1,w_2]$ at $\zeta _l~(l=1,2),$ we obtain the inequality

$$\begin{aligned} |\zeta _1-\zeta _2|\le B_2 h_X(w_1,w_2) \end{aligned}$$

(5.4)

in the same way as in case (ii). Let $\alpha$ be a shortest hyperbolic geodesic joining $w_1$ and $w_2$ in X. Let $\zeta _l'$ be an intersection point of $\alpha$ with $\beta _l$ for $l=1,2.$

First assume that $D_j(w_1,\zeta _1)\ge 4\rho _j.$ Since $D_j(\zeta _1,\zeta _1')=|\zeta _1-\zeta _1'|\le 2\rho _j\le D_j(w_1,\zeta _1)/2,$ we obtain $D_j(w_1,\zeta _1)\le 2D_j(w_1,\zeta _1').$ By the first part of the theorem, we now observe that

$$\begin{aligned} D_j(w_1,\zeta _1')=d_X(w_1,\zeta _1')\le M_0\rho _{n}h_X(w_1,\zeta _1') \le M_0\rho _{n}h_X(w_1,w_2). \end{aligned}$$

Thus $D_j(w_1,\zeta _1)\le 2M_0\rho _{n}h_X(w_1,w_2).$ Next assume that $D_j(w_1,\zeta _1)<4\rho _j.$ By (4.1), we have

$$\begin{aligned} |\zeta _1-\zeta _2|\ge {{\text {dist}}}(E_j,E_k)\ge (1-e^{-1}-e^{-2})\rho _j \end{aligned}$$

Thus, with the help of (5.4), we have

$$\begin{aligned} D_j(w_1,\zeta _1)<\frac{4}{1-e^{-1}-e^{-2}}\,|\zeta _1-\zeta _2| \le \frac{4B_2}{1-e^{-1}-e^{-2}}\, h_X(w_1,w_2). \end{aligned}$$

We can deal with $D_k(\zeta _2,w_2)$ in the same way. Therefore, letting

$$\begin{aligned} N_2''=B_2+2\max \left\{ 2M_0\rho _{n},~ \frac{4B_2}{1-e^{-1}-e^{-2}}\right\} , \end{aligned}$$

we obtain the inequality $e_X(w_1,w_2)\le N_2' h_X(w_1,w_2).$

Summarising all the cases, we now conclude that the right-most inequality in the assertion of the theorem holds with the choice $N_2=\max \{N_2',N_2''\}.$ $\square$

We end the paper with the observation that by the proof we can take the bounds $N_1$ and $N_2$ in the last theorem under the convention $a_n=\infty$ as follows:

$$\begin{aligned} N_1=\frac{2\rho _0}{\pi }{\quad \text {and}\quad }N_2=C\, \frac{\rho _n}{\rho _0}\log \frac{\rho _n}{\rho _0}, \end{aligned}$$

where $C>0$ is an absolute constant and

$$\begin{aligned} \rho _0=\min _{1\le j<n}\rho _j. \end{aligned}$$