††footnotetext:  File: Landen20221219.tex, printed: 2024-8-27, 23.36

#  Landen transformations applied to approximation

Rahim Kargar∗ R. Kargar: Department of Mathematics and Statistics, University
of Turku, Turku, Finland<EMAIL_ADDRESS>
ORCID ID: http://orcid.org/0000-0003-1029-5386 ,  Oona Rainio  O. Rainio:
Department of Mathematics and Statistics, University of Turku, Turku, Finland
<EMAIL_ADDRESS>
ORCID ID: http://orcid.org/0000-0002-7775-7656 and  Matti Vuorinen  M.
Vuorinen: Department of Mathematics and Statistics, University of Turku,
Turku, Finland<EMAIL_ADDRESS>
ORCID ID: http://orcid.org/0000-0002-1734-8228

###### Abstract.

We study computational methods for the approximation of special functions
recurrent in geometric function theory and quasiconformal mapping theory. The
functions studied can be expressed as quotients of complete elliptic integrals
and as inverses of such quotients. In particular, we consider the distortion
function $\varphi_{K}(r)$ which gives a majorant for $|f(x)|$ when
$f:\mathbb{B}^{2}\to\mathbb{B}^{2},f(0)=0,$ is a quasiconformal mapping of the
unit disk $\mathbb{B}^{2}.$ It turns out that the approximation method is very
simple: five steps of Landen iteration is enough to achieve machine precision.

###### Key words and phrases:

Landen transformation; Elliptic integral; Quasiconformal mappings;
Quasiconformal Schwarz lemma; Grötzsch capacity.
∗Corresponding author

###### 2010 Mathematics Subject Classification:

Primary: 30C62; Secondary: 33C75

In memoriam: Academician Yu. G. Reshetnyak 1929-2021

##  1\. Introduction

The Landen transformation

$r\mapsto\frac{2\sqrt{r}}{1+r},\quad r\in(0,1),$

and the Landen sequences of functions, recursively defined in terms of this
transformation, are closely related to elliptic integrals and elliptic
functions. For instance, the complete integrals $\mathchoice{\hbox{\,\fFt
K}}{\hbox{\,\fFt K}}{\hbox{\,\fFa K}}{\hbox{\,\fFp K}}(r)$ satisfy functional
identities involving the Landen transformation, and these integrals can be
expressed as infinite products, where the factor functions are expressed in
terms of the Landen sequences.

Our goal here is to show that some of the well-known special functions of
geometric function theory can be efficiently computed using a few steps of the
Landen iteration. These functions include the function $\mu(r)$ related to the
conformal modulus of the Grötzsch ring domain, defined as a quotient of
complete elliptic integrals, and

$\varphi_{K}:[0,1]\rightarrow[0,1];\quad\varphi_{K}(r)=\mu^{-1}\left(\mu(r)/K\right),\quad
K>0,$

the special function in the quasiconformal Schwarz lemma.

In recent years many papers have been published with information about the
asymptotic behavior of the function $\mathchoice{\hbox{\,\fFt K}}{\hbox{\,\fFt
K}}{\hbox{\,\fFa K}}{\hbox{\,\fFp K}}(r)$ at the singularity $r=1$, see Ref.
[4, p. 53]. We give here a recursive scheme for the numerical approximation
for the several functions we consider. In five iteration steps we obtain
approximations with errors close to machine epsilon. The code is just a few
lines and can be implemented in every programming language — no programming
libraries are needed. For instance, the first iteration for the function
$\varphi_{K}$ improves the classical majorant function $4^{1-1/K}r^{1/K}$ for
$\varphi_{K}(r),K>1.$

The above functions have been extensively studied in the monograph [4] and the
associated software for computation is given on the attached diskette. We use
this software as a reference when we study the precision of our algorithms.
The function $\varphi_{K}(r)$ for integer values of $K$ also occurs in the
study of so called modular equations [4, p. 92], [2, 1]. Previously the Landen
iteration applied to $\varphi_{K}(r)$ has been studied in [4, p. 93] and in
Partyka’s paper [10].

The structure of this article is as follows. In Section 2 we define the
ascending and descending Landen sequences and investigate their application to
the aforementioned approximation problems. In Section 3 we analyse the
algorithms in detail and study their numerical performance.

##  2\. Ascending and Descending Landen Sequences

In this section we review the Landen transformation and its applications to
compute elliptic integrals and related special functions. These facts will be
applied to quasiconformal mappings in the next section.

######  2.1.

Landen sequences. For $r\in(0,1)$ let $L(r,0)=r$ and

(2.2) $L(r,p+1)=\frac{2\sqrt{L(r,p)}}{1+L(r,p)};\quad
L(r,-p-1)=\left(\frac{L(r,-p)}{1+\sqrt{1-L(r,-p)^{2}}}\right)^{2},$

for $p=0,1,2,3,\ldots$. The recursively defined sequences $\\{L(r,p)\\}$ and
$\\{L(r,-p)\\}$ are called ascending and descending Landen sequences,
respectively. It is clear that each of the Landen functions
$L(\cdot,p):(0,1)\rightarrow(0,1)$ is an increasing homeomorphism with

$L(r,p)<L(r,p+1)\quad{\rm and}\quad L(r,p+q)=L(L(r,p),q)$

for all $r\in(0,1)$ and $p,q\in\mathbb{Z}:=\\{0,\pm 1,\pm 2,\pm 3,\ldots\\}$.
In particular,

$y=L(r,p)\Leftrightarrow r=L(y,-p).$

Therefore, $L(\cdot,-p)$ is the inverse of $L(\cdot,p)$.

Throughout the paper we use the following abbreviations

$r^{\prime}=\sqrt{1-r^{2}},\quad
w=1-\frac{r^{4}}{(1+r^{\prime})^{4}}=\frac{4(2-r^{2})r^{\prime}-8(1-r^{2})}{r^{4}}.$

As Table 1 suggests, the functions in the Landen sequence become more involved
when $|p|$ increases.

$L(r,-2)$ | $L(r,-1)$ | $L(r,0)$ | $L(r,1)$ | $L(r,2)$
---|---|---|---|---
$\displaystyle{\frac{r^{4}}{(1+\sqrt{w})^{2}(1+r^{\prime})^{4}}}$ | $\displaystyle{\frac{r^{2}}{(1+r^{\prime})^{2}}}$ | $r$ | $\displaystyle{\frac{2\sqrt{r}}{1+r}}$ | $\displaystyle{\frac{2\sqrt{2}r^{1/4}\sqrt{1+r}}{(1+\sqrt{r})^{2}}}$
Table 1. A few functions of the Landen sequence.

######  Proposition 2.3.

Let $r\in(0,1)$ and $p=0,1,2,\ldots$. Then
i)

(2.4) $L(r,-p-1)<r^{2^{p}};$

ii)

(2.5) $L(r,p+1)>r^{2^{-p}}\geq r.$

###### Proof.

i) We prove the assertion by induction. It is clear that it holds for $p=0$.
We assume that the inequality holds for $p=k-1$, i.e. $L(r,-k)<r^{2^{k-1}}$.
By the second identity of (2.2), and using the last inequality, we get

$L(r,-k-1)=\left(\frac{L(r,-k)}{1+\sqrt{1-L(r,-k)^{2}}}\right)^{2}\leq\frac{r^{2^{k}}}{\left(1+\sqrt{1-r^{2^{k}}}\right)^{2}}<r^{2^{k}}$

for all $r\in(0,1)$, and $k>1$.
ii) Due to the similarity of the proof to i) we omit the details. ∎

######  Proposition 2.6.

The following identities hold for all $r\in(0,1)$ and $p=0,1,2,3,\ldots$.

(2.7) $L(r,p)^{2}+L(r^{\prime},-p)^{2}=1;$ (2.8)
$L(r,p-1)=\frac{1-L(r^{\prime},-p)}{1+L(r^{\prime},-p)}.$

###### Proof.

In order to prove the first identity we use induction. It is clear that the
identity (2.7) holds true for $p=0$. Assume that (2.7) holds true when $p=k$,
i.e.

$L(r,k)^{2}+L(r^{\prime},-k)^{2}=1.$

By using (2.2) and the last identity we obtain

$\displaystyle L(r,k+1)^{2}+L(r^{\prime},-k-1)^{2}$
$\displaystyle=\left(\frac{2\sqrt{L(r,k)}}{1+L(r,k)}\right)^{2}+\left(\frac{L(r^{\prime},-k)}{1+\sqrt{1-L(r^{\prime},-k)^{2}}}\right)^{4}$
$\displaystyle=\frac{4L(r,k)(1+L(r,k))^{2}+(1-L(r,k)^{2})^{2}}{(1+L(r,k))^{4}}=1$

concluding the proof of (2.7).
The identity (2.8) can be proved by applying (2.7). We have

(2.9)
$\frac{1-L(r^{\prime},-p)}{1+L(r^{\prime},-p)}=\frac{1-\sqrt{1-L(r,p)^{2}}}{1+\sqrt{1-L(r,p)^{2}}}=\left(\frac{1-\sqrt{1-L(r,p)^{2}}}{L(r,p)}\right)^{2}.$

On the other hand, if we replace $p$ by $-p$ in the second identity of (2.2)
we get

(2.10) $L(r,p-1)=\left(\frac{L(r,p)}{1+\sqrt{1-L(r,p)^{2}}}\right)^{2}.$

A simple calculation shows that the right hand side of (2.9) is equal to the
last identity (2.10). The proof is now complete. ∎

######  Remark 2.11.

It follows from (2.7) and the first inequality of (2.4) that

$L(r,p+1)=\sqrt{1-L(r^{\prime},-p-1)^{2}}>\sqrt{1-(r^{\prime})^{2^{p+1}}}:=\ell(r,p).$

Computer experiment shows that this lower bound, i.e. $\ell(r,p)$, is better
than the lower bound in (2.5), i.e. $r^{2^{-p}}$, when $r$ is close to one.

The complete elliptic integral

$\mathchoice{\hbox{\,\fFt K}}{\hbox{\,\fFt K}}{\hbox{\,\fFa K}}{\hbox{\,\fFp
K}}(r)=\int^{1}_{0}\frac{dx}{\sqrt{(1-x^{2})(1-r^{2}x^{2})}},\quad r\in(0,1),$

defines a homeomorphism $\mathchoice{\hbox{\,\fFt K}}{\hbox{\,\fFt
K}}{\hbox{\,\fFa K}}{\hbox{\,\fFp K}}:(0,1)\to(\pi/2,\infty).$ The following
two identities due to Landen express important properties of the complete
elliptic integral $\mathchoice{\hbox{\,\fFt K}}{\hbox{\,\fFt K}}{\hbox{\,\fFa
K}}{\hbox{\,\fFp K}}(r)$ [6, p. 12] (see also [4, p. 51])

(2.12) $\left\\{\begin{array}[]{ll}\mathchoice{\hbox{\,\fFt K}}{\hbox{\,\fFt
K}}{\hbox{\,\fFa K}}{\hbox{\,\fFp
K}}\left(\displaystyle{\frac{2\sqrt{r}}{1+r}}\right)=(1+r)\mathchoice{\hbox{\,\fFt
K}}{\hbox{\,\fFt K}}{\hbox{\,\fFa K}}{\hbox{\,\fFp K}}(r);\\\ \\\
\mathchoice{\hbox{\,\fFt K}}{\hbox{\,\fFt K}}{\hbox{\,\fFa K}}{\hbox{\,\fFp
K}}\left(\displaystyle{\frac{1-r}{1+r}}\right)=\displaystyle{\frac{1}{2}(1+r)}\mathchoice{\hbox{\,\fFt
K}}{\hbox{\,\fFt K}}{\hbox{\,\fFa K}}{\hbox{\,\fFp
K}}(r^{\prime}),\end{array}\right.$

The first identity of (2.12) shows that $\mathchoice{\hbox{\,\fFt
K}}{\hbox{\,\fFt K}}{\hbox{\,\fFa K}}{\hbox{\,\fFp
K}}(r)=\mathchoice{\hbox{\,\fFt K}}{\hbox{\,\fFt K}}{\hbox{\,\fFa
K}}{\hbox{\,\fFp K}}(L(r,-1))(1+L(r,-1)).$ This observation is the basis of
the following classical result, see Ref. [6, p. 14]. Observe that
$\mathchoice{\hbox{\,\fFt K}}{\hbox{\,\fFt K}}{\hbox{\,\fFa K}}{\hbox{\,\fFp
K}}(L(r,-p))\to\pi/2$ when $p\to\infty.$

######  Lemma 2.13.

For $r\in(0,1)$ we have

$\mathchoice{\hbox{\,\fFt K}}{\hbox{\,\fFt K}}{\hbox{\,\fFa K}}{\hbox{\,\fFp
K}}(r)=\frac{\pi}{2}\prod_{n=0}^{\infty}\frac{2}{1+L(r^{\prime},-n)}=\frac{\pi}{2}\prod_{n=1}^{\infty}(1+L(r,-n)).$

Lemma 2.13 gives fast converging methods for numerical evaluation of
$\mathchoice{\hbox{\,\fFt K}}{\hbox{\,\fFt K}}{\hbox{\,\fFa K}}{\hbox{\,\fFp
K}}(r)$.

######  2.14.

Arithmetic-Geometric Mean. For $0<b<a$ define $a_{0}=a$, $b_{0}=b$,
$a_{n+1}=(a_{n}+b_{n})/2$ and $b_{n+1}=\sqrt{a_{n}b_{n}}$. The common limit of
these sequences

$AG(a,b)=\lim_{n\rightarrow\infty}a_{n}=\lim_{n\rightarrow\infty}b_{n},$

is the arithmetic geometric mean of $a$ and $b$. An in-depth discussion of
this is found in [6], see also Ref. [7].

One of the basic properties of $AG(a,b)$ is that (see Ref. [4, Lemma 4.3])

$AG(a,b)=aAG(1,b/a),\quad a>0,b\geq 0.$

######  Theorem 2.15.

(Gauss) For $r\in(0,1)$

$\mathchoice{\hbox{\,\fFt K}}{\hbox{\,\fFt K}}{\hbox{\,\fFa K}}{\hbox{\,\fFp
K}}(r)=\frac{\pi}{2AG(1,r^{\prime})}.$

From Lemma 2.13 and Theorem 2.15 we obtain the identity for $s\in(0,1)$

$AG(1,s)=\prod_{n=0}^{\infty}\left(\frac{1+L(s,-n)}{2}\right).$

######  Lemma 2.16.

Consider the arithmetic geometric mean iteration with $a=1$, $b\in(0,1)$,
$\alpha\in[0,1]$, and let $b_{n}$ be the $n^{\rm th}$ iterate of the
$b$-sequence. Then $b_{n}<L(b^{\alpha},n)$ for $n=1,2,3\ldots$.

###### Proof.

We use induction to prove this lemma. We have

$b_{1}=\sqrt{a_{0}b_{0}}=\sqrt{b}\quad{\rm and}\quad
L(b^{\alpha},1)=\frac{2\sqrt{b^{\alpha}}}{1+b^{\alpha}}.$

It is easy to see that $\sqrt{b}<2\sqrt{b^{\alpha}}/(1+b^{\alpha})$ holds for
all $b\in(0,1)$ and $\alpha\in[0,1]$. Let $b_{k}<L(b^{\alpha},k)$ for all
$k=n>1$. We need to show that $b_{k+1}<L(b^{\alpha},k+1)$. Due to the fact
that $a_{k}\in(0,1)$ for all $k>1$, and $t\mapsto 2\sqrt{t}/(1+t)$ is an
increasing function we get

$b_{k+1}=\sqrt{a_{k}b_{k}}<\sqrt{b_{k}}<\frac{2\sqrt{b_{k}}}{1+b_{k}}<\frac{2\sqrt{L(b^{\alpha},k)}}{1+L(b^{\alpha},k)}=L(b^{\alpha},k+1),$

concluding the proof. ∎

######  Remark 2.17.

(1) There is a large body of literature about the properties of
$\mathchoice{\hbox{\,\fFt K}}{\hbox{\,\fFt K}}{\hbox{\,\fFa K}}{\hbox{\,\fFp
K}}(r)$ due, in particular, to the authors of [9] and their students. See also
the literature survey [5].
(2) The logarithmic mean of $a,b>0,a\neq b$, is defined by (see [4, p. 77])

$\mathcal{L}(a,b)=\frac{a-b}{\log(a/b)}=\prod_{k=1}^{\infty}\frac{a^{2^{-k}}+b^{2^{-k}}}{2}.$

Denote $\mathcal{L}_{t}(a,b)=\mathcal{L}(a^{t},b^{t})^{1/t}$ for $t>0$. As
shown in [7, Proposition 2.7] the following very sharp inequality holds for
$x\in(0,1)$

$\mathcal{L}_{3/2}(1,x)>AG(1,x)>\mathcal{L}(1,x).$

For what follows the following decreasing homeomorphism
$\mu:(0,1)\to(0,\infty),$

(2.18) $\mu(r)=\frac{\pi}{2}\frac{\mathchoice{\hbox{\,\fFt K}}{\hbox{\,\fFt
K}}{\hbox{\,\fFa K}}{\hbox{\,\fFp K}}(r^{\prime})}{\mathchoice{\hbox{\,\fFt
K}}{\hbox{\,\fFt K}}{\hbox{\,\fFa K}}{\hbox{\,\fFp K}}(r)},\quad r\in(0,1),$

is crucial. From (2.12) we obtain [4, p. 51]

(2.19)
$\mu\left(\frac{2\sqrt{r}}{1+r}\right)=\frac{1}{2}\mu(r);\quad\mu\left(\frac{1-r^{\prime}}{1+r^{\prime}}\right)=2\mu(r).$

In terms of the Landen sequences we can write (2.19) in the following form for
$0<r<1$ and $p\in\mathbb{Z}$

(2.20) $\mu(r)=2^{p}\mu(L(r,p)).$

By (2.18) it is clear that

$\mu(r)\mu(r^{\prime})=\frac{\pi^{2}}{4}.$

By Jacobi’s work [8, p. 462, (B.25)] the inverse of $\mu$ can be expressed in
terms of theta-functions as follows for $y>0$

$\mu^{-1}(y)=\left(\frac{2\sum^{\infty}_{n=0}q^{(n+1/2)^{2}}}{1+2\sum^{\infty}_{n=1}q^{n^{2}}}\right)^{2},\quad
q=\exp(-2y).$

Jacobi also proved formulas for $\mu^{-1}(y)$ as infinite products, see [4, p.
91].

Below we also study the special function

(2.21) $\varphi_{K}(r)=\mu^{-1}\left(\mu(r)/K\right),\quad r\in(0,1),K>0,$

It defines a homeomorphism $\varphi_{K}:[0,1]\rightarrow[0,1]$ with limit
values $\varphi_{K}(0)=0$ and $\varphi_{K}(1)=1$. The basic estimate for
$\varphi_{K}(r)$, $K\geq 1$, $r\in(0,1)$ is [4, Thorem 10.9(1)]

(2.22) $r^{1/K}<\varphi_{K}(r)<4^{1-1/K}r^{1/K}.$

For information about this and other related inequalities the reader is
referred to [8, p.319, 16.51]. By (2.20) it is clear that for
$r\in(0,1),p\in\mathbb{Z},$

(2.23) $\varphi_{2^{p}}(r)=L(r,p).$

It is noteworthy that the functions $\mu,\mu^{-1},\varphi_{K}$ satisfy many
functional identities [3, 4]. For instance, the Pythagorean type identity of
the Landen transformation (2.7) has the following counterparts for these
functions [8, p. 463, p. 125]

$\left(\mu^{-1}(y)\right)^{2}+\left(\mu^{-1}\left(\frac{\pi^{2}}{4y}\right)\right)^{2}=1,\quad
y>0;$ (2.24) $\varphi_{K}(r)^{2}+\varphi_{1/K}(r^{\prime})^{2}=1,\quad
K>0,r\in(0,1).$

The following inequalities hold for $r\in(0,1)$ (see [8, p. 122, (7.21)]):

(2.25)
$\log\frac{1}{r}<\log\frac{1+3r^{\prime}}{r}<\log\frac{(1+\sqrt{r^{\prime}})^{2}}{r}<\mu(r)<\log\frac{2(1+r^{\prime})}{r}<\log\frac{4}{r}.$

As a result of Jacobi’s work, the following inequalities hold also for $0<r<1$
[4, p. 91, (5.30)]

(2.26) $\log\frac{(1+\sqrt{r^{\prime}})^{2}}{r}<{\rm
arth}\sqrt[4]{r^{\prime}}<\mu(r)<\frac{\pi^{2}}{4{\rm arth}\sqrt[4]{r}}.$

We summarize the lower and upper bounds of $\mu(r)$ with their inverses in
Tables 2 and 3, respectively.

$j$ | $u_{j}(r)$ | $u_{j}^{-1}(y)$
---|---|---
$1$ | $\displaystyle{{\rm arth}\sqrt[4]{r^{\prime}}}$ | $\displaystyle{\sqrt{1-{\rm th}^{8}(y)}}$
$2$ | $\displaystyle{\log\frac{(1+\sqrt{r^{\prime}})^{2}}{r}}$ | $-$
$3$ | $\displaystyle{\log\frac{1+3r^{\prime}}{r}}$ | $\displaystyle{\frac{\exp(y)+3\sqrt{8+\exp(2y)}}{9+\exp(2y)}}$
$4$ | $\displaystyle{\log(1/r)}$ | $\displaystyle{\exp(-y)}$
$5$ | $\displaystyle{\log\frac{1+r^{\prime}}{r}}$ | $\displaystyle{\frac{2\exp(y)}{1+\exp(2y)}}$
Table 2. Lower bounds of $\mu$ and their inverses. $j$ | $v_{j}(r)$ | $v_{j}^{-1}(y)$
---|---|---
$1$ | $\displaystyle{\frac{\pi^{2}}{4{\rm arth}\sqrt[4]{r}}}$ | $\displaystyle{{\rm th}^{4}(\pi^{2}/4y)}$
$2$ | $\displaystyle{\log\frac{2(1+r^{\prime})}{r}}$ | $\displaystyle{\frac{4\exp(\max\\{y,\log 2\\})}{4+\exp(2\max\\{y,\log 2\\})}}$
$3$ | $\displaystyle{\log(4/r)}$ | $\displaystyle{4\exp(-\max\\{y,\log 4\\})}$
Table 3. Upper bounds of $\mu$ and their inverses.  Lower bounds | Upper bounds
---|---
$\displaystyle{\sqrt{1-{\rm th}^{8}(y)}}$ | $\displaystyle{{\rm th}^{4}(\pi^{2}/4y)}$
$\displaystyle{\frac{\exp(y)+3\sqrt{8+\exp(2y)}}{9+\exp(2y)}}$ | $\displaystyle{\frac{4\exp(\max\\{y,\log 2\\})}{4+\exp(2\max\\{y,\log 2\\})}}$
$\displaystyle{\exp(-y)}$ | $\displaystyle{4\exp(-\max\\{y,\log 4\\})}$
$\displaystyle{\frac{2\exp(y)}{1+\exp(2y)}}$ | $-$
Table 4. Upper and lower bounds for $\mu^{-1}(y)$.

We ignore the inverse of $u_{2}$ in Table 2 since it is a very complicated
formula.

######  Lemma 2.27.

Assume that $u,v:(0,1)\rightarrow(0,\infty)$ are decreasing homeomorphisms
with

(2.28) $u(r)<\mu(r)<v(r),\quad 0<r<1.$

Then

$u^{-1}(v(r)/K)<\varphi_{K}(r)<v^{-1}(u(r)/K)$

for all $K>1$ and $r\in(0,1)$.

###### Proof.

Because $\mu$ is decreasing, and also by (2.28) we can obtain

$\mu^{-1}(v(r)/K)<\varphi_{K}(r)=\mu^{-1}(\mu(r)/K)<\mu^{-1}(u(r)/K).$

It follows from $u^{-1}(y)<\mu^{-1}(y)$, $y>0$, that
$u^{-1}(v(r)/K)<\mu^{-1}(v(r)/K)$. Also, since $\mu^{-1}(y)<v^{-1}(y)$, $y>0$,
we get $\mu^{-1}(u(r)/K)<v^{-1}(u(r)/K)$ for all $K>1$ and $r\in(0,1)$. The
proof is now complete. ∎

######  Corollary 2.29.

Let $u:(0,1)\rightarrow(0,\infty)$ and $v:(0,1]\rightarrow[c,\infty)$, $c>0$,
be decreasing homeomorphisms which satisfy (2.28). Then

$u^{-1}(v(r)/K)<\varphi_{K}(r)<v^{-1}(\max\\{u(r)/K,c\\})$

for all $K>1$ and $r\in(0,1)$.

######  Example 2.30.

Consider $u_{1}$ and $v_{3}$ as above. Since $v_{3}$ is a decreasing
homeomorphism from $(0,1]$ onto $[\log 4,\infty)$, therefore,

$u_{1}^{-1}(v_{3}(r)/K)<\varphi_{K}(r)<v_{3}^{-1}(\max\\{u_{1}(r)/K,\log
4\\})$

for all $K>1$ and $r\in(0,1)$.

We recall the following lemma from [4, p. 17]:

######  Lemma 2.31.

Let $f$ be a decreasing homeomorphism from $(0,1)$ onto $(0,\infty)$, and let
$g,h$ be strictly decreasing continuous functions from $(0,1)$ into
$(0,\infty)$ with $h(0+)=\infty$ such that $g(r)<f(r)<h(r)$. Let $C>1$ and
$s=f^{-1}(f(r)/C)$. Then

$g(r)>Cg(s)\quad and\quad h(r)<Ch(s),\quad 0<r<1,$

if and only if $f(r)/g(r)$ and $h(r)/f(r)$ are strictly increasing on $(0,1)$.
In particular, if both $h^{-1}(h(r)/C)$ and $g^{-1}(g(r)/C)$ are defined, then

$g^{-1}(g(r)/C)<s<h^{-1}(h(r)/C),\quad 0<r<1.$

As an application of Lemma 2.31 we have:

######  Lemma 2.32.

Let $u_{1}(r)$ and $v_{2}(r)$ be defined as in Tables 2 and 3, respectively,
where $r\in(0,1)$. Then

$u_{1}^{-1}(u_{1}(r)/K)<\varphi_{K}(r)<v_{2}^{-1}(v_{2}(r)/K)$

for all $K>1$ and $r\in(0,r_{0})$, where $r_{0}\in(0,1)$ is such that both
$u_{1}^{-1}(u_{1}(r)/K)$ and $v_{2}^{-1}(v_{2}(r)/K)$ are defined. Moreover,
the first inequality is sharp in the sense that $u_{1}^{-1}(u_{1}(r)/K)\to r$
as $K\rightarrow 1$.

###### Proof.

It follows from (2.25) and (2.26) that $u_{1}(r)<\mu(r)<v_{2}(r)$ for all
$r\in(0,1)$. It is enough to show that both $\mu(r)/u_{1}(r)$ and
$v_{2}(r)/\mu(r)$ are defined and strictly increasing on $(0,1)$. By using the
first identity of (2.19) and letting $u=2\sqrt{r}/(1+r)$, we have

$\frac{v_{2}(r)}{\mu(r)}\quad{\rm
incr.}\,\Leftrightarrow\frac{v_{2}(u)}{\mu(u)}\quad{\rm
incr.}\,\Leftrightarrow\frac{\log(2/\sqrt{r})}{\mu(2\sqrt{r}/(1+r))}\quad{\rm
incr.}\,\Leftrightarrow\frac{\frac{1}{2}\log(4/r)}{\frac{1}{2}\mu(r)}\quad{\rm
incr.}$

which is valid by [4, Theorem 2.16(2)]. It follows also from [4, Theorem
5.13(6)] that $\mu(r)/u_{1}(r)$ is strictly increasing from $(0,1)$ onto
$(1,\infty)$. For illustration see Figure 1. The proof is now complete. ∎

Figure 1. 1: The graph of $\mu(r)/u_{1}(r)$, where $0<r<1$ 1: The graph of
$v_{2}(r)/\mu(r)$, where $0<r<1$.

######  Remark 2.33.

It is worth mentioning that the new upper and lower bounds for $\varphi_{K}$
are more accurate than those bounds in (2.22). Indeed,

$r^{1/K}<u_{1}^{-1}(u_{1}(r)/K)<\varphi_{K}(r)<v_{2}^{-1}(v_{2}(r)/K)<4^{1-1/K}r^{1/K}$

for all $K>1$ and $r\in(0,1)$.

##  3\. Landen Approximations

The three functions $\mu,\mu^{-1},\varphi_{K}$ were extensively investigated
in [4], with computer implementations in languages, Mathematica, MATLAB, C on
the accompanying diskette. Here our goal is to show that for a large range of
the arguments we obtain results with accuracy to those in [4], now only using
Mathematica. The methods applied in [4] for the numerical evaluation of  K and
$\mu$ were based on the arithmetic-geometric mean while for $\mu^{-1}$ and
$\varphi_{K}$ a Newton iteration was used. Here we show that the Landen
sequences yield approximations with errors close to machine epsilon agreement
with the results of [4] when the recursion level is moderate, 4 or 5.

Our starting point is the following lemma.

######  Lemma 3.1.

(1) The function $\mu(r)+\log r$ is a monotone decreasing function from
$(0,1)$ onto $(0,\log 4).$

(2) For $0<r<1,p\in\mathbb{Z}$

$2^{-p}\log\frac{1}{L(r,-p)}<\mu(r)<2^{-p}\log\frac{4}{L(r,-p)},$

and, in particular, for $p=1$ we have

$u_{5}(r)<\mu(r)<v_{2}(r),$

where $u_{5}$ and $v_{2}$ are defined as in Tables 2 and 3, respectively.

###### Proof.

(1) This is well-known, see [4, p. 84, Thm 5.13].

(2) The proof follows from (2.20) and part (1). ∎

######  Proposition 3.2.

Let $u,v:(0,1)\rightarrow(0,\infty)$ be continuous functions with
$|u(r)-v(r)|<M$ for some constant $M$ and for all $r\in(0,1)$. Also let
$u(r)<\mu(r)<v(r)$ for all $r\in(0,1)$. Then

$\lim_{p\rightarrow-\infty}2^{p}u(L(r,p))=\lim_{p\rightarrow-\infty}2^{p}v(L(r,p))=\mu(r).$

###### Proof.

By (2.20)

$2^{p}u(L(r,p))<\mu(r)=2^{p}\mu(L(r,p))<2^{p}v(L(r,p))$

which implies that

$|2^{p}v(L(r,p))-2^{p}u(L(r,p))|<2^{p}M,$

concluding the proof. ∎

######  Remark 3.3.

By (2.25) we can apply Proposition 3.2, for example, with $u_{2}$ and $v_{2}$,
where $u_{2}$ and $v_{2}$ are defined as in Tables 2 and 3, respectively.

Below we will find the best and simplest approximation for $\mu^{-1}(y)$,
based on the Landen transformation.

######  Proposition 3.4.

Let $u,v:(0,1)\rightarrow(0,\infty)$ be decreasing homeomorphism with
$u(r)<\mu(r)<v(r)$ for all $r\in(0,1)$. Then for $y>0$ and $r=\mu^{-1}(y)$ we
have

$L(u^{-1}(2^{-p}y),-p)<r=\mu^{-1}(y)<L(v^{-1}(2^{-p}y),-p).$

###### Proof.

By (2.20)

$y=\mu(r)=2^{p}\mu(L(r,p))<2^{p}v(L(r,p))$

and because $v$ is decreasing $L(r,p)<v^{-1}(2^{-p}y)$. Hence
$r<L(v^{-1}(2^{-p}y),-p)$. The proof of the lower bound is similar, so we omit
the details. ∎

######  Remark 3.5.

We know by (2.26) that

$u_{1}(r)<\mu(r)<v_{1}(r)$

for all $r\in(0,1)$, where $u_{1}$ and $v_{1}$ are defined as in Tables 2 and
3, respectively. It is easy to see that $u_{1}$ and $v_{1}$ are a
homeomorphism of $(0,1)$ onto $(0,\infty)$. Applying Proposition 3.4 with
$u_{1}(r)$, $v_{1}(r)$, and their inverses we obtain

$\mu^{-1}(y)\approx L\left(u_{1}^{-1}(2^{-p}y),-p\right)=:f_{1}(y,p)$

and

$\mu^{-1}(y)\approx L\left(v_{1}^{-1}(2^{-p}y),-p\right)=:g_{1}(y,p).$

It also follows from (2.25) that the following inequalities

$u_{3}(r)<\mu(r)<v_{2}(r),$

hold true for $r\in(0,1)$, where $u_{3}$ and $v_{2}$ are a homeomorphism of
$(0,1)$ onto $(0,\infty)$ and $(\log 2,\infty)$, respectively. If we apply
Proposition 3.4 for $u_{3}$ and $v_{2}$, we get

$\mu^{-1}(y)\approx L\left(u_{3}^{-1}(2^{-p}y),-p\right)=:f_{2}(y,p)$

and

$\mu^{-1}(y)\approx L\left(v_{2}^{-1}(2^{-p}y),-p\right)=:g_{2}(y,p).$

Finally, applying

$u_{4}(r)<\mu(r)<v_{3}(r)$

and applying Proposition 3.4 with $u_{4}$ and $v_{3}$ (which are a
homeomorphism of $(0,1)$ onto $(0,\infty)$ and $(\log 4,\infty)$,
respectively) we obtain

$\mu^{-1}(y)\approx L\left(u_{4}^{-1}(2^{-p}y),-p\right)=:f_{3}(y,p)$

and

$\mu^{-1}(y)\approx L\left(v_{3}^{-1}(2^{-p}y),-p\right)=:g_{3}(y,p).$

Computational results for some values $y$ in the range $(0.2,20)$ are
summarized in Table 5. Only the cases $p=-4,-5$ are taken into account in this
table. When $p=\ldots,-6,-3,-2,-1,\ldots$ the error is large.

$y$ | $\mu^{-1}(y)-g_{2}(y,-4)$ | $\mu^{-1}(y)-g_{3}(y,-5)$
---|---|---
$0.5$ | $0$ | $0$
$1.5$ | $-2.22045\times 10^{-16}$ | $-2.22045\times 10^{-16}$
$2.5$ | $1.11022\times 10^{-16}$ | $1.11022\times 10^{-16}$
$3.5$ | $-1.38778\times 10^{-17}$ | $-1.38778\times 10^{-17}$
$4.5$ | $2.08167\times 10^{-17}$ | $2.08167\times 10^{-17}$
$5.5$ | $0$ | $0$
$6.5$ | $0$ | $0$
$7.5$ | $-1.30104\times 10^{-18}$ | $-1.30104\times 10^{-18}$
$8.5$ | $-4.33681\times 10^{-19}$ | $-4.33681\times 10^{-19}$
$9.5$ | $-7.58942\times 10^{-19}$ | $-7.58942\times 10^{-19}$
$10.5$ | $-2.1684\times 10^{-19}$ | $-2.1684\times 10^{-19}$
$11.5$ | $3.38813\times 10^{-20}$ | $3.38813\times 10^{-20}$
$12.5$ | $-1.18585\times 10^{-20}$ | $-1.18585\times 10^{-20}$
$13.5$ | $-3.38813\times 10^{-21}$ | $-3.38813\times 10^{-21}$
$14.5$ | $-4.23516\times 10^{-22}$ | $0$
$15.5$ | $-1.90582\times 10^{-21}$ | $-1.90582\times 10^{-21}$
$16.5$ | $-5.82335\times 10^{-22}$ | $-5.82335\times 10^{-22}$
$17.5$ | $2.24993\times 10^{-22}$ | $2.24993\times 10^{-22}$
$18.5$ | $7.27919\times 10^{-23}$ | $7.27919\times 10^{-23}$
$19.5$ | $1.65436\times 10^{-23}$ | $1.65436\times 10^{-23}$
Table 5. The error between $\mu^{-1}(y)$ and $g_{2}(y,-4)$, and $\mu^{-1}(y)$
and $g_{3}(y,-5)$ for some values in range $y\in(0.2,20)$. For the computation
of $\mu^{-1}(y)$ we have used the Mathematica "cip.m" file from [4, Appendix
B].

Computer experiments show that $g_{2}(y,-4)$ and $g_{3}(y,-5)$ are the best
approximations for $\mu^{-1}(y)$. We note that $\mu^{-1}(y)-g_{2}(y,-4)$ and
$\mu^{-1}(y)-g_{3}(y,-5)$ have an error value of order
$10^{-14},\ldots,10^{-24}$ in the interval $(0.2,20)$, see Figure 2. For the
computation of $\mu^{-1}(y)$ we use the Mathematica "cip.m" file from [4,
Appendix B].

Figure 2. 1: The graph of $\mu^{-1}(y)-g_{2}(y,-4)$, where $0<y<10$ 1: The
graph of $\mu^{-1}(y)-g_{3}(y,-5)$, where $0<y<10$.

$p$ | $\mu^{-1}(y)$ | $\mu(r)$
---|---|---
$1$ | $\displaystyle{\frac{4\sqrt{\exp(-\max\\{2r,\log 4\\})}}{1+4\exp(-\max\\{2r,\log 4\\})}}$ | $\displaystyle{\log\frac{2(1+r^{\prime})}{r}}$
$2$ | $\displaystyle{\frac{4\sqrt{\frac{\sqrt{\exp(-\max\\{4r,\log 4\\})}}{1+4\exp(-\max\\{4r,\log 4\\})}}}{1+\frac{4\sqrt{\exp(-\max\\{4r,\log 4\\})}}{1+4\exp(-\max\\{4r,\log 4\\})}}}$ | $\displaystyle{\frac{1}{4}\log\frac{4\left(1+r^{\prime}\right)^{4}\left(1+\sqrt{w}\right)^{2}}{r^{4}}}$
Table 6. Two steps of Landen approximations of $\mu^{-1}$ by $g_{3}(y,-p))$,
and $\mu$ by Lemma 3.1(2).

######  3.6.

The special function $\varphi_{K}(r)$. In the study of Hölder continuity of
quasiconformal mappings of the plane, the special function $\varphi_{K}(r)$
defined as (2.21) has an important role. Based on Proposition 3.2 and Remark
3.5 we study the approximation [4, Theorem 5.43]

(3.7) $\varphi_{K}(r)=L(\varphi_{K}(L(r,-p)),p)\approx
L\left(4^{1-1/K}L(r,-p)^{1/K},p\right)=:L_{\varphi}(K,r,p)$

for various values of $p$. Table 7 shows a structural formula for
$L_{\varphi}(K,r,p)$, where $p=0,1$.

$p$ | $L_{\varphi}(K,r,p)$ | c
---|---|---
$0$ | $4^{1-1/K}r^{1/K}$ | $-$
$1$ | ${\displaystyle{\frac{2\sqrt{4^{1-1/K}\,\,c^{1/K}}}{1+4^{1-1/K}\,\,c^{1/K}}}}$ | $\displaystyle{\left(\frac{r}{1+\sqrt{1-r^{2}}}\right)^{2}}$
Table 7. The function $L_{\varphi}(K,r,p)$ for $p=0,1$.

Here we note that $L_{\varphi}(K,r,p)$ is a majorant for the function
$\varphi_{K}(r)$. We also study the following approximation by applying Remark
3.5 and Lemma 3.1(2)

(3.8) $\varphi_{K}(r)\approx
g_{3}\left(2^{-p}\log(4/L(r,-p))/K,-5\right)=:LM(K,r,p),$

where $K>1$, and $r\in(0,1)$. Computer experiments show that $LM(K,r,5)$ is
the best approximation for $\varphi_{K}(r)$, see Figure 3.

Figure 3. The 3D plot of $\varphi_{K}(r)-LM(K,r,5)$ for $1<K<4$ and $0<r<1$.

######  3.9.

Remark. The next few lines of Mathematica code

    L[s_, p_] :=  Module[{j = 0, y = s},
       While[((j < Abs[p])), If[p < 0, y = (y / (1 + Sqrt[1 - y^2]))^2,
         y = 2 * Sqrt[y] / (1 + y) ]; j++]; y];
    LPhi[K_, r_, p_] := L[4 * Exp[(1 / K) * Log[L[r, -p] / 4]], p];

define an approximation for $\varphi_{K}$. This function satisfies

(3.10) $4^{1-1/K}r^{1/K}>{\tt LPhi[K,r,1]}$

as we see using the command

     Plot3D[{0, 4^(1 - 1/K) r^(1/K) - LPhi[K, r, 1]}, {r, 0, 1}, {K, 1, 3}]

The LHS function of (3.10) here is a majorant for $\varphi_{K},K>1,$ i.e.
$\varphi_{K}(r)<4^{1-1/K}r^{1/K}$ for $K>1,r\in(0,1)$ by [8, Thm 9.32].

The RHS function LPhi[K, r, 1] of (3.10) is not a majorant for
$\varphi_{K},K>1,$ because for the values $K=2,r=0.9$ it is smaller than
$\varphi_{2}(r)=2\sqrt{r}/(1+r),$ that is
$\varphi_{2}(0.9)=0.998614>0.993649={\tt LPhi[2,0.9,1]}$.

######  3.11.

Conclusion. For $K\in(1,20)$ and $r\in(0,1)$ the approximations (3.7) and
(3.8) with $p=5$ yield maximal error of the order $10^{-14}$. The reported
error is based on the identity (2.24). The approximation (3.7) based only on
the Landen transformation is remarkably simple and precise as it makes no use
of elliptic integrals. One could also use this identity (2.23) to test the
above algorithm.

######  3.12.

Some open problems. Computational experiments have led us to formulate the
following questions:
(1) Let $L_{\varphi}(K,r,p)$ be defined as in (3.7). Then

$L_{\varphi}(K,r,p)\leq 4^{1-\frac{1}{K}}r^{\frac{1}{K}}$

for $1<K<4.6$, $r\in(0,0.7]$, and $p=0,1,2,\ldots$.

Motivation. Considering that $p=0$ is obvious, we may assume that
$p=1,2,3,\ldots$. Since $L(\cdot,p):(0,1)\rightarrow(0,1)$ is an increasing
homeomorphism, we are looking for $r\in(0,1)$ and $K>1$ such that
$4^{1-1/K}L(r,-p)^{1/K}<1$. Computer experiments show that
$4^{1-1/K}L(r,-p)^{1/K}<1$ holds true for all $r\in(0,0.7]$, $1<K<4.6$, and
$p=1,2,\ldots$.

(2) Remark 3.9 only deals with the case $p=1$. What about $p=2$? Can we find
some pair of functions $u,v$ where $u$ is a minorant of $\mu$ and $v$ a
majorant of $\mu$ such that the corresponding $L_{\varphi}(K,r,1)$ would be a
majorant of $\varphi_{K}$?

Funding. The first author received financial support provided by the Doctoral
Programme (formerly MATTI and now EXACTUS) of the Department of Mathematics
and Statistics of the University of Turku. The research of the second author
was supported by the Finnish Cultural Foundation.

## References

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  * [2] G.D. Anderson, S.-L. Qiu, M.K. Vamanamurthy and M. Vuorinen,  Generalized elliptic integrals and modular equations. Pacific J. Math. 192, 1–37 (2000)
  * [3] G.D. Anderson, M.K. Vamanamurthy and M. Vuorinen,  Distortion functions for plane quasiconformal mappings. Israel J. Math. 62 (1988), no. 1, 1–16.
  * [4] G.D. Anderson, M.K. Vamanamurthy and M. Vuorinen,  _Conformal invariants, inequalities and quasiconformal maps._ J. Wiley, 1997.
  * [5] G.D. Anderson, M. Vuorinen, and X. Zhang, Topics in special functions III. Analytic number theory, approximation theory, and special functions, 297–345, Springer, New York, 2014.
  * [6] J.W. Borwein and P.B. Borwein,  _Pi and the AGM._ Canadian Mathematical Society Series of Monographs and Advanced Texts, 4. John Wiley & Sons, Inc., New York, 1998. A study in analytic number theory and computational complexity, Reprint of the 1987 original, A Wiley-Interscience Publication.
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# Packing measure and dimension of the limit sets of IFSs of generalized
complex continued fractions1112010 Mathematics Subject Classification: 28A78,
28A80

Kanji INUI222corresponding author
Center for Mathematics, School of Fundamental Science and Technology,
Faculty of Science and Technology, Keio University
3-14-1, Hiyoshi, Kohoku-ku, Yokohama, 223-8522, JAPAN
E-mail<EMAIL_ADDRESS>

Hiroki SUMI
Course of Mathematical Science, Department of Human Coexistence,
Graduate School of Human and Environmental Studies, Kyoto University
Yoshida-nihonmatsu-cho, Sakyo-ku, Kyoto, 606-8501, JAPAN
E-mail<EMAIL_ADDRESS>
Homepage: http://www.math.h.kyoto-u.ac.jp/~sumi/index.html

###### Abstract

We consider a family of conformal iterated function systems (for short, CIFSs)
of generalized complex continued fractions which is a generalization of the
CIFS of complex continued fractions. We show the packing dimension and the
Hausdorff dimension of the limit set of each CIFS in the family are equal and
the packing measure of the limit set with respect to the packing dimension of
the limit set is finite in order to present new and interesting examples of
infinite CIFSs. Note that the Hausdorff measure of the limit set with respect
to the Hausdorff dimension is zero. To prove the above results, we consider
three cases (essentially two cases) and define a ‘nice’ subset of the index
set of the CIFS in each case. In addition, we estimate the cardinality of the
‘nice’ subsets and the conformal measure of the CIFSs. 333Keywords: infinite
conformal iterated function systems, fractal geometry, packing measures,
packing dimension, generalized complex continued fractions.

##  1 Introduction and the main results

Fractal geometry has been developed in order to study the geometrical
properties of fractals. One of the major studies in Fractal geometry is the
study of estimating the dimensions and measures of fractals. By estimating the
dimensions and measures of fractals, it is possible to explain phenomena that
appear in fractals which are different from the ones that appear in ‘usual
figures’ (see [4]). For this reason, the study of estimating the dimensions
and measures of fractals has been the major topic since Fractal geometry began
to attract attention not only in mathematics but also in many other fields.

Mathematically speaking, iterated function systems are a powerful method to
construct fractals (more precisely, limit set) and in many papers the study of
estimating the dimensions and measures of the limit sets has been studied. For
example, Mauldin’s and Urbański’s paper [5] presents the general theory of
estimating the dimensions and measures of the limit set constructed by
conformal iterated function systems with finitely many mappings (for short,
finite CIFSs). Note that by the formula and results in [5], we obtain some
estimations of the dimensions and measures of the limit sets. Indeed, by the
formula on the Hausdorff dimension of the limit set constructed by finite
CIFSs in [5] we obtain the estimation of the Hausdorff dimension of the limit
sets, and by the theorem in [5] we deduce the positiveness and finiteness of
the Hausdorff measure and the packing measure of the limit sets with respect
to the Hausdorff dimension of the limit sets. By these results, we obtain that
the packing dimension of the limit set constructed by finite CIFSs equals the
Hausdorff dimension of the limit set, we obtain the positiveness and
finiteness of the Hausdorff measure of the limit set constructed by finite
CIFSs with respect to the Hausdorff dimensions of the limit set, and we obtain
the positiveness and finiteness of the packing measure of the limit set with
respect to the packing dimension of the limit set.

In addition, Mauldin’s and Urbański’s paper [5] presents the general theory of
estimating the dimensions and measures of the fractals (more precisely, limit
sets too) constructed by conformal iterated function systems with infinitely
many mappings (for short, infinite CIFSs), and now, there are many results of
the limit set constructed by infinite CIFSs (for example, see [6], [8], [3],
[7], [9], [10] and so on). Note that they generalized the above formula and
theorems, and by the generalized theorems in their paper we may obtain non-
positiveness of the Hausdorff measure of the limit set constructed by infinite
CIFSs with respect to the Hausdorff dimension of the limit sets. This theorem
indicates we may find a new phenomenon of infinite CIFSs which cannot hold in
finite CIFSs.

Moreover, in Mauldin’s and Urbański’s paper [5], they constructed an
interesting example of an infinite CIFS and the limit set which is related to
the complex continued fractions. The precise construction of the example is
the following. Let $X:=\\{z\in\mathbb{C}\ |\ |z-1/2|\leq 1/2\\}$. We call
$\hat{S}:=\\{\hat{\phi}_{(m,n)}(z)\colon X\to X\ |\
(m,n)\in\mathbb{Z}\times\mathbb{N}\\}$ the CIFS of complex continued
fractions, where $\mathbb{Z}$ is the set of integers, $\mathbb{N}$ is the set
of positive integers and

$\displaystyle\hat{\phi}_{(m,n)}(z):=\frac{1}{z+m+ni}\quad(z\in X).$

Let $\hat{J}$ be the limit set of $\hat{S}$ (see Definition 2.1) and $\hat{h}$
be the Hausdorff dimension of $\hat{J}$. For each $s\geq 0$, we denote by
$\mathcal{H}^{s}$ the $s$-dimensional Hausdorff measure and denoted by
$\mathcal{P}^{s}$ the $s$-dimensional packing measure. For this example,
Mauldin and Urbański showed the following theorem.

######  Theorem 1.1 (D. Mauldin, M. Urbanski (1996)).

Let $\hat{S}$ be the CIFS of complex continued fractions as above. Then, we
have that $\mathcal{H}^{\hat{h}}(\hat{J})=0$ and
$0<\mathcal{P}^{\hat{h}}(\hat{J})<\infty$.

Note that they obtained an example of infinite CIFS for which the Hausdorff
measure of the limit set with respect to the Hausdorff dimension of the limit
set is zero and the packing measure of the limit set with respect to the
packing dimension of the limit set is positive and finite. That is, they found
a new phenomenon of infinite CIFSs which cannot hold in finite CIFSs.

In our previous papers [1] and [2], we considered a family of CIFSs of
generalized complex continued fractions which is a generalization of the CIFS
$\hat{S}$ of complex continued fractions, in order to present new and
interesting examples of infinite CIFSs. We estimated the Hausdorff dimension
of the limit set of each CIFS of the generalized complex continued fractions
and showed non-positiveness of the Hausdorff measure and positiveness of the
packing measure of the limit set with respect to the Hausdorff dimension of
the limit set. Note that the family of the CIFSs introduced in the papers [1]
and [2] has uncountably many elements. On the other hand, we did not obtain
results on the relationship between the Hausdorff dimension and the packing
dimension and the finiteness of the packing measure of the limit set with
respect to the packing dimensions of the limit set in [1] and [2], and we have
been interested in the relationship and the finiteness of the packing measure
of the limit set. The aim of this paper is to show the relationship between
the Hausdorff dimension and the packing dimension of each limit set and the
finiteness of the packing measure of each limit set with respect to the
packing dimensions in the family of the CIFSs of the generalized complex
continued fractions, in order to find new, interesting and uncountably
infinite examples of infinite CIFSs with the phenomenon which cannot hold in
finite CIFSs.

The precise statement is the following. Let

$\displaystyle A_{0}:=\\{\tau=u+iv\in\mathbb{C}\ |\ u\geq 0\ \mathrm{and}\
v\geq 1\\}\quad\text{and}\quad X:=\\{z\in\mathbb{C}\ |\ |z-1/2|\leq 1/2\\},$

and we set $I_{\tau}:=\\{m+n\tau\in\mathbb{C}\ |\ m,n\in\mathbb{N}\\}$ for
each $\tau\in A_{0}$, where $\mathbb{N}$ is the set of the positive integers.

######  Definition 1.2 (The CIFS of generalized complex continued fractions).

Let $\tau\in A_{0}$. Then, we say that $S_{\tau}:=\\{\phi_{b}\colon
X\rightarrow X\ |\ b\in I_{\tau}\\}$ is the CIFS of generalized complex
continued fractions. Here, for each $\tau\in A_{0}$ and $b\in I_{\tau}$,
$\phi_{b}$ is defined by

$\displaystyle\phi_{b}(z):=\frac{1}{z+b}\quad(z\in X).$

We call $\\{S_{\tau}\\}_{\tau\in A_{0}}$ the family of CIFSs of generalized
complex continued fractions. For each $\tau\in A_{0}$, let $J_{\tau}$ be the
limit set of the CIFS $S_{\tau}$ (see Definition 2.1) and let $h_{\tau}$ be
the Hausdorff dimension of the limit set $J_{\tau}$. We remark that this
family of the CIFSs is a generalization of $\hat{S}$ in some sense. Indeed,
$S_{\tau}$ is related to ‘generalized’ complex continued fractions since each
point of the limit set $J_{\tau}$ of $S_{\tau}$ is of the form

$\cfrac{1}{b_{1}+\cfrac{1}{b_{2}+\cfrac{1}{b_{3}+\cdots}}}$

for some sequence $(b_{1},b_{2},b_{3},\ldots)$ in $I_{\tau}$ (See Definition
2.1). Note that there are many kinds of general theories for continued
fractions and related iterated function systems ([3], [5], [6], [8]). In [2],
we showed the following theorem.

######  Theorem 1.3 ([2, Theorem 1.3]).

Let $\\{S_{\tau}\\}_{\tau\in A_{0}}$ be the family of CIFSs of generalized
complex continued fractions. Then, for each $\tau\in A_{0}$, we have
$\mathcal{H}^{h_{\tau}}(J_{\tau})=0$ and $0<\mathcal{P}^{h_{\tau}}(J_{\tau})$.

We now present the main theorem in this paper.

######  Theorem 1.4 (the main theorem).

Let $\\{S_{\tau}\\}_{\tau\in A_{0}}$ be the family of CIFSs of generalized
complex continued fractions. Then, for each $\tau\in A_{0}$, we have
$\mathcal{P}^{h_{\tau}}(J_{\tau})<\infty$.

Combining Theorem 1.3 and Theorem1.4, we obtain the following corollary.

######  Corollary 1.5.

Let $\\{S_{\tau}\\}_{\tau\in A_{0}}$ be the family of CIFSs of generalized
complex continued fractions. Then, for each $\tau\in A_{0}$, we have
$0<\mathcal{P}^{h_{\tau}}(J_{\tau})<\infty$. In particular, for each $\tau\in
A_{0}$, the packing dimension of the limit set $J_{\tau}$ equals the Hausdorff
dimension $h_{\tau}$ of the limit set $J_{\tau}$.

######  Remark 1.6.

By the general theory of finite CIFSs, the Hausdorff measure of the limit set
of each finite CIFS with respect to the Hausdorff dimension of the limit set
and the packing measure of the limit set with respect to the Hausdorff
dimension of the limit set is positive and finite. However, Corollary 1.5
indicates that for each $S_{\tau}$ of the family of CIFSs of generalized
complex continued fractions which consists of uncountably many elements, the
packing dimension of the limit set equals the Hausdorff dimension of the limit
set and the Hausdorff measure of the limit set with respect to the Hausdorff
dimension of the limit set is zero and the packing measure of the limit set
with respect to the Hausdorff dimension (which equals the packing dimension of
the limit set) is positive and finite. This is also a new phenomenon which
cannot hold in the finite CIFSs.

######  Remark 1.7.

It was shown that for each $\tau\in A_{0}$, $\overline{J_{\tau}}\setminus
J_{\tau}$ is at most countable and
$h_{\tau}=\dim_{\mathcal{H}}(\overline{J_{\tau}}$) ([1]). Thus, for each
$\tau\in A_{0}$, we have
$0<\mathcal{P}^{h_{\tau}}(\overline{J_{\tau}})=\mathcal{P}^{h_{\tau}}(J_{\tau})<\infty$.
Also, for each $\tau\in A_{0}$, since the set of attracting fixed points of
elements of the semigroup generated by $S_{\tau}$ is dense in $J_{\tau}$,
Theorem 1.1 of [11] implies that $\overline{J_{\tau}}$ is equal to the Julia
set of the rational semigroup generated by $\\{\phi_{b}^{-1}\mid b\in
I_{\tau}\\}$.

The ideas and strategies to prove the main theorem are the following. To prove
the finiteness of the packing measure of the limit set $J_{\tau}$, we apply
Lemma 4.10 in the paper [5] to $S_{\tau}$ for each $\tau\in A_{0}$. That is,
it suffices to show that for each $r>0$ (which is sufficiently small) and
$b\in I_{\tau}$ with $\mathrm{diam}\phi_{b}(X)/r\ll 1$, we have

$\displaystyle m_{\tau}(B(x,r))\gtrsim r^{h_{\tau}},$ (1.0.1)

where $x:=1/b$, $B(x,r)$ is the open ball with the center $x$ and the radius
$r$ with respect to the Euclidean distance in $\mathbb{C}$, $m_{\tau}$ is
$h_{\tau}$-conformal measure of $S_{\tau}$ (see Theorem 2.8) and ‘$f(r)\gtrsim
g(r)$’ means that there exists a ‘small’ constant $c>0$ such that $f(r)\geq
cg(r)$ for all $r>0$. Note that there is a useful inequality for the conformal
measure $m_{\tau}$ :

$\displaystyle m_{\tau}\left(\bigcup_{a\in I}\phi_{a}(X)\right)=\sum_{a\in
I}m_{\tau}\left(\phi_{a}(X)\right)=\sum_{a\in
I}\int_{X}|\phi_{a}^{\prime}(y)|^{h_{\tau}}m_{\tau}(\mathrm{d}y)\gtrsim\sum_{a\in
I}\ |a|^{-2h_{\tau}}m_{\tau}(X)=\sum_{a\in I}\ |a|^{-2h_{\tau}}$ (1.0.2)

for each $I\subset I_{\tau}$, where we use the property on the $S_{\tau}$ (see
Lemma 3.2). To prove the above inequality (1.0.1), we essentially consider the
following two cases:

  1. 1.

$r\ll|x|$ and

  2. 2.

$r\gg|x|$.

In the first case, by the assumptions, we deduce that $|x|^{2}\leq r$ and
$r<|x|$. We next define $I_{\tau,1}\subset I_{\tau}$ and show the following
inclusions and inequality:

$\displaystyle B(x,r)\supset\bigcup_{a\in
I_{\tau,1}}\phi_{a}(X),\quad|I_{\tau,1}|\gtrsim\left(\frac{r}{|x|^{2}}\right)^{2}\quad\text{and}\quad|a|\lesssim|x|^{-1}$

for each $a\in I_{\tau,1}$. Here, for any set $A$, we denote by $|A|$ the
cardinality of $A$. To prove the above inclusion and inequalities, we prove
some additional lemmas. Therefore, by the above inclusion, the inequality
(1.0.2) and the above inequalities, we have

$\displaystyle m_{\tau}(B(x,r))$ $\displaystyle\geq
m_{\tau}\left(\bigcup_{a\in I_{\tau,1}}\phi_{a}(X)\right)\gtrsim\sum_{a\in
I_{\tau,1}}\ |a|^{-2h_{\tau}}\gtrsim|I_{\tau,1}|\cdot|x|^{2h_{\tau}}$
$\displaystyle\gtrsim r^{2}\cdot|x|^{2h_{\tau}-4}\gtrsim r^{2}\cdot
r^{h_{\tau}-2}=r^{h_{\tau}},$

where we use the inequality $|x|^{2}\leq r$ and $h_{\tau}<2$.

In the second case, since $r\gg|x|$, we have $B(0,\tilde{r})\subset B(x,r)$,
where $\tilde{r}=cr$ and $c>0$ is a ‘small’ positive number. Next, we define
$I_{\tau}(\tilde{r})\subset I_{\tau}$ and show the following inclusion and
inequalities:

$\displaystyle B(0,\tilde{r})\supset\bigcup_{a\in
I_{\tau}(\tilde{r})}\phi_{a}(X),\quad|I_{\tau}(\tilde{r})|\gtrsim
r^{-2}\quad\text{and}\quad|a|\lesssim r^{-1}$

for each $a\in I_{\tau}(\tilde{r})$ (see Lemma 3.5). Therefore, by the above
inclusion, the inequality (1.0.2) and the above inequalities, we have

$\displaystyle m_{\tau}(B(x,r))$ $\displaystyle\geq
m_{\tau}(B(0,\tilde{r}))\geq m_{\tau}\left(\bigcup_{a\in
I_{\tau}(\tilde{r})}\phi_{a}(X)\right)\gtrsim\sum_{a\in I_{\tau}(\tilde{r})}\
|a|^{-2h_{\tau}}$ $\displaystyle\gtrsim|I_{\tau}(\tilde{r})|\cdot
r^{2h_{\tau}}\gtrsim r^{2h_{\tau}-2}>r^{h_{\tau}},$

where we use the inequalities $r^{2h_{\tau}-2}>r^{h_{\tau}}$ since $r$ is
sufficiently small and $h_{\tau}<2$.

The rest of the paper is organized as follows. In Section 2, we summarize the
general theory of the CIFSs and recall some definitions and theorems in the
theory. In Section 3, we present some results for the CIFSs of generalized
complex continued fractions in the paper [1] and [2]. Also, we prove a slight
modification of lemmas in the paper [1] and [2] to prove Theorem 1.4. In
Section 4, we prove the main theorem (Theorem 1.4). In this section, we first
show some additional lemmas to prove Theorem 1.4 and next present the setting
for the proof of the main theorem. Then, we finally show Theorem 1.4. To prove
Theorem 1.4, we consider three cases.

##  2 Conformal iterated function systems

In this section, we summarize the general theory of CIFSs ([1], [2], [5],
[6]). We first recall the definition of CIFSs and the limit set of the CIFSs.

######  Definition 2.1 (Conformal iterated function system).

Let $X\subset\mathbb{R}^{d}$ be a non-empty compact and connected set with the
Euclidean norm $|\cdot|$ and let $I$ be a finite set or bijective to
$\mathbb{N}$. Suppose that $I$ has at least two elements. We say that
$S:=\\{\phi_{i}\colon X\to X\ |\ i\in I\\}$ is a conformal iterated function
system (for short, CIFS) if $S$ satisfies the following conditions.

  1. 1.

Injectivity: $\phi_{i}\colon X\to X$ is injective for each $i\in I$.

  2. 2.

Uniform Contractivity: There exists $c\in(0,1)$ such that, for all $i\in I$
and $x,y\in X$, the following inequality holds:

$\displaystyle|\phi_{i}(x)-\phi_{i}(y)|\leq c|x-y|.$

  3. 3.

Conformality: There exists $\epsilon>0$ and an open and connected subset
$V\subset\mathbb{R}^{d}$ with $X\subset V$ such that for all $i\in I$,
$\phi_{i}$ extends to a $C^{1+\epsilon}$diffeomorphism on $V$ and $\phi_{i}$
is conformal on $V$ i.e. for each $x\in V$ and $i\in I$, there exists
$C_{i}(x)>0$ such that for each $u,v\in\mathbb{R}^{d}$,

$\displaystyle|\phi_{i}^{\prime}(x)u-\phi_{i}^{\prime}(x)v|=C_{i}(x)|u-v|.$

Here, $\phi_{i}(x)$ denotes the derivative of $\phi_{i}$ at $x\in V$.

  4. 4.

Open Set Condition (OSC): For all $i,j\in I\ (i\neq j)$,
$\phi_{i}(\mathrm{Int}(X))\subset\mathrm{Int}(X)$ and
$\phi_{i}(\mathrm{Int}(X))\cap\phi_{j}(\mathrm{Int}(X))=\emptyset$. Here,
$\mathrm{Int}(X)$ denotes the set of interior points of $X$ with respect to
the topology in $\mathbb{R}^{d}$.

  5. 5.

Bounded Distortion Property(BDP): There exists $\tilde{K}\geq 1$ such that for
all $x,y\in V$ and for all $w\in I^{*}:=\bigcup_{n=1}^{\infty}I^{n}$, the
following inequality holds:

$\displaystyle|\phi^{\prime}_{w}(x)|\leq\tilde{K}\cdot|\phi^{\prime}_{w}(y)|.$

Here, for each $n\in\mathbb{N}$ and $w=w_{1}w_{2}\cdots w_{n}\in I^{n}$, we
set $\phi_{w}:=\phi_{w_{1}}\circ\phi_{w_{2}}\circ\cdots\circ\phi_{w_{n}}$ and
$\displaystyle|\phi^{\prime}_{w}(x)|$ denotes the norm of the derivative of
$\phi_{w}$ at $x\in X$ with respect to the Euclidean norm on $\mathbb{R}^{d}$.

  6. 6.

Cone Condition: For all $x\in\partial X$, there exists an open cone
$\mathrm{Cone}(x,u,\alpha)$ with a vertex $x$, a direction $u$, an altitude
$|u|$ and an angle $\alpha$ such that $\mathrm{Cone}(x,u,\alpha)$ is a subset
of $\mathrm{Int}(X)$.

We endow $I$ with the discrete topology and endow $I^{\infty}:=I^{\mathbb{N}}$
with the product topology. Note that $I^{\infty}$ is Polish in general and
$I^{\infty}$ is a compact metrizable space if $I$ is a finite set.

Let $S$ be a CIFS and we set $w|_{n}:=w_{1}w_{2}\cdots w_{n}\in I^{n}$ and
$\phi_{w|_{n}}:=\phi_{w_{1}}\circ\phi_{w_{2}}\circ\cdots\circ\phi_{w_{n}}$ for
each $w=w_{1}w_{2}w_{3}\cdots\in I^{\infty}$. Note that
$\bigcap_{n\in\mathbb{N}}\phi_{w|_{n}}(X)$ is a singleton (denoted by
$\\{x_{w}\\}$) and the coding map $\pi_{S}\colon I^{\infty}\rightarrow X$ of
$S$ defined by $\pi_{S}(w):=x_{w}$ is well-defined. Then, the limit set
$J_{S}$ of $S$ is defined by

$\displaystyle J_{S}:=\pi(I^{\infty})=\bigcup_{w\in
I^{\infty}}\bigcap_{n\in\mathbb{N}}\phi_{w|_{n}}(X)(\subset
X\subset\mathbb{R}^{d}).$

We set $h_{S}:=\dim_{\mathcal{H}}J_{S}$, where we denote by
$\dim_{\mathcal{H}}A$ the Hausdorff dimension of a set
$A\subset\mathbb{R}^{d}$ with respect to the Euclidean distance.

We next recall the pressure function of CIFS $S$ as follows in order to define
the regularity of CIFSs.

######  Definition 2.2.

For each $n\in\mathbb{N}$, $[0,\infty]$-valued function $\psi^{n}_{S}$ is
defined by

$\displaystyle\psi^{n}_{S}(t):=\sum_{w\in
I^{n}}||\phi_{w}^{\prime}||_{X}^{t}\quad(t\geq 0).$

Here, for a $C^{1}$ map $f\colon Y\to\mathbb{R}^{d}$
($Y\subset\mathbb{R}^{d}$), we set

$\displaystyle|f^{\prime}(y)|:=\sup\\{|f^{\prime}(y)u|\ |\
u\in\mathbb{R}^{d},|u|=1\\}\ (y\in
Y)\quad\text{and}\quad||f^{\prime}||_{Y}:=\sup\\{|f^{\prime}(y)|\ |\ y\in
Y\\}.$

We set $\theta_{S}:=\inf\\{t\geq 0|\ \psi_{S}^{1}(t)<\infty\\}$ and
$F(S):=\\{t\geq 0\ |\ \psi^{1}_{S}(t)<\infty\\}$. Note that by the following
lemma, we deduce that $F(S)=(\theta_{S},\infty)$ or
$F(S)=[\theta_{S},\infty)$.

######  Lemma 2.3 ([5]).

Let $S$ be a CIFS. Then, $\psi_{S}^{1}(t)$ is non-increasing on $[0,\infty)$,
and decreasing and convex on $F(S)$. In addition, we have
$\psi_{S}^{1}(d)\leq\tilde{K}^{d}$. In particular, $\theta_{S}\leq d$.

In addition, by the following proposition, we deduce the basic properties of
$\psi_{S}^{n}$.

######  Proposition 2.4 ([5]).

Let $S$ be a CIFS. For all $m,n\in\mathbb{N}$ and $t\geq 0$, we have

$\tilde{K}^{-2t}\psi_{S}^{k}(t)\psi_{S}^{n}(t)\leq\psi_{S}^{m+n}(t)\leq\psi_{S}^{m}(t)\psi_{S}^{n}(t).$

In particular, $\psi_{S}^{n}(t)<\infty$ for each $n\in\mathbb{N}$ if and only
if $\psi_{S}^{n}(t)<\infty$ for some $n\in\mathbb{N}$ (or $n=1$), and the
function $n\mapsto\log\psi_{S}^{n}(t)$ is subadditive for all $t\geq 0$. By
the subadditivity of $\log\psi_{S}^{n}(t)$, we now define the pressure
function of $S$ as follows.

######  Definition 2.5 (Pressure function).

The pressure function of $S$ is the function
$P_{S}\colon[0,\infty)\to(-\infty,\infty]$ defined by

$\displaystyle\displaystyle
P_{S}(t):=\lim_{n\to\infty}\frac{1}{n}\log\psi^{n}_{S}(t)\in(-\infty,\infty]\quad(t\geq
0).$

######  Proposition 2.6 ([5]).

Let $S$ be a CIFS and $P_{S}$ be the pressure function of $S$. Then,
$P_{S}(t)<\infty$ if and only if $\psi_{S}^{1}(t)<\infty$ for each $t\geq 0$.
In particular, $\theta_{S}=\inf\\{t\geq 0\ |\ P_{S}(t)<\infty\\}$. In
addition, $P_{S}$ is non-increasing on $[0,\infty)$, and decreasing and convex
on $F(S)$.

Note that $P_{S}(0)=\infty$ if and only if $I$ is infinite. By using the
pressure function in Definition 2.5, we define the regularity of CIFSs.

######  Definition 2.7 (Regular, Strongly regular, Hereditarily regular).

Let $S$ be a CIFS. We say that

  * •

$S$ is regular if there exists $t\geq 0$ such that $P_{S}(t)=0$,

  * •

$S$ is strongly regular if there exists $t\geq 0$ such that
$P_{S}(t)\in(0,\infty)$ and

  * •

$S$ is hereditarily regular if, for all $I^{\prime}\subset I$ with
$|I\setminus I^{\prime}|<\infty$, $S^{\prime}:=\\{\phi_{i}\colon X\to X\ |\
i\in I^{\prime}\\}$ is regular.

Here, for any set $A$, we denote by $|A|$ the cardinality of $A$.

Note that if a CIFS $S$ is hereditarily regular then $S$ is strong regular,
and if $S$ is strong regular then $S$ is regular.

We finally recall the $h_{S}$-conformal measure of $S$. If a CIFS $S$ is
regular, there is the following ‘nice’ probability measure $m_{S}$
($h_{S}$-conformal measure of $S$) on $J_{S}$. Indeed, we often use $m_{S}$ in
order to estimate the packing measure of the limit set of CIFSs.

######  Theorem 2.8 ([5] Lemma 3.13).

Let $S$ be a CIFS. If $S$ is regular, then there exists the unique Borel
probability measure $m_{S}$ on $X$ such that the following properties hold.

  1. 1.

$m_{S}(J_{S})=1$.

  2. 2.

For all Borel subset $A$ on $X$ and $i\in I$,
$m_{S}(\phi_{i}(A))=\int_{A}|\phi_{i}^{\prime}(y)|^{h_{S}}m_{S}(\mathrm{d}y)$.

  3. 3.

For all $i,j\in I$ with $i\neq j$, $m_{S}(\phi_{i}(X)\cap\phi_{j}(X))=0$.

We call $m_{S}$ the $h_{S}$-conformal measure of $S$. As we mentioned above,
by the existence of the conformal measure of CIFSs, we estimate the packing
measure and obtain the following key theorem to prove Theorem 1.4.

######  Theorem 2.9 ([5] Lemma 4.10).

Let $S$ be a regular CIFS and $m_{S}$ be the $h_{S}$-conformal measure of $S$.
Suppose that there exist $L>0$, $\xi>0$ and $\gamma\geq 1$ such that for all
$b\in I$ and $r>0$ with $\gamma\cdot\mathrm{diam}\phi_{b}(X)\leq r\leq\xi$,
there exists $x\in\phi_{b}(V)$ such that $m_{S}(B(x,r))\geq Lr^{h_{S}}$,
$B(x,r)$ is the open ball with the center $x$ and the radius $r$ with respect
to the Euclidean distance in $\mathbb{R}^{d}$. Then, we have
$\mathcal{P}^{h_{S}}(J_{S})<\infty$.

##  3 CIFSs of generalized complex continued fractions

In this section, we present some results on the CIFSs of generalized complex
continued fractions introduced in the papers [1] and [2], which are needed to
prove the results of this paper. Note that these CIFSs are important and
interesting examples of infinite CIFSs. Rest of this paper, we denote by
$B(y,r)\subset\mathbb{R}^{d}(d\in\mathbb{N})$ the open ball with center
$y\in\mathbb{R}^{d}$ and radius $r>0$, with respect to the $d$-dimensional
Euclidean norm and we identify $\mathbb{C}$ with $\mathbb{R}^{2}$.

We first present the following lemma shown in [1] and [2] in order to prove
Theorem 1.4.

######  Lemma 3.1 (Lemmas 3.1, 3.3 and 3.4 in [2]).

For all $\tau\in A_{0}$, $S_{\tau}$ is a hereditarily regular CIFS . In
addition, we have $1<h_{\tau}<2$.

In addition, in order to prove Theorem 1.4, we next prove the following lemma
(a slight modification of Lemma 3.2 in [2]). For the readers, we give a proof
of Lemma 3.2.

######  Lemma 3.2.

Let $\tau\in A_{0}$. Then, there exists $K_{0}\geq 1$ such that for all $K\geq
K_{0}$ and $a\in I_{\tau}$, the following properties hold.

  1. 1.

$\phi_{a}(X)\subset B(0,K|a|^{-1})$.

  2. 2.

$K^{-1}|a|^{-2}\leq|\phi_{a}^{\prime}(z)|\leq K|a|^{-2}$ for each $z\in X$.

  3. 3.

$K^{-1}|a|^{-2}\leq\mathrm{diam}{\phi_{a}(X)}$.

###### Proof.

Let $\tau\in A_{0}$. Note that by using the BDP, there exists a constant
$C\geq 1$ such that for all $z,w\in X$,

$\displaystyle|\phi^{\prime}_{a}(z)|\leq C\cdot|\phi^{\prime}_{a}(w)|.$
(3.0.1)

We set $K_{0}:=C(\geq 1)$ and let $K\geq K_{0}$ and $a\in I_{\tau}$. Then, by
the inequality (3.0.1) with $w=0\in X$, we have

$\displaystyle|\phi_{a}(z)|\cdot|a|=\frac{|a|}{|a+z|}=\sqrt{\frac{|a|^{2}}{|a+z|^{2}}}=\sqrt{\frac{|\phi_{a}^{\prime}(z)|}{|\phi_{a}^{\prime}(0)|}}\leq\sqrt{C}\leq
K_{0}\leq K.$

for each $z\in X$. It follows that $\phi_{a}(X)\subset B(0,K|a|^{-1})$. Also,
by the inequality (3.0.1), we have

$\displaystyle K^{-1}|a|^{-2}=K^{-1}|\phi_{a}^{\prime}(0)|\leq
C^{-1}|\phi_{a}^{\prime}(0)|\leq|\phi_{a}^{\prime}(z)|\quad\text{and}\quad|\phi_{a}^{\prime}(z)|\leq
C|\phi_{a}^{\prime}(0)|\leq K|\phi_{a}^{\prime}(0)|=K|a|^{-2}$

for each $z\in X$, which deduce that
$K^{-1}|a|^{-2}\leq|\phi_{a}^{\prime}(z)|\leq K|a|^{-2}$. Moreover, by the
inequality (3.0.1), we have

$\displaystyle\mathrm{diam}{\phi_{a}(X)}\cdot|a|^{2}$
$\displaystyle\geq|\phi_{a}(z)-\phi_{a}(w)|\cdot|a|^{2}=\left|\frac{1}{z+a}-\frac{1}{w+a}\right|\cdot|a|^{2}=\frac{|w-z|}{|z+a||w+a|}\cdot|a|^{2}$
$\displaystyle=|w-z|\cdot\sqrt{\frac{|a|^{2}}{|a+z|^{2}}}\cdot\sqrt{\frac{|a|^{2}}{|a+w|^{2}}}=|w-z|\cdot\sqrt{\frac{|\phi_{a}^{\prime}(z)|}{|\phi_{a}^{\prime}(0)|}}\cdot\sqrt{\frac{|\phi_{a}^{\prime}(w)|}{|\phi_{a}^{\prime}(0)|}}\geq|w-z|\cdot
C^{-1}$

for all $z,w\in X=B(1/2,1/2)$. Since $\mathrm{diam}X=\sup\\{|z-w|\ |\ z,w\in
X\\}=1$, we obtain that $\mathrm{diam}{\phi_{a}(X)}\geq C^{-1}|a|^{-2}\geq
K^{-1}|a|^{-2}$. Therefore, we have proved our lemma. ∎

We recall the following notations used in the paper [2]. We identify
$I_{\tau}$ with $\\{{}^{t}(s,t)\in\mathbb{R}^{2}\ |\ s+it\in I_{\tau}\\}$ and
$\mathbb{N}^{2}$ with $\\{{}^{t}(m,n)\in\mathbb{R}^{2}\ |\
m,n\in\mathbb{N}\\}$, where for any matrix $A$, we denote by ${}^{t}A$ the
transpose of $A$. For each $\tau=u+iv\in A_{0}$, we set

$\displaystyle E_{\tau}:=\left(\begin{array}[]{cc}1&u\\\
0&v\end{array}\right)\quad\text{and}\quad
F_{\tau}:={}^{t}E_{\tau}E_{\tau}=\left(\begin{array}[]{cc}1&u\\\
u&|\tau|^{2}\end{array}\right).$

Note that by direct calculations, $E_{\tau}\mathbb{N}^{2}=I_{\tau}$,
$E_{\tau}$ is invertible and there exist the eigenvalues $\lambda_{1}>0$ and
$\lambda_{2}>0$ of $F_{\tau}$ with $\lambda_{1}<\lambda_{2}$. Note that since
$F_{\tau}$ is symmetric, there exist an eigenvector $v_{1}\in\mathbb{R}^{2}$
of $F_{\tau}$ with respect to $\lambda_{1}$ and an eigenvector
$v_{2}\in\mathbb{R}^{2}$ of $F_{\tau}$ with respect to $\lambda_{2}$ such that
$V_{\tau}:=(v_{1},v_{2})$ is an orthogonal matrix.

For each $\tau\in A_{0}$, we set
$N_{\tau}:=\sqrt{2\lambda_{2}}/\sqrt{\lambda_{1}}+1\ (>2)$. In addition, for
each $\tau\in A_{0}$ and $R>0$, we set

$\displaystyle D_{1}(\tau,R):=\\{^{t}(x,y)\in\mathbb{R}^{2}\ |\
R^{2}/\lambda_{1}<x^{2}+y^{2}\leq(N_{\tau}R)^{2}/\lambda_{2}\\}\quad\text{and}$
$\displaystyle D_{2}(\tau,R):=\\{^{t}(x,y)\in\mathbb{R}^{2}\ |\
R^{2}<x^{2}+y^{2}\leq(N_{\tau}R)^{2}\\}.$

Note that $R/\sqrt{\lambda_{1}}<(N_{\tau}R)/\sqrt{\lambda_{2}}$ for each $R>0$
since $\sqrt{\lambda_{2}}/\sqrt{\lambda_{1}}<N_{\tau}$ and $R<N_{\tau}R$ for
each $R>0$ since $1<N_{\tau}$. By these notations, we now present the
following proposition and lemma shown in the paper [2].

######  Proposition 3.3 (Proposition 4.3 in [2]).

Let $R>0$. Then, for each $R\geq 6$,

$\displaystyle 0<\frac{R^{2}-7R+7}{2}\leq|\\{^{t}(m,n)\in\mathbb{N}^{2}\ |\
m^{2}+n^{2}\leq R^{2}\\}|\leq R^{2}.$

######  Lemma 3.4 (Lemmas 4.2 and 4.4 in [2]).

Let $\tau\in A_{0}$. Then, there exist $\tilde{C}_{\tau}>0$ and
$\tilde{L}_{\tau}>0$ such that for all $R>\tilde{C}_{\tau}$,

$\displaystyle|I_{\tau}\cap
D_{2}(\tau,R)|\geq\tilde{L}_{\tau}R^{2}-\frac{7N_{\tau}}{2\sqrt{\lambda_{2}}}R.$

We finally prove the following lemma (a slight modification of Lemma 3.4). For
the readers we give a proof of Lemma 3.5.

######  Lemma 3.5.

Let $\tau\in A_{0}$. Then, there exist $Q_{\tau}>0$ and $C_{\tau}>0$ such that
for all $R\geq C_{\tau}$, we have

$|I_{\tau}\cap D_{2}(\tau,R)|>Q_{\tau}R^{2}.$ (3.0.2)

###### Proof.

Let $\tau\in A_{0}$ and we set $Q_{\tau}:=\tilde{L}_{\tau}/2>0$, where
$\tilde{L}_{\tau}>0$ is the number in Lemma 3.4. Note that there exists
$C_{\tau}>\tilde{C}_{\tau}$ such that for all $R\geq C_{\tau}$,
$(\tilde{L}_{\tau}R^{2})/2-(7N_{\tau}R)/(2\sqrt{\lambda_{2}})>0$, which is
equivalent to

$\displaystyle\tilde{L}_{\tau}R^{2}-\frac{7N_{\tau}}{2\sqrt{\lambda_{2}}}R>\frac{\tilde{L}_{\tau}}{2}R^{2},$
(3.0.3)

where $\tilde{C}_{\tau}>0$ is the number in Lemma 3.4. By Lemma 3.4 and the
inequality (3.0.3), we deduce that

$\displaystyle|I_{\tau}\cap
D_{2}(\tau,R)|\geq\tilde{L}_{\tau}R^{2}-\frac{7N_{\tau}}{2\sqrt{\lambda_{2}}}R>\frac{\tilde{L}_{\tau}}{2}R^{2}=Q_{\tau}R^{2}$

for all $R\geq C_{\tau}$. Therefore, we have proved our lemma. ∎

##  4 Proof of the main theorem

In this section, we prove the main theorem (Theorem 1.4). Rest of this paper,
we use the notations in Section 3.

###  4.1 Lemmas for the proof of the main theorem

In this subsection, we prove the following lemmas to prove Theorem 1.4.

######  Lemma 4.1.

Let $f(z):=1/z\ (z\in\mathbb{C}\setminus\\{0\\})$. Then, for each
$B(x,r)(\subset\mathbb{C})$ with $r<|x|$, we have

$f(B(x,r))=B\left(\frac{|x|^{2}}{|x|^{2}-r^{2}}\cdot\frac{1}{x},\
\frac{r}{|x|^{2}-r^{2}}\right).$

###### Proof.

Let $a\in\mathbb{C}$. Then, $|1/a\ -\
\overline{x}/(|x|^{2}-r^{2})|=r/(|x|^{2}-r^{2})$ if and only if
$|r^{2}-\overline{x}(x-a)|=r|a|$, which is also equivalent to the following
equation:

$r^{4}-r^{2}(\overline{x}(x-a)+x(\overline{x}-\overline{a})+a\overline{a})+x\overline{x}(x-a)(\overline{x}-\overline{a})=0.$
(4.1.1)

Since
$(\overline{x}(x-a)+x(\overline{x}-\overline{a})+a\overline{a})=x\overline{x}+(x-a)(\overline{x}-\overline{a})$,
the equation (4.1.1) is equivalent to $(r^{2}-|x-a|^{2})(r^{2}-|x|^{2})=0$.
Since $0<r<|x|$, we deduce that
$|1/a-\overline{x}/(|x|^{2}-r^{2})|=r/(|x|^{2}-r^{2})$ if and only if
$r=|x-a|$. In addition, since $f(x)=1/x$ and $r<|x|$, we obtain that

$\left|f(x)-\frac{|x|^{2}}{|x|^{2}-r^{2}}\cdot\frac{1}{x}\right|=\frac{r}{|x|^{2}-r^{2}}\frac{r}{|x|}<\frac{r}{|x|^{2}-r^{2}}.$

That is, $f(x)\in B(\bar{x}/(|x|^{2}-r^{2}),r/(|x|^{2}-r^{2}))$ ). Therefore,
we have proved our lemma. ∎

######  Lemma 4.2.

Let $\tau\in A_{0}$, $\tilde{x}\in\mathbb{R}^{2}$ and $\tilde{R}>0$. Then, we
have

$E_{\tau}(B(E_{\tau}^{-1}\tilde{x},\tilde{R}/\sqrt{\lambda_{2}}))\subset
B(\tilde{x},\tilde{R}).$

###### Proof.

Let ${}^{t}(x,y)\in B(E_{\tau}^{-1}\tilde{x},\tilde{R}/\sqrt{\lambda_{2}})$.
We set $\tilde{y}:={}^{t}(x,y)-E_{\tau}^{-1}\tilde{x}$. Since $V_{\tau}$ is
orthogonal and
$|\tilde{y}|^{2}=|{}^{t}(x,y)-E_{\tau}^{-1}\tilde{x}|^{2}<\tilde{R}^{2}/\lambda_{2}$,
we have

$\displaystyle|E_{\tau}{}^{t}(x,y)-\tilde{x}|^{2}=|E_{\tau}\tilde{y}|^{2}={}^{t}\tilde{y}^{t}E_{\tau}E_{\tau}\tilde{y}={}^{t}\tilde{y}F_{\tau}\tilde{y}$
$\displaystyle={}^{t}\tilde{y}V_{\tau}\left(\begin{array}[]{cc}\lambda_{1}&0\\\
0&\lambda_{2}\end{array}\right){}^{t}V_{\tau}\tilde{y}=\lambda_{1}z_{1}^{2}+\lambda_{2}z_{2}^{2}\leq\lambda_{2}|V_{\tau}\tilde{y}|^{2}=\lambda_{2}|\tilde{y}|^{2}<\tilde{R}^{2},$

where ${}^{t}(z_{1},z_{2}):={}^{t}V_{\tau}\tilde{y}$. Therefore, we have
proved our lemma. ∎

######  Lemma 4.3.

Let $\tau\in A_{0}$, let $w\in\mathbb{R}^{2}$ and $\bar{R}>0$ with
$|w|>\bar{R}$. Then, for each $M\geq 2$, we have

$\displaystyle
E_{\tau}B\left(E_{\tau}^{-1}\left(w-\frac{\bar{R}}{M|w|}w\right),\frac{\bar{R}}{\sqrt{\lambda_{2}}M}\right)\subset
B(0,|w|)\cap B(w,\bar{R})\quad\text{and}$ $\displaystyle
E_{\tau}\left(\mathbb{N}^{2}\cap
B\left(E_{\tau}^{-1}\left(w-\frac{\bar{R}}{M|w|}w\right),\frac{\bar{R}}{\sqrt{\lambda_{2}}M}\right)\right)\subset
I_{\tau}\cap B(0,|w|)\cap B(w,\bar{R}).$

In particular, we have

$\displaystyle\left|\mathbb{N}^{2}\cap
B\left(E_{\tau}^{-1}\left(w-\frac{\bar{R}}{M|w|}w\right),\frac{\bar{R}}{\sqrt{\lambda_{2}}M}\right)\right|$
$\displaystyle\leq|I_{\tau}\cap B(0,|w|)\cap B(w,\bar{R})|.$

######  Proof.

Let $\tau\in A_{0}$, $w\in\mathbb{R}^{2}$, $\bar{R}>0$ and $M\geq 2$, and
assume that $|w|>\bar{R}$. We first show that the following inclusion:

$B\left(w-\frac{\bar{R}}{M|w|}w,\frac{\bar{R}}{M}\right)\subset B(0,|w|)\cap
B(w,\bar{R}).$

Indeed, Let $z\in B\left(w-(\bar{R}w)/(M|w|),\bar{R}/M\right)$. Since
$|w|>\bar{R}$ and $M\geq 2$, we have $1>\bar{R}/(M|w|)$ and

$\displaystyle|z|$
$\displaystyle\leq\left|z-\left(w-\frac{\bar{R}}{M|w|}w\right)\right|+\left|w-\frac{\bar{R}}{M|w|}w\right|$
$\displaystyle<\frac{\bar{R}}{M}+\left|1-\frac{\bar{R}}{M|w|}\right||w|=\frac{\bar{R}}{M}+\left(1-\frac{\bar{R}}{M|w|}\right)|w|=|w|.$

In addition, since $M\geq 2$, we have $(2\bar{R})/M\leq\bar{R}$ and

$\displaystyle|z-w|$
$\displaystyle\leq\left|z-\left(w-\frac{\bar{R}}{M|w|}w\right)\right|+\left|\frac{\bar{R}}{M}\frac{w}{|w|}\right|<\frac{\bar{R}}{M}+\frac{\bar{R}}{M}=\frac{2\bar{R}}{M}\leq\bar{R}.$

Hence, we have proved the desired inclusion. By Lemma 4.2 with
$\tilde{x}:=w-(\bar{R}w)/(M|w|)$ and $\tilde{R}:=\bar{R}/M$, we have

$\displaystyle
E_{\tau}B\left(E_{\tau}^{-1}\left(w-\frac{\bar{R}}{M|w|}w\right),\frac{\bar{R}}{\sqrt{\lambda_{2}}M}\right)\subset
B\left(w-\frac{\bar{R}}{M|w|}w,\frac{\bar{R}}{M}\right)\subset B(0,|w|)\cap
B(w,\bar{R}).$

Moreover, since $I_{\tau}=E_{\tau}(\mathbb{N}^{2})$ it follows that

$\displaystyle E_{\tau}(\mathbb{N}^{2})\cap
E_{\tau}\left(B\left(E_{\tau}^{-1}\left(w-\frac{\bar{R}}{M|w|}w\right),\frac{\bar{R}}{\sqrt{\lambda_{2}}M}\right)\right)\subset
I_{\tau}\cap B(0,|w|)\cap B(w,\bar{R}).$

Finally, since $E_{\tau}$ is injective, we have proved our lemma. ∎

We finally show the following lemma.

######  Lemma 4.4.

Let $\tau\in A_{0}$. Then, there exist $C_{\tau}^{\prime}>0$ and
$Q_{\tau}^{\prime}>0$ such that for all $w\in\mathbb{R}^{2}$ and
$R^{\prime}>0$ with $|w|>R^{\prime}\geq C_{\tau}^{\prime}$, we have

$|I_{\tau}\cap B(0,|w|)\cap
B(w,R^{\prime})|>Q_{\tau}^{\prime}(R^{\prime})^{2}$

###### Proof.

Let $\tau\in A_{0}$. Also, let $w\in\mathbb{R}^{2}$ and $R^{\prime}>0$ with
$|w|>R^{\prime}>2\sqrt{2\lambda_{2}}$. We set
$\xi={}^{t}(\xi_{1},\xi_{2}):=E_{\tau}^{-1}\left(w-(R^{\prime}w)/(2|w|)\right)\in\mathbb{R}^{2}$
and let $\zeta_{1}$ and $\zeta_{2}$ be minimum integers with
$\zeta_{1}\geq\xi_{1}$ and $\zeta_{2}\geq\xi_{2}$. We first show that

$\displaystyle
B\left(\zeta,\frac{R^{\prime}}{2\sqrt{\lambda_{2}}}-\sqrt{2}\right)\subset
B\left(\xi,\frac{R^{\prime}}{2\sqrt{\lambda_{2}}}\right)=B\left(E_{\tau}^{-1}\left(w-\frac{R^{\prime}}{2|w|}w\right),\frac{R^{\prime}}{2\sqrt{\lambda_{2}}}\right),$
(4.1.2)

where $\zeta:={}^{t}(\zeta_{1},\zeta_{2})\in\mathbb{N}^{2}$. Indeed, note that
$R^{\prime}>2\sqrt{2\lambda_{2}}$ if and only if
$R^{\prime}/(2\sqrt{\lambda_{2}})-\sqrt{2}>0$. Since $|\zeta_{1}-\xi_{1}|\leq
1$ and $|\zeta_{2}-\xi_{2}|\leq 1$, we have

$\displaystyle|\zeta-\xi|^{2}=|\zeta_{1}-\xi_{1}|^{2}+|\zeta_{2}-\xi_{2}|^{2}\leq
2.$

It follows that for each $a\in
B\left(\zeta,R^{\prime}/(2\sqrt{\lambda_{2}})-\sqrt{2}\right)$,

$\displaystyle|a-\xi|\leq|a-\zeta|+|\zeta-\xi|<\frac{R^{\prime}}{2\sqrt{\lambda_{2}}}-\sqrt{2}+\sqrt{2}=\frac{R^{\prime}}{2\sqrt{\lambda_{2}}}.$

Therefore, we have proved the inclusion (4.1.2).

We now set $Q_{\tau}^{\prime}:=1/(32\lambda_{2})$ and let
$C_{\tau}^{\prime}\geq 34\sqrt{\lambda_{2}}$ be a number such that

$\displaystyle\frac{(R^{\prime}-2\sqrt{2\lambda_{2}})^{2}}{16\lambda_{2}}-\frac{(R^{\prime})^{2}}{32\lambda_{2}}>0$
(4.1.3)

for each $R^{\prime}\geq C_{\tau}^{\prime}$.

Let $R^{\prime}\geq C_{\tau}^{\prime}$. Note that
$l:=R^{\prime}/(2\sqrt{\lambda_{2}})-\sqrt{2}>17-2=15>6$ and by using a
geometric observation, we see that

$\displaystyle|\\{^{t}(m,n)\in\mathbb{N}^{2}\ |\
(m-\zeta_{1})^{2}+(n-\zeta_{2})^{2}\leq
l^{2}\\}|=|\\{^{t}(m,n)\in\mathbb{N}^{2}\ |\ m^{2}+n^{2}\leq l^{2}\\}|.$

By proposition 3.3, we deduce that

$\displaystyle|\\{^{t}(m,n)\in\mathbb{N}^{2}\ |\
(m-\zeta_{1})^{2}+(n-\zeta_{2})^{2}\leq l^{2}\\}|$
$\displaystyle=|\\{^{t}(m,n)\in\mathbb{N}^{2}\ |\ m^{2}+n^{2}\leq
l^{2}\\}|\geq\frac{l^{2}-7l+7}{2}>\frac{l^{2}-7l}{2}>\frac{l^{2}}{4},$

where we use the following inequality
$(l^{2}-7l)/2-l^{2}/4=(l^{2}-14l)/4=l(l-14)/4>0$ for each $l>15$.

Finally, let $w\in\mathbb{R}^{2}$ with $|w|>R^{\prime}$. Note that
$R^{\prime}\geq C_{\tau}^{\prime}>2\sqrt{2\lambda_{2}}$. By Lemma 4.3 with
$\bar{R}:=R^{\prime}$ and $M=2$, the inclusion (4.1.2) and the definition of
$Q_{\tau}^{\prime}$ and $C_{\tau}^{\prime}$, we have

$\displaystyle|I_{\tau}\cap B(0,|w|)\cap B(w,R^{\prime})|$
$\displaystyle\geq\left|\mathbb{N}^{2}\cap
B\left(E_{\tau}^{-1}\left(w-\frac{R^{\prime}}{2|w|}w\right),\frac{R^{\prime}}{2\sqrt{\lambda_{2}}}\right)\right|\geq\left|\mathbb{N}^{2}\cap
B\left(\zeta,\frac{R^{\prime}}{2\sqrt{\lambda_{2}}}-\sqrt{2}\right)\right|$
$\displaystyle=|\\{^{t}(m,n)\in\mathbb{N}^{2}\ |\
(m-\zeta_{1})^{2}+(n-\zeta_{2})^{2}\leq l^{2}\\}|$
$\displaystyle>\frac{l^{2}}{4}=\frac{(R^{\prime}-2\sqrt{2\lambda_{2}})^{2}}{16\lambda_{2}}>\frac{(R^{\prime})^{2}}{32\lambda_{2}}=Q_{\tau}^{\prime}(R^{\prime})^{2}.$

Thus, we have proved our lemma. ∎

###  4.2 Proof for Theorem 1.4

We now prove Theorem 1.4. Rest of this section, Let $K\geq 1$ be a number
which satisfies 1. $\sim$ 3\. in Lemma 3.2.

###### Proof of Theorem 1.4.

It suffices to show that $S_{\tau}$ satisfies the assumption of Theorem 2.9
for each $\tau\in A_{0}$.

Let $\tau\in A_{0}$ and we set $r_{0}:=\min\\{1/8,KC_{\tau}^{-1}\\}(>0)$,
where $C_{\tau}>0$ is the number in Lemma 3.5. Note that there exists
$R_{0}>\max\\{C_{\tau}^{\prime},1\\}(>0)$ such that $(R-1)/R\geq 1/2$ for each
$R>R_{0}$, where $C_{\tau}^{\prime}>0$ is the number in Lemma 4.4. Recall that
$N_{\tau}=\sqrt{2\lambda_{2}}/\sqrt{\lambda_{1}}+1(>2)$. We define constants
$L_{\tau}^{\prime}$, $\xi$, $\gamma$ and $L_{\tau}$ as follows:

$\displaystyle L^{\prime}_{\tau}$
$\displaystyle:=\min\\{Q_{\tau}^{\prime}/4,(R_{0}+1)^{-2}\\}\ (>0),$
$\displaystyle\xi$ $\displaystyle:=r_{0}^{2}(>0),$ $\displaystyle\gamma$
$\displaystyle:=K(\geq 1)\quad\text{and}$ $\displaystyle L_{\tau}$
$\displaystyle:=\min\left\\{L_{\tau}^{\prime}(8K)^{-h_{\tau}},Q_{\tau}K^{2-3h_{\tau}}N_{\tau}^{-2h_{\tau}}2^{2-2h_{\tau}}\right\\}(>0).$

Here, $Q_{\tau}$ and $Q_{\tau}^{\prime}$ are the numbers in Lemmas 3.5 and 4.4
respectively.

Let $b:=m+n\tau\in I_{\tau}$ and $r>0$ with
$\gamma\cdot\mathrm{diam}(\phi_{b}(X))\leq r\leq\xi$. We set
$x:=1/b=\phi_{b}(0)\in\phi_{b}(X)\subset\phi_{b}(V)$. To prove Theorem 1.4, it
suffices to show the following claim:

######  Claim ($\star$).

$\displaystyle m_{\tau}(B(x,r))\geq L_{\tau}r^{h_{\tau}}.$

Rest of this paper, we consider the following three cases.

  1. Case 1. $r\leq|x|/2$

  2. Case 2. $2|x|\geq r>|x|/2$ and

  3. Case 3. $r>2|x|$.

We now consider Case 1. Note that by the assumption and Lemma 3.2, we have

$0<r\ (\leq|x|/2)<|x|\quad\text{and}\quad|x|^{2}=K\cdot
K^{-1}|b|^{-2}\leq\gamma\cdot\mathrm{diam}\phi_{b}(X)\leq r.$ (4.2.1)

We set $f(z):=1/z\ (z\in\mathbb{C}\setminus\\{0\\})$. We set

$w:=\frac{|x|^{2}}{|x|^{2}-r^{2}}\cdot\frac{1}{x}\quad\text{and}\quad
R:=\frac{r}{|x|^{2}-r^{2}}$

for simplicity. Also, we set

$I_{\tau}(x,r):=\\{a\in I_{\tau}\ |\ \phi_{a}(X)\subset B(x,r)\\}=\\{a\in
I_{\tau}\ |\ B(a+1/2,1/2)\subset f(B(x,r))=B(w,R)\\}$ (4.2.2)

and $I_{\tau,1}:=I_{\tau}(x,r)\cap B(0,|w|)$, where we use Lemma 4.1 in the
equation (4.2.2). Note that $|w|=|x|/(|x|^{2}-r^{2})$.

We next show that $|I_{\tau,1}|>L^{\prime}_{\tau}R^{2}$ for each $R>0$. To
prove this inequality, we consider the following two cases (Cases 1-1, 1-2).
Recall that $R_{0}>\max\\{C_{\tau}^{\prime},1\\}(>0)$ is the positive real
number such that $(R-1)/R\geq 1/2$ for each $R>R_{0}$.

Case 1-1. $R\geq R_{0}+1$.
Note that by the equation (4.2.2), we have

$\displaystyle I_{\tau,1}$ $\displaystyle=\\{a\in I_{\tau}\ |\
B(a+1/2,1/2)\subset B(w,R)\\}\cap B(0,|w|)$ $\displaystyle\supset\\{a\in
I_{\tau}\ |\ a\in B(w,R-1)\\}\cap B(0,|w|)=I_{\tau}\cap B(w,R-1)\cap
B(0,|w|).$ (4.2.3)

Since $|x|>r$, we have $|w|>R\ (>R_{0}>0)$. By the inclusion (4.2.3) and Lemma
4.4 with $R^{\prime}:=R-1(\geq R_{0}\geq C_{\tau}^{\prime})$, we have

$\displaystyle|I_{\tau,1}|$ $\displaystyle\geq|I_{\tau}\cap B(w,R-1)\cap
B(0,|w|)|>Q_{\tau}^{\prime}(R-1)^{2}\geq\frac{Q_{\tau}^{\prime}}{4}R^{2}\geq
L^{\prime}_{\tau}R^{2}.$

Case 1-2. $R_{0}+1>R$.
We show that $b\in I_{\tau,1}$. Note that since
$|b|=1/|x|\leq|x|/(|x|^{2}-r^{2})=|w|$, we have $b\in B(0,|w|)$. Therefore, we
have only to show that $b\in I_{\tau}(x,r)$. By using a geometric observation,
it suffices to show that $|w-(b+1/2)|\leq R-1/2$. To this end, we set
$A:=1-|b|^{2}r^{2}$ for simplicity. Note that by the inequality (4.2.1) and
$|x|=1/|b|$, $A=1-|b|^{2}r^{2}>0$ and $|b|^{2}r-1\geq 0$. Then, we have

$\displaystyle\left|2|b|^{2}r-A\right|^{2}-|2r^{2}|b|^{2}b-A|^{2}$
$\displaystyle=4|b|^{4}r^{2}-2A\cdot
2|b|^{2}r+A^{2}-\left(2r^{2}|b|^{2}b-A\right)\left(2r^{2}|b|^{2}\overline{b}-A\right)$
$\displaystyle=4|b|^{4}r^{2}-2A\cdot
2|b|^{2}r-(2|b|^{2}r^{2})^{2}\cdot|b|^{2}+2|b|^{2}r^{2}A\cdot(b+\overline{b})$
$\displaystyle=4|b|^{4}r^{2}-4A|b|^{2}r-4|b|^{6}r^{4}+4|b|^{2}r^{2}A\cdot\Re(b)$
$\displaystyle=4|b|^{4}r^{2}(1-|b|^{2}r^{2})+4A|b|^{2}r(r\Re(b)-1)$
$\displaystyle=4A|b|^{4}r^{2}+4A|b|^{2}r(r\Re(b)-1)=4A|b|^{2}r(|b|^{2}r-1+r\Re(b))\geq
0,$

where $\Re(b)\geq 0$ is the real part of $b$. In addition, since
$|b|^{2}r-1\geq 0$, we have

$2|b|^{2}r-A=|b|^{2}r^{2}+|b|^{2}r+|b|^{2}r-1\geq|b|^{2}r-1\geq 0.$

Therefore, we deduce that $|2r^{2}|b|^{2}b-A|\leq 2|b|^{2}r-A$ and it follows
that

$\displaystyle\left|w-\left(b+\frac{1}{2}\right)\right|$
$\displaystyle=\left|\frac{|x|^{2}}{|x|^{2}-r^{2}}\frac{1}{x}-b-\frac{1}{2}\right|=\left|\frac{b}{1-|b|^{2}r^{2}}-b-\frac{1}{2}\right|$
$\displaystyle=\frac{|2b-2b(1-|b|^{2}r^{2})-(1-|b|^{2}r^{2})|}{2(1-|b|^{2}r^{2})}=\frac{|2b|b|^{2}r^{2}-A|}{2(1-|b|^{2}r^{2})}$
$\displaystyle\leq\frac{2|b|^{2}r-A}{2(1-|b|^{2}r^{2})}=\frac{|b|^{2}r}{1-|b|^{2}r^{2}}-\frac{1-|b|^{2}r^{2}}{2(1-|b|^{2}r^{2})}=\frac{r}{|x|^{2}-r^{2}}-\frac{1}{2}=R-\frac{1}{2}.$

Thus, we have proved $B(b+1/2,1/2)\subset B(w,R)$ and $b\in I_{\tau,1}$. By
the assumption $R_{0}+1>R$, we deduce that

$\displaystyle|I_{\tau,1}|\geq 1>\left(\frac{R}{R_{0}+1}\right)^{2}\geq
L^{\prime}_{\tau}R^{2}.$

Hence, we have proved $|I_{\tau,1}|>L^{\prime}_{\tau}R^{2}$ for each $R>0$.

We now show the statement of Claim ($\star$) for Case 1. $r\leq|x|/2$. Rest of
this paper, we denote by $m_{\tau}$ the $h_{\tau}$-conformal measure for
$S_{\tau}$ for each $\tau\in A_{0}$. Note that by Theorem 2.8 and Lemma 3.2,
we have

$\displaystyle m_{\tau}\left(\bigcup_{a\in I_{\tau,1}}\phi_{a}(X)\right)$
$\displaystyle=\sum_{a\in
I_{\tau,1}}m_{\tau}\left(\phi_{a}(X)\right)=\sum_{a\in
I_{\tau,1}}\int_{X}|\phi_{a}^{\prime}(y)|^{h_{\tau}}m_{\tau}(\mathrm{d}y)$
$\displaystyle\geq\sum_{a\in
I_{\tau,1}}K^{-h_{\tau}}|a|^{-2h_{\tau}}m_{\tau}(X)=\sum_{a\in
I_{\tau,1}}K^{-h_{\tau}}|a|^{-2h_{\tau}}.$

Therefore, since $\phi_{a}(X)\subset B(x,r)$ and $|a|\leq|w|$ for each $a\in
I_{\tau,1}$ we deduce that

$\displaystyle m_{\tau}(B(x,r))$ $\displaystyle\geq
m_{\tau}\left(\bigcup_{a\in I}\phi_{a}(X)\right)=\sum_{a\in
I_{\tau,1}}K^{-h_{\tau}}|a|^{-2h_{\tau}}\geq\sum_{a\in
I_{\tau,1}}K^{-h_{\tau}}|w|^{-2h_{\tau}}\geq|I_{\tau,1}|\cdot
K^{-h_{\tau}}|w|^{-2h_{\tau}}.$

In addition, since $r\leq|x|/2$, we obtain that $|x|^{2}-r^{2}\geq 3|x|^{2}/4$
(i.e. $|w|=|x|/(|x|^{2}-r^{2})\leq 4/(3|x|)$ ) and

$\displaystyle m_{\tau}(B(x,r))$ $\displaystyle\geq
L^{\prime}_{\tau}R^{2}\cdot
K^{-h_{\tau}}\left(\frac{3|x|}{4}\right)^{2h_{\tau}}\geq
L^{\prime}_{\tau}\left(\frac{r}{|x|^{2}}\right)^{2}\cdot
K^{-h_{\tau}}\left(\frac{9}{16}\right)^{h_{\tau}}|x|^{2h_{\tau}}$
$\displaystyle\geq L^{\prime}_{\tau}K^{-h_{\tau}}2^{-h_{\tau}}\cdot r^{2}\
|x|^{2h_{\tau}-4}\geq L^{\prime}_{\tau}(2K)^{-h_{\tau}}\cdot
r^{2}r^{h_{\tau}-2}\geq L^{\prime}_{\tau}(2K)^{-h_{\tau}}r^{h_{\tau}}\ $
$\displaystyle\geq L_{\tau}r^{h_{\tau}},$

where we use the inequality $|I_{\tau,1}|>L^{\prime}_{\tau}R^{2}$ for each
$R>0$ and the inequality $|x|^{2(h_{\tau}-2)}\geq r^{h_{\tau}-2}$ since
$|x|^{2}\leq r$ and $h_{\tau}<2$ (see Lemma 3.1). Thus, we have proved the
statement of Claim ($\star$) for Case 1. $r\leq|x|/2$.

We now consider Case 2. We set $\tilde{r}:=r/4$. Then, by the assumption, we
have $\tilde{r}\leq|x|/2$. In addition, since $|x|^{2}=K\cdot
K^{-1}|b|^{-2}\leq\gamma\cdot\mathrm{diam}\phi_{b}(X)\leq r\leq\xi=r_{0}^{2}$,
we have

$|x|^{2}\leq r_{0}\cdot|x|\leq\frac{1}{8}\cdot 2r=\tilde{r},$

where used the definition of $r_{0}>0$. Therefore, the positive real number
$\tilde{r}$ satisfies the inequalities $\tilde{r}\leq|x|/2$ (the assumption of
Case 1.), $\tilde{r}<|x|$ and $|x|^{2}\leq\tilde{r}$ (the inequalities
(4.2.1)) instead of $r>0$. By the same argument as Case 1. with $\tilde{r}>0$
instead of $r>0$, we have

$\displaystyle m_{\tau}(B(x,r))$ $\displaystyle\geq
m_{\tau}(B(x,\tilde{r}))\geq
L^{\prime}_{\tau}(2K)^{-h_{\tau}}\tilde{r}^{h_{\tau}}\geq
L^{\prime}_{\tau}(2K)^{-h_{\tau}}4^{-h_{\tau}}r^{h_{\tau}}\geq
L^{\prime}_{\tau}(8K)^{-h_{\tau}}r^{h_{\tau}}\geq L_{\tau}r^{h_{\tau}}.$

Thus, we have proved the statement of Claim ($\star$) for Case 2. $2|x|\geq
r>|x|/2$.

We now consider Case 3. We set $\bar{r}:=r/2$. Note that $B(0,\bar{r})\subset
B(x,r)$ since $|y-x|\leq|x|+|y|<r/2+\bar{r}=r$ for each $y\in B(0,\bar{r})$.
We set $r_{\tau}:=KC_{\tau}^{-1}(>0)$ for simplicity, where $C_{\tau}$ is the
number in Lemma 3.5. We set

$\displaystyle I_{\tau}(\bar{r}):=\\{a\in I_{\tau}\ |\ \bar{r}/N_{\tau}\leq
K|a|^{-1}<\bar{r}\\}.$

We show that

$|I_{\tau}(\bar{r})|>Q_{\tau}K^{2}\bar{r}^{-2}.$ (4.2.4)

Indeed, note that by the definition of $r>0$, $\xi>0$ and $r_{0}>0$, we have
$\bar{r}=r/2<r_{0}\leq r_{\tau}$. We set $R:=K\bar{r}^{-1}$ for simplicity.
Note that $\bar{r}<r_{\tau}$ if and only if $R>C_{\tau}$ and

$\displaystyle I_{\tau}(\bar{r})=\\{a\in I_{\tau}\ |\ K\bar{r}^{-1}<|a|\leq
N_{\tau}K\bar{r}^{-1}\\}=I_{\tau}\cap D_{2}(\tau,K\bar{r}^{-1})=I_{\tau}\cap
D_{2}(\tau,R).$

By Lemma 3.5, we obtain that

$\displaystyle|I_{\tau}(\tilde{r})|=|I_{\tau}\cap
D_{2}(\tau,R)|>Q_{\tau}R^{2}=Q_{\tau}K^{2}\bar{r}^{-2}.$

Therefore, we have proved the inequality (4.2.4). Now, by Theorem 2.8 and
Lemma 3.2, we have

$\displaystyle m_{\tau}\left(\bigcup_{a\in
I_{\tau}(\bar{r})}\phi_{a}(X)\right)$ $\displaystyle=\sum_{a\in
I_{\tau}(\bar{r})}m_{\tau}\left(\phi_{a}(X)\right)=\sum_{a\in
I_{\tau}(\bar{r})}\int_{X}|\phi_{a}^{\prime}(y)|^{h_{\tau}}m_{\tau}(\mathrm{d}y)$
$\displaystyle\geq\sum_{a\in
I_{\tau}(\bar{r})}K^{-h_{\tau}}|a|^{-2h_{\tau}}m_{\tau}(X)=\sum_{a\in
I_{\tau}(\bar{r})}K^{-h_{\tau}}|a|^{-2h_{\tau}}.$

In addition, since $\phi_{a}(X)\subset B(0,\bar{r})$ (see Lemma 3.2) and
$|a|^{-1}\geq\bar{r}/(N_{\tau}K)$ for each $a\in I_{\tau}(\bar{r})$ we deduce
that

$\displaystyle m_{\tau}(B(x,r))$ $\displaystyle\geq m_{\tau}(B(0,\bar{r}))\geq
m_{\tau}\left(\bigcup_{a\in I_{\tau}(\bar{r})}\phi_{a}(X)\right)\geq\sum_{a\in
I_{\tau}(\bar{r})}K^{-h_{\tau}}\left(\frac{\bar{r}}{KN_{\tau}}\right)^{2h_{\tau}}$
$\displaystyle=|I_{\tau}(\bar{r})|\cdot
K^{-3h_{\tau}}N_{\tau}^{-2h_{\tau}}\bar{r}^{2h_{\tau}}>Q_{\tau}K^{2-3h_{\tau}}N_{\tau}^{-2h_{\tau}}\cdot\bar{r}^{2h_{\tau}-2}$
$\displaystyle=Q_{\tau}K^{2-3h_{\tau}}N_{\tau}^{-2h_{\tau}}2^{2-2h_{\tau}}\cdot
r^{2h_{\tau}-2}\geq
Q_{\tau}K^{2-3h_{\tau}}N_{\tau}^{-2h_{\tau}}2^{2-2h_{\tau}}r^{h_{\tau}}\geq
L_{\tau}r^{h_{\tau}},$

where we use the inequalities (4.2.4) and $r^{2h_{\tau}-2}>r^{h_{\tau}}$ since
$r<r_{0}\leq 1/8<1$ and $h_{\tau}<2$ (see Lemma 3.1). Thus, we have proved the
statement of Claim ($\star$) for Case 3. $r>2|x|$.

Hence, by the three cases (Cases 1. $\sim$ 3\. ), we have proved Theorem 1.4.
∎

## Acknowledgment

The authors would like to thank Mariusz Urbański for helpful comments on [5].
The second author is partially supported by JSPS Grant-in-Aid for Scientific
Research (B) Grant Number JP 19H01790.

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