File: mtqm100323.tex, printed: 2024-8-27, 18.27

# On Mori’s theorem for
quasiconformal maps in the $n$-space

B.A. Bhayo  Department of Mathematics, University of Turku, FI-20014 Turku,
Finland<EMAIL_ADDRESS>and  M. Vuorinen  Department of Mathematics, University
of Turku, FI-20014 Turku, Finland<EMAIL_ADDRESS>
###### Abstract.

R. Fehlmann and M. Vuorinen proved in 1988 that Mori’s constant $M(n,K)$ for
$K$-quasiconformal maps of the unit ball in $\mathbf{R}^{n}$ onto itself
keeping the origin fixed satisfies $M(n,K)\to 1$ when $K\to 1\,.$ We give here
an alternative proof of this fact, with a quantitative upper bound for the
constant in terms of elementary functions. Our proof is based on a refinement
of a method due to G.D. Anderson and M. K. Vamanamurthy. We also give an
explicit version of the Schwarz lemma for quasiconformal self-maps of the unit
disk. Some experimental results are provided to compare the various bounds for
the Mori constant when $n=2\,.$

###### Key words and phrases:

Quasiconformal mappings, Hölder continuity

###### 2000 Mathematics Subject Classification:

Primary 30C65

In memoriam: M.K. Vamanamurthy, 5 September 1934– 6 April 2009

##  1\. Introduction

Distortion theory of quasiconformal and quasiregular mappings in the Euclidean
$n$-space $\mathbf{R}^{n}$ deals with estimates for the modulus of continuity
and change of distances under these mappings. Some of the examples are the
Hölder continuity, the quasiconformal counterpart of the Schwarz lemma, and
Mori’s theorem. The investigation of these topics started in the early 1950’s
for the case $n=2$ and ten years later for the case $n\geq 3\,.$ Many authors
have contributed to the distortion theory, for some historical remarks see
[Vu1, 11.50].

As in [FV] we define Mori’s constant $M(n,K)$ in the following way. Let
$QC_{K},\,K\geq 1,$ stand for the family of all $K$-quasiconformal maps of the
unit ball $\mathbf{B}^{n}$ onto itself keeping the origin pointwise fixed.
Note that it is a well-known basic fact that an element in the set $QC_{K}$
can be extended by reflection to a $K$-quasiconformal map of the whole space
$\overline{\mathbf{R}}^{n}={\mathbf{R}}^{n}\cup\\{\infty\\}$ onto itself
keeping the point $\infty$ fixed. Then for all $K\geq 1,\,n\geq 2\,,$ there
exists a least constant $M(n,K)\geq 1$ such that

(1.1) $|f(x)-f(y)|\leq M(n,K)|x-y|^{\alpha},\quad\alpha=K^{1/(1-n)}\,,$

for all $f\in QC_{K},x,y\in\mathbf{B}^{n}\,.$

L. V. Ahlfors [A1] proved in 1954 that $M(2,K)\leq 12^{K^{2}}$ and this
property was refined by A. Mori [Mo] in 1956 to the effect that $M(2,K)\leq
16$ and $16$ cannot be replaced by a smaller constant independent of $K\,.$
This result can also be found in [A2], [FM], and [LV]. On the other hand the
trivial observation that $16$ fails to be a sharp constant for $K=1$ led to
the following conjecture, which is still open in 2009.

######  1.2 The Mori Conjecture.

$M(2,K)=16^{1-1/K}.$

O. Lehto and K.I. Virtanen demonstrated in 1973 [LV, pp. 68] that $M(2,K)\geq
16^{1-1/K}$ (this lower bound was not given in the 1965 German edition of the
book). It is natural to expect that for a fixed $n\geq 2,$ $M(n,K)\to 1$ when
$K\to 1$ and this convergence result with an explicit upper bound for $M(n,K)$
was proved by R. Fehlmann and M. Vuorinen [FV]. A counterpart of this result
for the chordal metric was proved recently by P. Hästö in [H].

######  1.3 Theorem.

[FV, Theorem 1.3] Let $f$ be a $K$-quasiconformal mapping of $\mathbf{B}^{n}$
onto $\mathbf{B}^{n}$, $n\geq 2$, $f(0)=0$. Then

(1.4) $|f(x)-f(y)|\leq M(n,K)|x-y|^{\alpha}$

for all $x,y\in\mathbf{B}^{n}$ where $\alpha=K^{1/(1-n)}$ and the constant
$M(n,K)$ has the following three properties:

  1. (1)

$M(n,K)\to 1$ as $K\to 1$, uniformly in $n$ ,

  2. (2)

$M(n,K)$ remains bounded for fixed $K$ and varying $n$ ,

  3. (3)

$M(n,K)$ remains bounded for fixed $n$ and varying $K$ .

For $n=2\,,$ the first majorants with the convergence property in 1.3(1) were
proved only in the mid 1980s and for $n\geq 3$ in [FV]. In [FV] a survey of
the various known bounds for $M(n,K)$ when $n\geq 2$ can be found – that
survey reflects what was known at the time of publication of [FV]. Some
earlier results on Hölder continuity had been proved in [G], [MRV], [R], [S].
Step by step the bound for Mori’s constant was reduced during the past twenty
years. As far as we know, the best upper bound known today for $n=2$ is
$M(2,K)\leq 46^{1-1/K}$ due to S.-L. Qiu [Q] (1997). Refining the parallel
work [FV], G.D. Anderson and M. K. Vamanamurthy proved the following theorem
in [AV].

######  1.5 Theorem.

For $n\geq 2,K\geq 1$,

$M(n,K)\leq 4\lambda_{n}^{2(1-\alpha)}\,,$

where $\alpha=K^{1/(1-n)}\,$ and $\lambda_{n}\in[4,2e^{n-1})\,,\lambda_{2}=4,$
is the Grötzsch ring constant [AN], [Vu1, p.89].

The first main result of this paper is Theorem 1.6 which improves on Theorem
1.5.

######  1.6 Theorem.

(1) For $n\geq 2,K\geq 1$, $M(n,K)\leq T(n,K)$

(1.7) $T(n,K)\leq\inf\\{h(t):t\geq 1\\}\,,\quad
h(t)=(3+\lambda_{n}^{\beta-1}t^{\beta})t^{-\alpha}\lambda_{n}^{2(1-\alpha)},\;t\geq
1\,,$

where $\alpha=K^{1/(1-n)}=1/\beta,$ and $\lambda_{n}$ is as in Theorem 1.5.

(2) There exists a number $K_{1}>1$ such that for all $K\in(1,K_{1})$ the
function $h$ has a minimum at a point $t_{1}$ with $t_{1}>1$ and

(1.8) $T(n,K)\leq
h(t_{1})=\left[\frac{3^{1-\alpha^{2}}(\beta-\alpha)^{\alpha^{2}}}{\alpha^{\alpha^{2}}}\lambda_{n}^{\alpha-\alpha^{2}}+\lambda_{n}^{\beta-1}\left(\frac{(3\alpha)^{\alpha}\lambda_{n}^{\alpha-1}}{(\beta-\alpha)^{\alpha}}\right)^{\beta-\alpha}\right]\lambda_{n}^{2(1-\alpha)}\,.$

Moreover, for $\beta\in(1,2)$ we have

(1.9) $h(t_{1})\leq
3^{1-\alpha^{2}}2^{5(1-\alpha)}K^{5}\left(\frac{3}{2}\sqrt[4]{\beta-\alpha}+\exp(\sqrt{\beta^{2}-1})\right).$

In particular, $h(t_{1})\to 1$ when $K\to 1\,.$

The last statement shows that Theorem 1.6 is better than the result of
Anderson and Vamanamurthy, Theorem 1.5, at least for values of $K$ close to
the critical value $1$, because the constant of Theorem 1.5 satisfies
$4\lambda_{n}^{2(1-\alpha)}\geq 4.$

The main method of our proof is to replace the argument of Anderson and
Vamanamurthy by a more refined inequality from [Vu2] and to introduce an
additional parameter ($t$ in the above theorem) which will be chosen in an
optimal way. The fact that this refined inequality is essentially sharp for
values of $t$ large enough, was recently proved by V. Heikkala and M. Vuorinen
in [HV]. This gave us a hint that the inequality from [Vu2] might lead to an
improvement of the results in [AV]. For the case $n=2$ a numerical comparison
of our bound (1.8) to Mori’s conjectured bound, to the bound in Theorem 1.5
and to the bound in [FV] is presented in tabular and graphical form at the end
of the paper.

We conclude this paper by discussing the Schwarz lemma for plane
quasiconformal self-mappings of the unit disk, formulated in terms of the
hyperbolic metric. The long history of this result is summarized in [Vu1,
p.152, 11.50]. An up-to-date form of the Schwarz lemma was given in [Vu1,
Theorem 11.2] and it will be stated for convenient reference also below as
Theorem 4.4. A particular case, formula (4.6), was rediscovered by D.B.A.
Epstein, A. Marden and V. Markovic [EMM, Thm 5.1].

We use the notations ch, th, arch and arth as in [Vu1], to denote the
hyperbolic cosine, tangent and their inverse functions, resp. The second main
result of this paper is an explicit form of the Schwarz lemma for quasiregular
mappings, Theorem 1.10. We believe that in this simple form the result is new
and perhaps of independent interest. The constant $c(K)$ below involves the
transcendental function $\varphi_{K}$ defined in Section 4.

######  1.10 Theorem.

If $f:\mathbf{B}^{2}\to\mathbf{R}^{2}$ is a non-constant $K$-quasiregular
mapping with $f\mathbf{B}^{2}\subset\mathbf{B}^{2}$, and $\rho$ is the
hyperbolic metric of $\mathbf{B}^{2}\,,$ then

$\rho(f(x),f(y))\leq c(K)\max\\{\rho(x,y),\rho(x,y)^{1/K}\\}$

for all $x,y\in\mathbf{B}^{2}$ where $c(K)=2{\rm arth}(\varphi_{K}({\rm
th}\frac{1}{2}))\,$ and

$K\leq u(K-1)+1\leq\log({\rm ch}(K{\rm arch}(e)))\leq c(K)\leq v(K-1)+K$

with $u={\rm arch}(e){\rm th}({\rm arch}(e))>1.5412$ and
$v=\log(2(1+\sqrt{1-1/e^{2}}))<1.3507$. In particular, $c(1)=1\,.$

Acknowledgments. The first author is indebted to the Graduate School of
Mathematical Analysis and its Applications for support. Both authors wish to
acknowledge the kind help of Prof. G.D. Anderson in the proof of Lemma 4.8,
the valuable help of the referee for the improvement of the manuscript, as
well as the expert help of Dr. H. Ruskeepää in the use of Mathematica [Ru].

##  2\. The main results

We shall follow here the standard notation and terminology for
$K$-quasiconformal and $K$-quasiregular mappings in the Euclidean $n$-space
$\mathbf{R}^{n}\,,$ see e.g. [V], [Vu1], and we also recall some basic
notation. For the modulus $M(\Gamma)$ of a curve family $\Gamma$ and its basic
properties see [V] and [Vu1].

Let $D$ and $D^{{}^{\prime}}$ be domains in $\overline{\mathbf{R}}^{n},K\geq
1$, and let $f:D\to D^{{}^{\prime}}$ be a homeomorphism. Then $f$ is
$K$-quasiconformal if

$M(\Gamma)/K\leq M(f\Gamma)\leq KM(\Gamma)$

for every curve family $\Gamma$ in $D$ [V].

For subsets $E,F,D\subset\overline{\mathbf{R}}^{n}$ we denote by
$\Delta(E,F;D)$ the family of all curves joining $E$ and $F$ in $D$. For
brevity we write $\Delta(E,F)=\Delta(E,F;{\mathbf{R}}^{n})\,.$ A ring is a
domain in ${\mathbf{R}}^{n}$, whose complement consists of two compact and
connected sets. If these sets are $E$ and $F$, then the ring is denoted by
$R(E,F)\,.$ The capacity of a ring $R(E,F)$ is

${\rm cap}R(E,F)=M(\Delta(E,F)).$

The complementary components of the Grötzsch ring $R_{G,n}(s)$ are
$\overline{\mathbf{B}}^{n}$ and $[se_{1},\infty],s>1$, while those of the
Teichmüller ring $R_{T,n}(t)$ are $[-e_{1},0]$ and $[te_{1},\infty],t>0$. The
conformal capacities of $R_{G,n}(s)$ and $R_{T,n}(t)$ are denoted by

$\left\\{\begin{array}[]{lll}\gamma_{n}(s)=M(\Delta(\overline{\mathbf{B}}^{n},[se_{1},\infty]))\\\
\tau_{n}(t)=M(\Delta([-e_{1},0],[te_{1},\infty]))\end{array}\right.$

respectively. Here $\gamma_{n}:(1,\infty)\to(0,\infty)$ and
$\tau_{n}:(0,\infty)\to(0,\infty)$ are decreasing homeomorphisms and they
satisfy the fundamental identity

(2.1) $\gamma_{n}(s)=2^{n-1}\tau_{n}(s^{2}-1),\quad t>1\,,$

see e.g. [Vu1, 5.53].

For $n\geq 2$ and $K>0$, the distortion function $\varphi_{K,n}:[0,1]\to[0,1]$
is a homeomorphism. It is defined by

(2.2)
$\varphi_{K,n}(t)=\displaystyle\frac{1}{\gamma_{n}^{-1}(K\gamma_{n}(1/t))},\quad
t\in(0,1),$

and $\varphi_{K,n}(0)=0\,,$ $\varphi_{K,n}(1)=1\,.$ For $n\geq 2,K\geq 1$ and
$0\leq r\leq 1$

(2.3)
$\varphi_{K,n}(r)\leq\lambda_{n}^{1-\alpha}r^{\alpha},\quad\alpha=K^{1/(1-n)}\,,$
(2.4)
$\varphi_{1/K,n}(r)\geq\lambda_{n}^{1-\beta}r^{\beta},\quad\beta=K^{1/(n-1)}\,,$

by [Vu1, Theorem 7.47] and where $\lambda_{n}\geq 4$ is as in Theorem 1.5.

######  2.5 Lemma.

Suppose that $f:\mathbf{B}^{n}\to\mathbf{B}^{n}$ is a $K$-quasiconformal
mapping with $f\mathbf{B}^{n}=\mathbf{B}^{n}$, $f(0)=0,$ and let
$h:\overline{\mathbf{R}}^{n}\to\overline{\mathbf{R}}^{n}$ be the inversion
$h(x)=x/|x|^{2}\,,h(\infty)=0,h(0)=\infty,$ and define
$g:\overline{\mathbf{R}}^{n}\to\overline{\mathbf{R}}^{n}$ by $g(x)=f(x)$ for
$x\in\mathbf{B}^{n},g(x)=h(f(h(x)))$ for
$x\in\mathbf{R}^{n}\setminus\overline{\mathbf{B}}^{n}$ and $g(x)=\lim_{z\to
x}f(z)$ for $x\in\partial\mathbf{B}^{n},g(\infty)=\infty$. Then $g$ is a
$K$-quasiconformal mapping, and we have for $x\in\mathbf{B}^{n}$

(2.6) $\varphi_{1/K,n}(|x|)\leq|f(x)|\leq\varphi_{K,n}(|x|).$

For $x\in\mathbf{R}^{n}\setminus\overline{\mathbf{B}}^{n}$

(2.7) $1/\varphi_{K,n}(1/|x|)\leq|g(x)|\leq 1/\varphi_{1/K,n}(1/|x|).$

###### Proof.

It is well-known that the above definition defines $g$ as a $K$-quasiconformal
homeomorphism. The formula (2.6) is well-known (see [AVV2, Theorem 4.2]) and
(2.7) follows easily. ∎

######  2.8 Lemma.

[Vu1, Lemma 7.35] Let $R=R(E,F)$ be a ring in $\overline{\mathbf{R}}^{n}$ and
let $a,b\in E,c,d\in F$ be distinct points. Then

$\text{cap}R=M(\Delta(E,F))\geq\tau_{n}\left(\frac{|a-c||b-d|}{|a-b||c-d|}\right).$

Equality holds if $b=t_{1}e_{1},a=t_{2}e_{1},c=t_{3}e_{1},d=t_{4}e_{1}$ and
$t_{1}<t_{2}<t_{3}<t_{4}$.

We consider Teichmüller’s extremal problem, which will be used to provide a
key estimate in what follows. For
$x\in\mathbf{R}^{n}\setminus\\{0,e_{1}\\},n\geq 2$, define

$p_{n}(x)=\inf_{E,F}M(\Delta(E,F))$

where the infimum is taken over all the pairs of continua $E$ and $F$ in
$\overline{\mathbf{R}}^{n}$ with $0,e_{1}\in E$ and $x,\infty\in F$. Note that
Lemma 2.8 gives the lower bound for $p_{n}(x)$ in Lemma 2.9.

######  2.9 Lemma.

[Vu2, Theorem 1.5] For $z\in\mathbf{R}^{n},|z|>1$, the following inequalities
hold:

$\tau_{n}(|z|)=p_{n}(-|z|e_{1})\leq p_{n}(z)\leq
p_{n}(|z|e_{1})=\tau_{n}(|z|-1)$

where $p_{n}(z)$ is the Teichmüller function. Furthermore, for
$z\in\mathbf{R}^{n}\setminus\\{0,e_{1}\\}$, there exists a circular arc $E$
with $0,e_{1}\in E$ and a ray $F$ with $z,\infty\in F$ such that

(2.10)
$p_{n}(z)\leq\tau_{n}\left(\frac{|z|+|z-e_{1}|-1}{2}\right)=M(\Delta(E,F))\leq\tau_{n}(|z|-1)$

with equality in the first inequality both for $z=-se_{1},s>0$, and for
$z=se_{1},s>1\,.$

######  2.11.

Notation. For $t>0,x,y\in\mathbf{B}^{n}\,,$ we write

$D(t,x,y)=|x+t\frac{y}{|y|}|\,\,\,\mathrm{if\,\,}y\neq 0,\quad
D(t,x,0)=|x+e_{1}|\,.$

By the triangle inequality we have

(2.12) $t-|x|\leq D(t,x,y)\leq t+|x|\,.$

######  2.13 Theorem.

For $n\geq 2,K\geq 1$, let
$f:\overline{\mathbf{R}}^{n}\to\overline{\mathbf{R}}^{n}$ be a
$K$-quasiconformal mapping, with $f\mathbf{B}^{n}\subset\mathbf{B}^{n}$,
$f(0)=0$ and $f(\infty)=\infty$. Then for $t\geq 1\,,$
$x,y\in\mathbf{B}^{n}\setminus\\{0\\}\,,$ we have

$\displaystyle|f(x)-f(y)|$ $\displaystyle\leq$
$\displaystyle(3+\varphi_{1/K,n}(1/t)^{-1})\varphi_{K,n}^{2}\left(\left(\frac{2|x-y|}{s_{1}+|x-y|}\right)^{1/2}\right)$
$\displaystyle\leq$
$\displaystyle(3+\lambda_{n}^{(\beta-1)}t^{\beta})\lambda_{n}^{2(1-\alpha)}\left(\frac{2|x-y|}{s_{1}+|x-y|}\right)^{\alpha}\,,\;\alpha=K^{1/(1-n)}=1/\beta,$

where $s_{1}=\displaystyle\max\\{a,b\\},a=t+|x|+D(t,y,x),b=t+|y|+D(t,x,y)$.

###### Proof.

Let $\Gamma$ be the family $\Delta(E,F)$ and let $E$ and $F$ be connected sets
as in Lemma 2.9 with $x,y\in E,z,\infty\in F$, where $z=-tx/|x|$ and
$\Gamma^{{}^{\prime}}=f(\Gamma)$. By Lemma 2.8 and (2.10), we have

$\displaystyle\tau_{n}\left(\frac{|f(z)-f(x)|}{|f(x)-f(y)|}\right)\leq
M(\Gamma^{{}^{\prime}})\leq KM(\Gamma)$ $\displaystyle\leq$ $\displaystyle
K\tau_{n}(u-1)\,,\quad
u=\displaystyle\frac{|x-z|+|z-y|-|x-y|+2|x-y|}{2|x-y|}\,.$

The basic identity ($\ref{1})$ yields

(2.14)
$\gamma_{n}\left(\left(\displaystyle\frac{|f(z)-f(y)|+|f(x)-f(y)|}{|f(x)-f(y)|}\right)^{1/2}\right)\leq
K\gamma_{n}\left(\left(u\right)^{1/2}\right)$
$=K\gamma_{n}\left(\left(\displaystyle\frac{t+|x|+D(t,y,x)+|x-y|}{2|x-y|}\right)^{1/2}\right).$

Applying $\gamma_{n}^{-1}$ to (2.14) we have

Figure 1.  Geometrical meaning of the proof of Theorem 2.13.

$\displaystyle\frac{|f(z)-f(y)|+|f(x)-f(y)|}{|f(x)-f(y)|}\geq\displaystyle\left(\gamma_{n}^{-1}\left(K\gamma_{n}\left(\left(\frac{a+|x-y|}{2|x-y|}\right)^{1/2}\right)\right)\right)^{2}=v.$

Because $f\mathbf{B}^{n}\subset\mathbf{B}^{n}$, by (2.6) and (2.4) we know
that

$|f(z)-f(y)|+|f(x)-f(y)|\leq 3+\varphi_{1/K,n}(1/t)^{-1}\leq
3+\lambda_{n}^{(\beta-1)}t^{\beta},$ (2.15)
$\frac{|f(x)-f(y)|}{3+\varphi_{1/K,n}(1/t)^{-1}}\leq\frac{|f(x)-f(y)|}{|f(z)-f(y)|+|f(x)-f(y)|}\leq
1/v,$

also

$\displaystyle|f(x)-f(y)|$ $\displaystyle\leq$
$\displaystyle(3+\varphi_{1/K,n}(1/t)^{-1})\varphi_{K,n}^{2}\left(\left(\frac{2|x-y|}{a+|x-y|}\right)^{1/2}\right)$
$\displaystyle\leq$
$\displaystyle(3+\lambda_{n}^{(\beta-1)}t^{\beta})\lambda_{n}^{2(1-\alpha)}\left(\frac{2|x-y|}{a+|x-y|}\right)^{\alpha}$

by inequalities (2.2) and (2.3). Exchanging the roles of $x$ and $y$ we see
that

$\displaystyle|f(x)-f(y)|$ $\displaystyle\leq$
$\displaystyle(3+\varphi_{1/K,n}(1/t)^{-1})\varphi_{K,n}^{2}\left(\left(\frac{2|x-y|}{s_{1}+|x-y|}\right)^{1/2}\right)$
$\displaystyle\leq$
$\displaystyle(3+\lambda_{n}^{(\beta-1)}t^{\beta})\lambda_{n}^{2(1-\alpha)}\left(\frac{2|x-y|}{s_{1}+|x-y|}\right)^{\alpha}.$

∎

Setting $t=1$, we get the following corollary.

######  2.16 Corollary.

For $n\geq 2,K\geq 1$, let
$f:\overline{\mathbf{R}}^{n}\to\overline{\mathbf{R}}^{n}$ be a
$K$-quasiconformal mapping, with $f\mathbf{B}^{n}\subset\mathbf{B}^{n}$,
$f(0)=0$ and $f(\infty)=\infty$. Then for all
$x,y\in\mathbf{B}^{n}\setminus\\{0\\}\,,$

$|f(x)-f(y)|\leq
4\lambda_{n}^{2(1-\alpha)}\left(\frac{2|x-y|}{s+|x-y|}\right)^{\alpha}\,,$

where $\alpha=K^{1/(1-n)}$ and
$s=\displaystyle\max\\{a,b\\},a=1+|x|+D(1,y,x),b=1+|y|+D(1,x,y)\,.$

###### Proof.

The proof is similar to the above proof except that here we consider the
particular case $t=1$. Because $f\mathbf{B}^{n}\subset\mathbf{B}^{n}$, we know
that $|f(z)-f(y)|+|f(x)-f(y)|\leq 4$,

$\displaystyle\frac{|f(x)-f(y)|}{4}$ $\displaystyle\leq$
$\displaystyle\frac{|f(x)-f(y)|}{|f(z)-f(y)|+|f(x)-f(y)|}$ $\displaystyle\leq$
$\displaystyle\displaystyle\frac{1}{\left(\gamma_{n}^{-1}\left(K\gamma_{n}\left(\left(\displaystyle\frac{a+|x-y|}{2|x-y|}\right)^{1/2}\right)\right)\right)^{2}},$

or

$\displaystyle|f(x)-f(y)|$ $\displaystyle\leq$ $\displaystyle
4\varphi_{K,n}^{2}\left(\left(\frac{2|x-y|}{a+|x-y|}\right)^{1/2}\right)$
$\displaystyle\leq$ $\displaystyle
4\lambda_{n}^{2(1-\alpha)}\left(\frac{2|x-y|}{a+|x-y|}\right)^{\alpha}$

by inequalities (2.2) and (2.3). Exchanging the roles of $x$ and $y$ we get

$|f(x)-f(y)|\leq
4\lambda_{n}^{2(1-\alpha)}\left(\frac{2|x-y|}{\max\\{a,b\\}+|x-y|}\right)^{\alpha}\,.$

∎

######  2.17 Corollary.

For $n\geq 2,K\geq 1,t\geq 1$, let $f$ be as in Theorem 2.13. Then

(2.18)
$|f(x)-f(y)|\leq(3+\lambda_{n}^{(\beta-1)}t^{\beta})\lambda_{n}^{2(1-\alpha)}\left(\frac{2|x-y|}{2t+||x|-|y||+|x-y|}\right)^{\alpha},$

for all $x,y\in\mathbf{B}^{n}\,,$

(2.19)
$|f(x)-f(y)|\leq(3+\lambda_{n}^{\beta-1}t^{\beta})\lambda_{n}^{2(1-\alpha)}\left(\frac{|x-y|}{\max\\{t+|x|,t+|y|\\}}\right)^{\alpha},$

for all $x,y\in\mathbf{B}^{n}\,,$ and

(2.20)
$|f(x)-f(y)|\leq(3+\lambda_{n}^{(\beta-1)}t^{\beta})\lambda_{n}^{2(1-\alpha)}\left(\frac{|x-y|}{t+|x|+(|x-y|)/2}\right)^{\alpha},$

if $D(t,y,x)>t+|x|,x,y\in\mathbf{B}^{n}$.

###### Proof.

Inequality $(\ref{11a})$ follows because by (2.11) $D(t,y,x)>t-|y|$ and
$D(t,x,y)>t-|x|$ for $x,y\in\mathbf{B}^{n}$, and hence, in the notation of
Theorem 2.13,

$s_{1}\geq\max\\{2t+|x|-|y|,2t+|y|-|x|\\}=2t+||x|-|y||\,.$

It is also clear that $D(t,y,x)\geq t+|x|-|x-y|$, and this implies that

$s_{1}\geq\max\\{2(t+|x|)-|x-y|,2(t+|y|)-|x-y|\\}=2\max\\{t+|x|,t+|y|\\}-|x-y|$

and hence the inequality $(\ref{11aa})$ follows. In the case of $(\ref{12a})$
we have $D(t,y,x)>t+|x|$ and see that, in the notation of Corollary 2.16,
$s>2(t+|x|)$ and $(\ref{12a})$ holds. ∎

######  2.21 Corollary.

For $n\geq 2,K\geq 1$, let $f$ be as in Theorem 2.13. Then

(2.22) $|f(x)-f(y)|\leq
4\lambda_{n}^{2(1-\alpha)}\left(\frac{2|x-y|}{2+||x|-|y||+|x-y|}\right)^{\alpha},$

for all $x,y\in\mathbf{B}^{n}\setminus\\{0\\}\,.$

######  2.23 Remark.

(1) In several of the above results we have supposed that
$x,y\in\mathbf{B}^{n}\setminus\\{0\\}\,.$ If one of the points $x,y$ were
equal to $0\,,$ then we would have a better result from the Schwarz lemma
estimate (4.7).

(2) Corollary 2.21 is an improvement of the Anderson-Vamanamurthy theorem 1.5
.

##  3\. Comparison with earlier bounds

######  3.1.

Proof of Theorem 1.6. (1) The inequality (1.7) follows easily from the
inequality (2.19).

(2) We see that the function $h$ has a local minimum at
$t_{1}=(3\alpha)^{\alpha}\lambda_{n}^{\alpha-1}(\beta-\alpha)^{-\alpha}\,.$ If
$t_{1}\geq 1\,,$ then the inequality (2.19) yields the desired conclusion. The
upper bound for $T(n,K)$ follows by substituting the argument $t_{1}$ in the
expression of $h\,.$

We next show that the value $K_{1}=4/3$ will do. Fix $K\in(1,K_{1})\,.$ Then
$\alpha=K^{1/(1-n)}\geq 3/4$ and $\alpha/(1-\alpha^{2})>1$.

Because $\lambda_{n}^{\alpha-1}\geq 2^{1/K-1}K^{-1}$ by [Vu1, Lemma 7.50(1)],
with $d=(6/K)^{1/K}/2K$ we have

$t_{1}=(3\alpha)^{\alpha}\lambda_{n}^{\alpha-1}(\beta-\alpha)^{-\alpha}\geq(3/K)^{1/K}2^{1/K-1}K^{-1}\left(\frac{\alpha}{1-\alpha^{2}}\right)^{\alpha}$
$=d\left(\frac{\alpha}{1-\alpha^{2}}\right)^{\alpha}\geq
d\left(\frac{\alpha}{1-\alpha^{2}}\right)^{3/4}$
$=\left(2r(K)\frac{\alpha}{1-\alpha^{2}}\right)^{3/4}\,;r(K)=d^{4/3}/2\,.$

It suffices to observe that $t_{1}>1$ certainly holds if
$2r(K)(\frac{\alpha}{1-\alpha^{2}})>1$ which holds for
$\alpha>1/(r(4/3)+\sqrt{1+r(4/3)^{2}})=0.53...\,,$ in particular, $t_{1}>1$
holds in the present case $\alpha>3/4\,.$

For the proof of (1.9) we give the following inequalites

(3.2) $\lambda_{n}^{\alpha-\alpha^{2}}\leq 2^{\alpha(1-\alpha)}K^{\alpha}\leq
2^{1-\alpha}K^{\alpha},\quad K\geq 1\,,$ (3.3)
$\lambda_{n}^{\beta-\alpha}=\lambda_{n}^{\beta+1-1-\alpha}=\lambda_{n}^{\beta(1-\alpha)+1-\alpha}=\lambda_{n}^{(\beta+1)(1-\alpha)}\leq(2^{1-\alpha}K)^{3},\quad\beta\in(1,2)\,,$

see [Vu1, Lemma 7.50(1)]. The formula (1.8) for $h(t_{1})$ has two terms. We
estimate separately each term as follows

$\displaystyle\frac{3^{1-\alpha^{2}}(\beta-\alpha)^{\alpha^{2}}}{\alpha^{\alpha^{2}}}\lambda_{n}^{\alpha-\alpha^{2}}\lambda_{n}^{2(1-\alpha)}$
$\displaystyle\leq$
$\displaystyle\frac{3^{(1-\alpha)(1+\alpha)}2^{\alpha(1-\alpha)}2^{2(1-\alpha)}K^{2}(\beta-\alpha)^{\alpha^{2}}}{\alpha^{\alpha^{2}}}K^{\alpha}$
$\displaystyle\leq$ $\displaystyle\frac{(9\cdot 2\cdot
4)^{1-\alpha}K^{2}(\beta-\alpha)^{\alpha^{2}}}{\alpha^{\alpha^{2}}}K^{\alpha}$
$\displaystyle=$ $\displaystyle
72^{1-\alpha}(\beta-\alpha)^{\alpha^{2}}K^{2}K^{\alpha}\exp(-\alpha^{2}\log\alpha)$
$\displaystyle\leq$ $\displaystyle
72^{1-\alpha}(\beta-\alpha)^{\alpha^{2}}K^{2}K^{\alpha}\exp(-\alpha\log\alpha)$
$\displaystyle=$ $\displaystyle
72^{1-\alpha}(\beta-\alpha)^{\alpha^{2}}K^{2}\exp((\log K-\log\alpha)\alpha)$
$\displaystyle=$ $\displaystyle
72^{1-\alpha}(\beta-\alpha)^{\alpha^{2}}K^{2}\exp\left(\left(1+\frac{1}{n-1}\log
K\right)\alpha\right)$ $\displaystyle=$ $\displaystyle
72^{1-\alpha}(\beta-\alpha)^{\alpha^{2}}K^{2}\exp\left(\frac{n}{n-1}\alpha\log
K\right)$ $\displaystyle\leq$ $\displaystyle
72^{1-\alpha}(\beta-\alpha)^{\alpha^{2}}K^{2}\exp(2\log K)$ $\displaystyle=$
$\displaystyle 72^{1-\alpha}(\beta-\alpha)^{\alpha^{2}}K^{4}$

by inequality (3.2),

$\displaystyle\lambda_{n}^{2(1-\alpha)}\lambda_{n}^{\beta-1}\left(\frac{(3\alpha)^{\alpha}\lambda_{n}^{\alpha-1}}{(\beta-\alpha)^{\alpha}}\right)^{\beta-\alpha}$
$\displaystyle=$
$\displaystyle\lambda_{n}^{2(1-\alpha)}\lambda_{n}^{\beta-1}\left((3\alpha)^{\alpha}\lambda_{n}^{\alpha-1}\right)^{\beta-\alpha}(\beta-\alpha)^{-\alpha(\beta-\alpha)}$
$\displaystyle\leq$
$\displaystyle(2^{1-\alpha}K)^{2}\lambda_{n}^{\beta-\alpha}\left((3\alpha)^{\alpha}\lambda_{n}^{\alpha-1}\right)^{\beta-\alpha}\left(\frac{\beta^{2}-1}{\beta}\right)^{-\alpha((\beta^{2}-1)/\beta)}$
$\displaystyle\leq$
$\displaystyle(2^{1-\alpha}K)^{2}\left(3^{\alpha}\lambda_{n}\right)^{\beta-\alpha}\beta^{\alpha^{2}}(\beta^{2}-1)^{-\alpha^{2}(\beta^{2}-1)}$
$\displaystyle\leq$
$\displaystyle(2^{1-\alpha}K)^{2}3^{\alpha(\beta-\alpha)}\lambda_{n}^{(\beta+1)(1-\alpha)}\exp\left(\frac{2\alpha^{2}}{e}\sqrt{\beta^{2}-1}\right)$
$\displaystyle\leq$ $\displaystyle
3^{1-\alpha^{2}}(2^{1-\alpha}K)^{2}(2^{1-\alpha}K)^{(\beta+1)}\exp\left(\frac{2\alpha^{2}}{e}\sqrt{\beta^{2}-1}\right)$
$\displaystyle\leq$ $\displaystyle
3^{1-\alpha^{2}}(2^{1-\alpha}K)^{5}\exp(\sqrt{\beta^{2}-1}),$

here we assume that $\beta\in(1,2)$ which implies that $\alpha\in(1/2,1)$.
Also the inequalities $(K-1)^{-(K-1)}\leq\exp((2/e)\sqrt{K-1})$ and (3.3) were
used, and we get

(3.4)
$h(t_{1})\leq\left[72^{1-\alpha}(\beta-\alpha)^{\alpha^{2}}K^{4}+3^{\beta-\alpha}(2^{1-\alpha}K)^{5}\exp(\sqrt{\beta^{2}-1})\right].$

Because $(\beta-\alpha)\in(0,\frac{3}{2})$ this implies that
$\frac{2}{3}(\beta-\alpha)\in(0,1)$ and $\alpha^{2}\in(\frac{1}{4},1)$ and
further
$(\frac{2}{3}(\beta-\alpha))^{\alpha^{2}}\leq(\frac{2}{3}(\beta-\alpha))^{1/4}$,
and finally

$(\beta-\alpha)^{\alpha^{2}}\leq(2/3)^{-\alpha^{2}}(\frac{2}{3}(\beta-\alpha))^{1/4}\leq(3/2)^{3/4}\sqrt[4]{\beta-\alpha}$
$=(3/2)^{3/4}\sqrt[4]{\beta-\alpha}<(3/2)\sqrt[4]{\beta-\alpha}\,.$

Next we prove that

(3.5) $72^{1-\alpha}\leq 3^{1-\alpha^{2}}2^{5(1-\alpha)}K\,.$

This inequality is equivalent to

$2^{2(\alpha-1)}3^{(1-\alpha)^{2}}\leq K\Longleftrightarrow-(1-\alpha)\log
4+(1-\alpha)^{2}\log 3\leq\log K\,.$

This last inequality holds because the left hand side is negative. Now from
(3.4) and (3.5) we get the desired inequality (1.9).  $\square$

######  3.6.

Graphical and numerical comparision of various bounds. The above bounds
involve the Grötzsch ring constant $\lambda_{n},$ which is known only for
$n=2,\lambda_{2}=4.$ Therefore only for $n=2$ we can compute the values of the
bounds. Solving numerically the equation ${4\cdot 16}^{1-1/K}=h(t_{1})$ for
$K$ we obtain $K=1.3089\,.$ We give numerical and graphical comparison of the
various bounds for the Mori constant.

Tabulation of the various upper bounds for Mori’s constant when $n=2$ and
$\lambda_{2}=4$ as a function of $K$: (a) Mori’s conjectured bound
$16^{1-1/K}$, (b) the Anderson-Vamanamurthy bound $4\cdot 16^{1-1/K}$, (c) the
bound from (1.8). For $K\in(1,1.3089)$ the upper bound in (1.8) is better than
the Anderson-Vamanamurthy bound and for $K>1.5946$ the upper bound in (1.8) is
better than the bound of Fehlmann and Vuorinen. Numerical values of the [FV]
bound given in the table were computed with the help of the algorithm for
$\varphi_{K,2}(r)$ attached with [AVV1, p. 92, 439].

Figure 2. Graphical illustration of the various upper bounds for Mori’s
constant when $n=2$ and $\lambda_{2}=4$ as a function of $K$: (a) Mori’s
conjectured bound $16^{1-1/K}$, (b) the Anderson-Vamanamurthy bound $4\cdot
16^{1-1/K}$, (c) the bound from (1.8). For $K\in(1,1.3089)$ the upper bound in
(1.8) is better than the Anderson-Vamanamurthy bound.

$\begin{array}[]{|c|c|c|c|c|}\hline\cr K&\log({16}^{1-1/K})&\log({4\cdot
16}^{1-1/K})&\log(FV)&\displaystyle\log(h(t_{1}))\\\ \hline\cr
1.1&0.2521&1.6384&0.7051&1.0188\\\ 1.2&0.4621&1.8484&1.2485&1.6058\\\
1.3&0.6398&2.0261&1.7046&2.0107\\\ 1.4&0.7922&2.1785&2.0913&2.3061\\\
1.5&0.9242&2.3105&2.4221&2.5296\\\ 1.6&1.0397&2.4260&2.7094&2.7031\\\
1.7&1.1417&2.5280&2.9633&2.8409\\\ 1.8&1.2323&2.6186&3.1921&2.9521\\\
1.9&1.3133&2.6996&3.4020&3.0433\\\ 2.0&1.3863&2.7726&3.5979&3.1192\\\
\hline\cr\end{array}$

For graphing and tabulation purposes we use the logarithmic scale. Note that
the upper bound for $M(2,K)$ given in [FV, Theorem 2.29] also has the
desirable property that it converges to $1$ when $K\to 1\,,$ see Figure 2.

Figure 3. Graphical comparison of various bounds when $n=2$ and
$\lambda_{2}=4\,,$ as a function of $K$: (a) the bound from (1.8), (b) the
Fehlmann and Vuorinen bound [FV]
$M(2,K)\leq\left(1+\varphi_{K,2}\left(\frac{K^{2}-1}{K^{2}+1}\right)\right)2^{2K-3/K}\frac{(K^{2}+1)^{(K+1/K)/2}}{(K^{2}-1)^{(K-1/K)/2}}.$
For $K>1.5946$ the upper bound in (1.8) is better than the Fehlmann-Vuorinen
bound.

######  3.7.

Comparison of estimates for the Hölder quotient. For a $K$-quasiconformal
mapping $f:\mathbf{B}^{n}\to f\mathbf{B}^{n}=\mathbf{B}^{n}\,,$ we call the
expression

$HQ(f)=\sup\\{|f(x)-f(y)|/|x-y|^{\alpha}:\;x,y\in\mathbf{B}^{n},f(0)=0\,\;x\neq
y\\},$

the Hölder coefficient of $f$. Clearly $HQ(f)\leq M(n,K)$. Theorem 2.13
yields, after dividing the both sides of the inequality in 2.13 by
$|x-y|^{\alpha}\,,$ the upper bound $HQ(f)\leq HQ(K)$ for the Hölder quotient
with

(3.8) $HQ(K)=\sup\\{\inf\\{U(t,x,y):\;t\geq 1\\}:\;x,y\in\mathbf{B}^{n}\\}\,,$
$U(t,x,y)=(3+\varphi_{1/K,n}(1/t)^{-1})\varphi_{K,n}^{2}\left(\left(\frac{2|x-y|}{s_{1}+|x-y|}\right)^{1/2}\right)\frac{1}{|x-y|^{\alpha}}\,.$

For $n=2$ we compare $HQ(K)$ to several other bounds (a) Mori’s conjectured
bound, (b) the FV bound, (c) the AV bound and give the results as a table and
Figure 3. Because the supremum and infimum in (3.8) cannot be explicitly found
we use numerical methods that come with Mathematica software. For the
numerical tests we used for the supremum a sample of $100,000$ random points
of the unit disk.

Figure 4. Graphical comparison of various bounds when $n=2$ and
$\lambda_{2}=4\,,$ as a function of $K$: (a) the bound from (3.8), (b) the
Fehlmann and Vuorinen bound [FV]
$M(2,K)\leq\left(1+\varphi_{K,2}\left(\frac{K^{2}-1}{K^{2}+1}\right)\right)2^{2K-3/K}\frac{(K^{2}+1)^{(K+1/K)/2}}{(K^{2}-1)^{(K-1/K)/2}},$
(c) the bound of the Mori conjecture. Note that the bound (3.8), based on a
simulation with $100,000$ random points, gives the best estimate in the cases
considered in the picture.

$\begin{array}[]{|c|c|c|c|c|}\hline\cr K&\log({16}^{1-1/K})&\log({4\cdot
16}^{1-1/K})&\log(FV)&\log(HQ(K))\\\ \hline\cr
1.1&0.2521&1.6384&0.7051&1.0171\\\ 1.2&0.4621&1.8484&1.2485&1.5940\\\
1.3&0.6398&2.0261&1.7046&1.9712\\\ 1.4&0.7922&2.1785&2.0913&2.1668\\\
1.5&0.9242&2.3105&2.4221&2.2928\\\ 1.6&1.0397&2.4260&2.7094&2.4003\\\
1.7&1.1417&2.5280&2.9633&2.4922\\\ 1.8&1.2323&2.6186&3.1921&2.5706\\\
1.9&1.3133&2.6996&3.4020&2.6371\\\ 2.0&1.3863&2.7726&3.5979&2.6934\\\
\hline\cr\end{array}$

##  4\. An explicit form of the Schwarz lemma

Recall that the hyperbolic metric $\rho(x,y),x,y\in\mathbf{B}^{n}\,,$ of the
unit ball is given by (cf. [KL], [Vu1])

(4.1) ${\rm
th}^{2}\frac{\rho(x,y)}{2}=\frac{|x-y|^{2}}{|x-y|^{2}+t^{2}}\,,\quad
t^{2}=(1-|x|^{2})(1-|y|^{2})\,.$

Next, we consider a decreasing homeomorphism
$\mu:(0,1)\longrightarrow(0,\infty)$ defined by

(4.2) $\mu(r)=\frac{\pi}{2}\,\frac{{\mathchoice{\hbox{\,\fFt K}}{\hbox{\,\fFt
K}}{\hbox{\,\fFa K}}{\hbox{\,\fFp K}}}(r^{\prime})}{{\mathchoice{\hbox{\,\fFt
K}}{\hbox{\,\fFt K}}{\hbox{\,\fFa K}}{\hbox{\,\fFp
K}}}(r)},\quad{\mathchoice{\hbox{\,\fFt K}}{\hbox{\,\fFt K}}{\hbox{\,\fFa
K}}{\hbox{\,\fFp
K}}}(r)=\int_{0}^{1}\frac{dx}{\sqrt{(1-x^{2})(1-r^{2}x^{2})}}\,,$

where ${\mathchoice{\hbox{\,\fFt K}}{\hbox{\,\fFt K}}{\hbox{\,\fFa
K}}{\hbox{\,\fFp K}}}(r)$ is Legendre’s complete elliptic integral of the
first kind and $r^{\prime}=\sqrt{1-r^{2}},$ for all $r\in(0,1)$.

The Hersch-Pfluger distortion function is an increasing homeomorphism
$\varphi_{K}:(0,1)\longrightarrow(0,1)$ defined by setting

(4.3) $\varphi_{K}(r)=\mu^{-1}(\mu(r)/K)\,,\,r\in(0,1),\,\,K>0.$

Note that with the notation of Section 2, $\gamma_{2}(1/r)=2\pi/\mu(r)$ and
$\varphi_{K}(r)=\varphi_{K,2}(r)\,$ for $r\in(0,1)\,.$

######  4.4 Theorem.

[Vu1, 11.2] Let $f:\mathbf{B}^{n}\to\mathbf{R}^{n}$ be a nonconstant
$K$-quasiregular mapping with $f\mathbf{B}^{n}\subset\mathbf{B}^{n}$ and let
$\alpha=K^{1/(1-n)}\,.$ Then

(4.5) ${\rm th}\frac{\rho(f(x),f(y))}{2}\leq\varphi_{K,n}({\rm
th}\frac{\rho(x,y)}{2})\leq\lambda_{n}^{1-\alpha}\left({\rm
th}\frac{\rho(x,y)}{2}\right)^{\alpha}\,,$ (4.6) $\rho(f(x),f(y))\leq
K(\rho(x,y)+\log 4)\,,$

for all $x,y\in\mathbf{B}^{n}\,,$ where $\lambda_{n}$ is the same constant as
in (1.5). If $f(0)=0\,,$ then

(4.7) $|f(x)|\leq\lambda_{n}^{1-\alpha}|x|^{\alpha}\,,$

for all $x\in\mathbf{B}^{n}\,.$

In the case of quasiconformal mappings with $n=2$ formulas (4.5) and (4.7)
also occur in [LV, p. 65] and formula (4.6) was rediscovered in [EMM, Theorem
5.1]. Comparing Theorem 4.4 to Theorem 1.10 we see that for $n=2$ the
expression $K(\rho(x,y)+\log 4)$ may be replaced with
$c(K)\max\\{\rho(x,y),\rho(x,y)^{1/K}\\}\,,$ which tends to $0$ when $x\to
y\,$ and to $\rho(x,y)$ when $K\to 1\,,$ as expected.

######  4.8 Lemma.

For $K>1$ the function

$t\mapsto\frac{2{\rm arth}(\varphi_{K}({\rm
th}\frac{t}{2}))}{\max\\{t,t^{1/K}\\}}\,,$

is monotone increasing on $(0,1)$ and decreasing on $(1,\infty)\,.$

###### Proof.

(1) Fix $K>1$ and consider

$f(t)=\frac{2{\rm arth}(\varphi_{K}({\rm th}\frac{t}{2}))}{t},\quad t>0.$

Let $r={\rm th}\frac{t}{2}$. Then $t/2={\rm arth}r$, and $t$ is an increasing
function of $r$ for $0<r<1$. Then

$f(t)=\frac{2{\rm arth}(\varphi_{K}({\rm th}\frac{t}{2}))}{t/2}=\frac{{\rm
arth}(\varphi_{K}(r))}{{\rm arth}r}=F(r).$

Then by [AVV1, Theorem 10.9(3)], $F(r)$ is strictly decreasing from $(0,1)$
onto $(K,\infty)$. Hence $f(t)$ is strictly decreasing from $(0,\infty)$ onto
$(K,\infty)$.

(2) Next consider

$g(t)=\frac{2{\rm arth}(\varphi_{K}({\rm th}\frac{t}{2}))}{t^{1/K}},$

and let $r={\rm th}\frac{t}{2}$. Then $t=2{\rm arth}r$ and

$g(t)=\frac{2{\rm arth}s}{2^{1/K}({\rm arth}r)^{1/K}}=\frac{2^{1-1/K}{\rm
arth}s}{({\rm arth}r)^{1/K}}\,,$

where $s=\varphi_{K}(r)$. We next apply [AVV1, Theorem 1.25]. We know
$\frac{d}{dr}({\rm arth}r)=1/(1-r^{2})$.

Writing $r^{\prime}=\sqrt{1-r^{2}},s^{\prime}=\sqrt{1-s^{2}},$ we obtain the
quotient of the derivatives

$\displaystyle\frac{2^{1-1/K}(1/(1-s^{2}))\frac{ds}{dr}}{\frac{1}{K}({\rm
arth}r)^{1/K-1}(1/(1-r^{2})}$ $\displaystyle=$ $\displaystyle
2^{1-1/K}\,K\,({\rm
arth}r)^{1-1/K}\frac{r^{{}^{\prime}2}}{s^{{}^{\prime}2}}\frac{1}{K}\frac{ss^{{}^{\prime}2}\mathchoice{\hbox{\,\fFt
K}}{\hbox{\,\fFt K}}{\hbox{\,\fFa K}}{\hbox{\,\fFp
K}}(s)^{2}}{rr^{{}^{\prime}2}\mathchoice{\hbox{\,\fFt K}}{\hbox{\,\fFt
K}}{\hbox{\,\fFa K}}{\hbox{\,\fFp K}}(r)^{2}}$ $\displaystyle=$ $\displaystyle
2^{1-1/K}({\rm arth}r)^{1-1/K}\frac{s\mathchoice{\hbox{\,\fFt K}}{\hbox{\,\fFt
K}}{\hbox{\,\fFa K}}{\hbox{\,\fFp K}}(s)^{2}}{r\mathchoice{\hbox{\,\fFt
K}}{\hbox{\,\fFt K}}{\hbox{\,\fFa K}}{\hbox{\,\fFp K}}(r)^{2}}$

by [AVV1, appendix E(23)]. By [AVV1, Lemma 10.7(3)],
$\frac{\mathchoice{\hbox{\,\fFt K}}{\hbox{\,\fFt K}}{\hbox{\,\fFa
K}}{\hbox{\,\fFp K}}(s)^{2}}{\mathchoice{\hbox{\,\fFt K}}{\hbox{\,\fFt
K}}{\hbox{\,\fFa K}}{\hbox{\,\fFp K}}(r)^{2}}$ is increasing, since $K>1$,
$({\rm arth}r)^{1/K-1}$ is increasing. Finally, $s/r$ is increasing by [AVV1,
Theorem 1.25] and E(23). So $g(t)$ is increasing in $t$ on $(0,\infty)$.

(3) Fix $K>1$. Clearly

$\max\\{t,t^{1/K}\\}=\left\\{\begin{array}[]{lll}t^{1/K}\quad{\rm for}\quad
0\leq t\leq 1\\\ t\quad{\rm for}\quad 1\leq t<\infty.\end{array}\right.$

Thus

$h(t)=\frac{2{\rm arth}(\varphi_{K}({\rm
th}\frac{t}{2}))}{\max\\{t,t^{1/K}\\}},\;$

increases on $(0,1)$ and decreases on $(1,\infty)$. ∎

Figure 5. Graphical comparison of lower and upper bounds for $c(K)$ with
$b(K)=\log({\rm ch}(K{\rm arch}(e)))$.

######  4.9.

Proof of Theorem 1.10. The maximum value of the function considered in Lemma
4.8 is $c(K)=2{\rm arth}(\varphi_{K}({\rm th}\frac{1}{2}))$. The inequality
now follows from Lemma 4.8.$\qquad\square$

######  4.10.

Bounds for the constant $c(K)$. In order to give upper and lower bounds for
$c(K)\,,$ we observe that the identity [AVV1, Theorem 10.5(2)] yields the
following formula

$c(K)=2{\rm
arth}\left(\varphi_{K}\left(\frac{1-1/e}{1+1/e}\right)\right)=2{\rm
arth}\left(\frac{1-\varphi_{1/K}(1/e)}{1+\varphi_{1/K}(1/e)}\right)\,.$

A simplification leads to

$c(K)=-\log\varphi_{1/K}(1/e)\,.$

Next, from the inequality $\varphi_{1/K}(r)\geq
2^{1-K}(1+r^{{}^{\prime}})^{1-K}r^{K}$ for $K\geq 1,r\in(0,1)$ (cf. [AVV1,
Corollary 8.74(2)]) we get with $v=\log(2(1+\sqrt{1-1/e^{2}}))<1.3507$

$\displaystyle c(K)$ $\displaystyle=$
$\displaystyle-\log\varphi_{1/K}(1/e)\leq-\log\left(2^{1-K}(1+\sqrt{1-1/e^{2}})^{1-K}e^{-K}\right)$
$\displaystyle=$ $\displaystyle v(K-1)+K<1.3507(K-1)+K.$

In order to estimate the constant $c(K)$ from below we need an upper bound for
$\varphi_{1/K,2}(r),\;K>1$, from above. For this purpose we prove the
following lemma.

######  4.11 Lemma.

For every integer $n\geq 2$ and each $K>1,\;r\in(0,1)$, there exists
$K$-quasiconformal maps $g:\mathbf{B}^{n}\to\mathbf{B}^{n}$ and
$h:\mathbf{B}^{n}\to\mathbf{B}^{n}$ with
$(a)\qquad
g(0)=0,\;g\mathbf{B}^{n}=\mathbf{B}^{n},\;h(0)=0,\;h\mathbf{B}^{n}=\mathbf{B}^{n}$
$(b)\qquad
g(re_{1})=\displaystyle\frac{2r^{\alpha}}{(1+r^{{}^{\prime}})^{\alpha}+(1-r^{{}^{\prime}})^{\alpha}},\;h(re_{1})=\displaystyle\frac{2r^{\beta}}{(1+r^{{}^{\prime}})^{\beta}+(1-r^{{}^{\prime}})^{\beta}}$
where $r^{{}^{\prime}}=\sqrt{1-r^{2}}$ and $\alpha=K^{1/(1-n)}=1/\beta$.
In particular, for $n=2$ and $K>1,\;r\in(0,1)$
$(c)\qquad\varphi_{1/K}(r)\leq\displaystyle\frac{2r^{K}}{(1+r^{{}^{\prime}})^{K}+(1-r^{{}^{\prime}})^{K}}\;;\;\;\varphi_{K}(r)\geq\displaystyle\frac{2r^{1/K}}{(1+r^{{}^{\prime}})^{1/K}+(1-r^{{}^{\prime}})^{1/K}}$.

###### Proof.

Fix $r\in(0,1)$. Let $T_{a}:\mathbf{B}^{n}\to\mathbf{B}^{n}$ be a Möbius
automorphism with $T_{a}(a)=0$ and $T_{a}(\mathbf{B}^{n})=\mathbf{B}^{n}$.
Choose $s\in(0,r)$ such that $T_{se_{1}}(0)=-T_{se_{1}}(re_{1})$. Then
$\rho(0,re_{1})=2\rho(0,se_{1})$ [Vu1, (2.17)], or equivalently,
$(1+r)/(1-r)=((1+s)/(1-s))^{2}$ and hence $s=r/(1+r^{{}^{\prime}})$. Consider
the $K$-quasiconformal mapping $f:\mathbf{B}^{n}\to\mathbf{B}^{n}$,
$f(x)=|x|^{\alpha-1}x,\;\alpha=K^{1/(1-n)}$. Then $f(\pm se_{1})=\pm
s^{\alpha}e_{1}$. The mapping $g=T_{-s^{\alpha}e_{1}}\circ f\circ
T_{se_{1}}:\mathbf{B}^{n}\to\mathbf{B}^{n}$ satisfies $g(0)=0$,
$g(re_{1})=te_{1}$ where
$\rho(-s^{\alpha}e_{1},s^{\alpha}e_{1})=\rho(0,te_{1})$ and hence
$t=2r^{\alpha}/((1+r^{{}^{\prime}})^{\alpha}+(1-r^{{}^{\prime}})^{\alpha})$ by
[Vu1, (2.17)]. The proof for $g$ is complete. For the map $h$ the proof is
similar except that we use the $K$-quasiconformal mapping
$m:x\mapsto|x|^{\beta-1}x,\;\beta=1/\alpha$. Note that $m=f^{-1}$ and
$t=1/{\rm ch}(\alpha\;{\rm arch}(1/r))$. For the proof of $(c)$ we apply
$(a),\;(b)$ together with [LV, (3.4), p.64]. ∎

######  4.12 Lemma.

For $K>1,$ $c(K)\geq\log({\rm ch}(K{\rm arch}(e)))\geq u(K-1)+1,$ where
$u={\rm arch}(e){\rm th}({\rm arch}(e))>1.5412$.

###### Proof.

From Lemma 4.11(c), we know that

$\displaystyle\varphi_{1/K}(1/e)$ $\displaystyle\leq$
$\displaystyle\frac{2/e^{K}}{(1+\sqrt{1-1/e^{2}})^{K}+(1-\sqrt{1-1/e^{2}})^{K}}$
$\displaystyle=$
$\displaystyle\frac{2}{(e+\sqrt{e^{2}-1})^{K}+(e-\sqrt{e^{2}-1})^{K}},$

hence

$\displaystyle c(K)$ $\displaystyle=$
$\displaystyle-\log\varphi_{1/K}(1/e)\geq-\log\left(\frac{2}{(e+\sqrt{e^{2}-1})^{K}+(e-\sqrt{e^{2}-1})^{K}}\right)$
$\displaystyle=$
$\displaystyle\log\left(\frac{(e+\sqrt{e^{2}-1})^{K}+(e-\sqrt{e^{2}-1})^{K}}{2}\right)$
$\displaystyle=$ $\displaystyle\log({\rm ch}(K{\rm arch}(e)))\geq u(K-1)+1,$

where the last inequality follows easily from the mean value theorem, applied
to the function $p(K)=\log({\rm ch}(K{\rm arch}(e)))\,.$ ∎

## References

  * [A1] L. V. Ahlfors: On quasiconformal mappings, J. Analyse Math. 3, (1954). 1–58; correction, 207–208, also: pp. 2-61 in Collected papers. Vol. 2. 1954–1979. Edited with the assistance of Rae Michael Shortt. Contemporary Mathematicians. Birkhäuser, Boston, Mass., 1982\. xix+515 pp. ISBN: 3-7643-3076-7.
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