On two Bloch type theorems for quaternionic slice regular functions

0.22 MB·

by  Zhenghua Xu

ON TWO BLOCH TYPE THEOREMS FOR QUATERNIONIC
SLICE REGULAR FUNCTIONS
6
1
ZHENGHUAXUANDXIEPINGWANG
0
2
n Abstract. Inthispaper weprovetwoBlochtype theorems forquaternionic
a slice regular functions. We first discuss the injective and covering properties
J of some classes of slice regular functions from slice regular Bloch spaces and
sliceregularBergmanspaces,respectively. Andthenweshowthatthereexits
1
auniversalballcontained intheimageoftheopenunitballBinquaternions
1 Hthroughthesliceregularrotationfeuofeachsliceregularfunctionf :B→H
withf′(0)=1forsomeu∈∂B.
]
V
C
.
h
1. Introduction
t
a
m Let C be the complex plane and D(z ,R) the open disc centred at z C with
0 0
radiusR>0. For simplicity, we denote by D the openunit disc D(0,1). L∈et H(D)
[
denote the class of holomorphic functions on D. Given a function F H(D), we

1 define B to be the least upper bound of all positive numbers R >∈0 such that
F
v there exists a number z C and a domain Ω D which is mapped conformally
8 0 ∈ ⊂
by F onto D(z ,r). The Bloch constant B is defined to be
3 0 h
3 B :=inf B :F H(D),F (0)=1 .
2 h F ∈ ′
0 The Bloch’s theorem asserts tha(cid:8)t B >0 in [3]. In [24], La(cid:9)ndau showed that
h
.
1
0 (1.1) Bh =inf BF :F ∈Bh,F(0)=0,F′(0)=1 ,
6 where denotes the class(cid:8)of holomorphic functions F H(cid:9)(D) with the Bloch
h
1 B ∈
seminorm
:
v F :=sup(1 z 2)F (z) 1.
Xi k kBh z∈D −| | | ′ |≤
In [1], Ahlfors proved that B 3/4 using his far-reaching generalization of
r h ≥
a the classical Schwarz-Pick lemma. The strict inequality B > 3/4 was established
h
by Heins [21] and Pommerenke [26], respectively. In [5], Bonk gave a new and
remarkableproofofthisresultandimprovedslightlyitusingthefollowingtheorem,
known as Bonk’s distortion theorem.
Theorem 1.1. Let F be such that F (0)=1. Then the inequality
h ′
∈B
1 √3z
ReF′(z) − | |
≥ (1 z /√3)3
−| |
holds for all z D(0,1/√3).
∈
2010 Mathematics Subject Classification. 30G35.
Key words and phrases. Quaternion,Sliceregularfunctions,Blochconstant.
ThisworkwassupportedbytheNNSFofChina(11071230), RFDP(20123402110068).
1
2 Z.XUANDX.WANG
ThefindingoftheprecisevalueofB isknowntobethenumberoneopenprob-
h
leminthegeometricfunctiontheoryofonecomplexvariablesincetheconfirmation
oftheBieberbachconjecturebydeBrangesin1985[10]. Totheauthors’knowledge,
the best lowerestimate for B is bynow in[6]. The Bloch’stheoremdoes nothold
h
for general holomorphic mappings of several complex variables. The counterexam-
plecanbefoundin[20]. Thus,oneneedstorestricttheclassofmappingstoamore
specific subclass to obtain a Bloch’s theorem. One of the well-known subclasses is
the class of K-quasiregular mappings [31]. Recently, the Bloch’s theorem in the
bicomplex number setting has been investigated successfully in [27].
In this paper, we first establish the quaternionic analogues of Bonk’s distortion
theorem and Bloch type theorem for slice regular functions. The theory of slice
regular functions over quaternions was initiated recently by Gentili and Struppa
[17, 18]. It is significantly different from the more classical theory of regular func-
tions in the sense of Cauchy-Fueter and has elegant applications to the functional
calculus for noncommutative operators [9], Schur analysis [2] and the construction
and classification of orthogonal complex structures on dense open subsets of R4
[13]. For the detailed up-to-date theory, we refer the reader to the monographs
[16, 9].
In order to formulate precisely our main results, we first introduce a few no-
tations. Let H be the skew field of quaternions and S the unit sphere of purely
imaginary quaternions, i.e.
S= q H:q2 = 1 .
{ ∈ − }
For each R>0 and each point q H, we set
0
∈
B(q ,R):= q H: q q <R ,
0 0
∈ | − |
the Euclidean ball centred at q of(cid:8)radius R, and for ea(cid:9)ch I S, we denote by
0
B(0,R) the intersection B(0,R) C . For simplicity, we den∈ote by B the open
I I
unitdisc B(0,1). Also,wedenote b∩y the slice regularBlochspaceonB, that is,
r
the space of slice regular functions f Bon B with the Bloch seminorm
kfkBr :=squpB(1−|q|2)|f′(q)|<∞,
∈
where f is the slice derivative of f. For each slice regular function f on B, we
′
define B to be the least upper bound of all positive numbers R > 0 such that
f
there exists a number q H and a domain Ω B such that the restriction
0
∈ ⊂
f :Ω B(q ,R)
|Ω → 0
is a homeomorphism. Similar to (1.1), we define the Bloch constant B for as
r r
B
B =inf B :f ,f(0)=0,f (0)=1, f 1 .
r f ∈Br ′ k kBr ≤
(cid:8) (cid:9)
Now we can show that B >0.23 by proving the following result.
r
Theorem 1.2. Let f be such that f(0)=0,f (0)=1 and f 1. Then
∈Br ′ || ||Br ≤
(a) the inequality
1 √3q
(1.2) Ref′(q) − | |
≥ (1 q /√3)3
−| |
holds for all q B(0,1/√3). Inparticular, f :B(0,1/√3) H
∈ |B(0,1/√3)I I →
is injective for every I S.
∈
ON TWO BLOCH TYPE THEOREMS FOR QUATERNIONIC SLICE REGULAR FUNCTIONS3
(b) f is injective on B(0,rB) and B(0,RB) f B(0,rB) , where
⊂
(cid:0) (cid:1)
1 1+r 1 1+r 2
rB = sup log log r2 0.3552,
0<r<1(cid:26)2 1 r −r4(cid:16) 1 r(cid:17) − (cid:27)≈
− −
and
2
1 1+r 1 1+r 1 1+r 2
RB = sup log log log r2 0.2308.
0<r<1(cid:26)2r2 1 r(cid:18)2 1 r −r4(cid:16) 1 r(cid:17) − (cid:19) (cid:27)≈
− − −
Moreover,
B(0,√3/4) f B(0,1/√3) .
⊂
(cid:0) (cid:1)
WecanalsoproveaBlochtypetheoremforfunctionsfromsliceregularBergman
spaceAp(B)(1 p<+ ) ofthe secondkindintroducedin[7],that is,the Hilbert
space of slice re≤gular fu∞nctions f on B with the norm
1
p
f Ap :=sup fI(z)pdσI(z) < ,
k k I∈S(cid:18)ZBI | | (cid:19) ∞
where dσ stands for the normalised Lebesgue measure on the plane C , i.e
I I
1
dσ (z)= dxdy
I
π
for all z =x+yI C .
I
∈
Theorem 1.3. Let p [1,+ ) and f Ap(B) be such that f(0) = 0,f (0) = 1
′
∈ ∞ ∈
and f Ap 1. Then f is injective on B(0,rp) and B(0,Rp) f B(0,rp) , where
k k ≤ ⊂
(cid:0) (cid:1)
2 4
rp = sup (1 r)−p (1 r)−p r2 ,
0<r<1(cid:26) − −q − − (cid:27)
and
2
Rp = sup r−2(1 r)−p2 (1 r)−p2 (1 r)−p2 r2 .
0<r<1(cid:26) − (cid:18) − −q − − (cid:19) (cid:27)
To prove Theorems 1.2 and 1.3, we need to establish a quaternionic analogue of
a classical result due to Landau (see [24, 23] or [22, pp. 36–39]) for slice regular
functions. Let α (0,1). Define to be a family of slice regular functions given
α
∈ F
by
α := f :B B f is slice regular such that f(0)=0, f′(0) =α ,
F → | |
(cid:8) (cid:12) (cid:9)
and for every f α, de(cid:12)note by rα(f) the radius of the largestball B(r) on which
∈F
f is injective. Namely,
r (f)=sup r : f is injective .
α |B(r)
(cid:8) (cid:9)
Set
r( )=inf r (f):f .
α α α
F ∈F
From[16,Theorem8.13]itfollowsthat(cid:8)foreachf (cid:9),therealdifferentialoff at
α
∈F
the origin 0 is non-degenerate, and thus the inverse function theorem implies that
r (f) is positive. Indeed, our result shows that r( ) is a positive number as well.
α α
F
Theorem 1.4. With notations as above, the following statements hold:
(a) for each α (0,1),
∈
α
r( )= ;
α
F 1+√1 α2
−
4 Z.XUANDX.WANG
(b) for each α (0,1) and each f ,
α
∈ ∈F
r (f) r( )
α α
≥ F
with equality if and only if
(1.3) f(q)=q 1 qαe−Iθ −∗ q αeIθ v
− ∗ −
(cid:0) (cid:1) (cid:0) (cid:1)
for some I S, θ R and v ∂B;
∈ ∈ ∈
(c) for each 0<r r( ),
α
≤ F
f B(0,r) =B 0,R (r) ,
α
f∈\Fα (cid:0) (cid:1) (cid:0) (cid:1)
where
α r
R (r)=r − ;
α
1 αr
−
(d) for each α (0,1), r 0,r( ) and f ,
α α
∈ ∈ F ∈F
(cid:0) (cid:1)
B 0,R (r) f B(0,r) .
α
⊂
(cid:0) (cid:1) (cid:0) (cid:1)
Moreover,B 0,R (r) isthelargestballcontainedintheimagesetf B(0,r)
α
if and only if(cid:0)f is of t(cid:1)he form (1.3). (cid:0) (cid:1)
In [3], Bloch discovered a theorem stating that if F is a holomorphic function
on the closed unit disc z C : z 1 such that F (0) = 1, then the image
′
{ ∈ | | ≤ } | |
domain contains discs of radius 3/2 √2. More recently, the corresponding result
−
has been established for square-integrable monogenic functions in the open ball of
the paravector space R3 with values in R3 H as well (see [25, Theorem 8]), and
for a kind of regulartranslationsofslice reg⊂ularfunctions onB in the quaternionic
setting (see [28, Theorem 6]), respectively. In this paper, we also present another
versionof Bloch type theoremfor slice regularfunctions. To this purpose, we need
to consider the slice regular rotation f , instead of the regular translation in [28],
u
of a slice regular function f(q)= ∞n=e0qnan on B given by
P
∞
f (q)= qnuna ,
u n
nX=0
e
for some constant u ∂B.
∈
Theorem 1.5. Let f be slice regular on B such that f (0) = 1. Then there exists
′
u ∂B such that the image of B under the slice regular rotation f of f contains
u
∈
an open ball of radius 5/2 √6.
− e
The remaining part of this paper is organized as follows. In Sect. 2, we set up
basic notations and give some preliminary results from the theory of slice regular
functions over quaternions. In Sect. 3, we shall give some useful lemmas, among
which the Landau’s lemma is established by means of one Lindelo¨f type inequality
for slice regular functions. Sect. 4 is devoted to the proofs of Theorems 1.2, 1.3,
and 1.4. Finally, Theorem 1.5 will be proved in Sect. 5.
ON TWO BLOCH TYPE THEOREMS FOR QUATERNIONIC SLICE REGULAR FUNCTIONS5
2. Preliminaries
In this section we recall some necessary definitions and preliminary results on
sliceregularfunctions. Tohaveamorecomplete insightonthetheory,wereferthe
reader to the monograph [16].
LetHdenotethenon-commutative,associative,realalgebraofquaternionswith
standard basis 1, i, j, k , subject to the multiplication rules
{ }
i2 =j2 =k2 =ijk= 1.
−
Everyelementq =x +x i+x j+x kinHiscomposedbythereal partRe(q)=x
0 1 2 3 0
and the imaginary part Im(q) =x i+x j+x k. The conjugate of q H is then
1 2 3
∈
q¯= Re(q) Im(q) and its modulus is defined by q 2 = qq = Re(q)2+ Im(q)2.
− | | | | | |
We cantherefore calculate the multiplicative inverseofeachq =0 as q 1 = q 2q.
− −
Every q H can be expressed as q =x+yI, where x,y R an6 d | |
∈ ∈
Im(q)
I =
Im(q)
| |
ifImq =0,otherwisewetakeI arbitrarilysuchthatI2 = 1. ThenI isanelement
6 −
of the unit 2-sphere of purely imaginary quaternions
S= q H:q2 = 1 .
∈ −
For every I S we will denote(cid:8)by C the plane(cid:9)R IR, isomorphic to C, and,
I
if Ω H, by Ω∈the intersection Ω C . Also, for R>⊕0, we will denote the open
I I
ball c⊂entred at q H with radius R∩by
0
∈
B(q ,R)= q H: q q <R .
0 0
∈ | − |
And let B denote the open unit ball(cid:8)B(0,1) for simplicity(cid:9).
We can now recall the definition of slice regularity.
Definition 2.1. Let Ω be a domain in H. A function f : Ω H is called slice
regular if,forallI S,itsrestrictionf toΩ isholomorphic, i.e→.,ithascontinuous
I I
∈
partial derivatives and satisfies
1 ∂ ∂
∂¯ f(x+yI):= +I f (x+yI)=0
I I
2(cid:18)∂x ∂y(cid:19)
for all x+yI Ω . We will denote by (B) the set of slice regular functions on
I
B. ∈ SR
A wide class ofexamples ofregularfunctions is givenby polynomials andpower
series. Indeed, a function f is slice regular on an open ball B(0,R) if and only if
f admits a power series expansion f(q) = ∞n=0qnan converging absolutely and
uniformlyoneverycompactsubsetofB(0,RP). Asshownin[8],thenaturaldomains
of definition of slice regular functions are the so-called symmetric slice domains.
Definition 2.2. Let Ω be a domain in H.
1. Ω is called a slice domain if it intersects the realaxis andif for any I S, Ω
I
is a domain in C . ∈
I
2. Ω is called an axially symmetric domain if for any point x+yI Ω, with
x,y R and I S, the entire two-dimensional sphere x+yS is contained∈in Ω.
∈ ∈
AdomaininHiscalledasymmetric slice domain ifitisnotonlyaslicedomain,
but also an axially symmetric domain. From now on, we will omit the term ‘slice’
when referring to slice regularfunctions and will focus mainly on regularfunctions
6 Z.XUANDX.WANG
on an open ball B(0,R) which is a typical axially symmetric slice domain. For
regular functions the natural definition of derivative is given by the following (see
[17, 18]).
Definition 2.3. Let f : B H be a regular function. For each I S, the
→ ∈
I-derivative of f at q =x+yI is defined by
1 ∂ ∂
∂ f(x+yI):= I f (x+yI)
I I
2(cid:18)∂x − ∂y(cid:19)
on B . The slice derivative of f is the function f defined by ∂ f on B for all
I ′ I I
I S.
∈
Thedefinitioniswell-definedbecause,bydirectcalculation,∂ f =∂ f inB B
I J I J
for any choice of I, J S. Furthermore, notice that the operators ∂ and∩∂¯
I I
commute, and ∂ f = ∂f∈for regular functions. Therefore, the slice derivative of a
I ∂x
regular function is still regular so that we can iterate the differentiation to obtain
the n-th slice derivative
∂nf
∂nf(x+yI)= (x+yI), n N.
I ∂xn ∀ ∈
Inwhatfollows,forthesakeofsimplicity,wewilldenotethen-thslicederivative
by f(n) for every n N.
∈
Inthetheoryofregularfunctions,thefollowingsplittinglemma(see[18])relates
closely slice regularity to classical holomorphy.
Lemma 2.4 (Splitting Lemma). Let f be a regular function on B. Then for any
I S and any J S with J I, there exist two holomorphic functions F,G:B
I
C∈such that ∈ ⊥ →
I
f (z)=F(z)+G(z)J, z =x+yI B .
I I
∀ ∈
Since the regularitydoes notkeepunder point-wiseproductoftworegularfunc-
tions a new multiplication operation, called the regular product (or -product),
∗
appears via a suitable modification of the usual one subject to noncommutative
setting. The regular product plays a key role in the theory of slice regular func-
tions. On open balls centred at the origin, the -product of two regular functions
∗
is defined by means of their power series expansions (see, e.g., [15, 8]).
Definition 2.5. Let f, g :B H be two regular functions and let
→
∞ ∞
f(q)= qna , g(q)= qnb
n n
nX=0 nX=0
be their series expansions. The regular product (or -product) of f and g is the
∗
function defined by
n
∞
f g(q)= qn a b
k n k
∗ nX=0 (cid:18)Xk=0 − (cid:19)
and it is regular on B.
Notice that the -product is associative and is not, in general, commutative.
∗
Its connection with the usual pointwise product is clarified by the following result
[15, 8].
ON TWO BLOCH TYPE THEOREMS FOR QUATERNIONIC SLICE REGULAR FUNCTIONS7
Proposition 2.6. Let f and g be regular on B. Then for all q B,
∈
f(q)g(f(q) 1qf(q)) if f(q)=0;
f g(q)= − 6
∗ (cid:26) 0 if f(q)=0.
We remark that if q = x+yI and f(q) = 0, then f(q) 1qf(q) has the same
−
6
modulus and same real part as q. Therefore f(q) 1qf(q) lies in the same 2-sphere
−
x+yS as q. Notice that a zero x +y I of the function g is not necessarily a zero
0 0
of f g, but some element on the same sphere x +y S does. In particular, a real
0 0
∗
zeroofg is stilla zerooff g. To presentacharacterizationofthe structureofthe
∗
zero set of a regular function f, we need to introduce the following functions.
Definition 2.7. Let f(q) = ∞ qna be a regular function on B. We define the
n
nP=0
regular conjugate of f as
∞
fc(q)= qna¯ ,
n
nX=0
and the symmetrization of f as
n
∞
fs(q)=f fc(q)=fc f(q)= qn a a¯ .
k n k
∗ ∗ nX=0 (cid:18)Xk=0 − (cid:19)
Both fc and fs are regular functions on B.
Wearenowabletodefinetheinverseelementofaregularfunctionf withrespect
to the -product. Let fs denote the zero set of the symmetrization fs of f.
∗ Z
Definition 2.8. Letf be a regularfunction onB. If f does not vanishidentically,
its regular reciprocal is the function defined by
f (q):=fs(q) 1fc(q)
−∗ −
and it is regular on B fs.
\Z
The following result shows that the regular quotient is nicely related to the
pointwise quotient (see [29, 30]).
Proposition 2.9. Let f and g be regular on B. Then for all q B fs,
∈ \Z
f g(q)=f(T (q)) 1g(T (q)),
−∗ f − f
∗
where Tf : B fs B fs is defined by Tf(q) = fc(q)−1qfc(q). Furthermore,
\Z → \Z
Tf and Tfc are mutual inverses so that Tf is a diffeomorphism.
We now recall a useful result for regular functions [12, 14].
Theorem 2.10. Let Ω be a symmetric slice domain and let f : Ω H be a
→
nonconstant regular function. If U is an axially symmetric open subset of Ω, then
f(U) is open. In particular, the image f(Ω) is open.
3. Some lemmas
In this section, we shall give some useful lemmas, which will be used in Sections
4 and 5. We begin with the following simple proposition.
8 Z.XUANDX.WANG
Proposition 3.1. Let Ω H be a bounded domain and f : Ω H a continuous
function such that f(Ω) is⊂open in H. Let a Ω be a point such→that
∈
(3.1) s:=liminf f(q) f(a) >0.
q ∂Ω | − |
→
Then B(f(a),s) f(Ω).
⊂
Proof. Foreachpointwontheboundary∂f(Ω)off(Ω),thereisasequence q
{ n}∞n=1
in Ω such that lim f(q ) = w. Since Ω is compact, we may assume that
n n
→∞
q converges to a point, say q Ω. If q Ω, then, by the continuity of
{ n}∞n=1 ∞ ∈ ∞ ∈
f, w = f(q ) f(Ω), which contradicts with the openness of f(Ω). Therefore,
∞ ∈
q ∂Ω. This together with (3.1) implies that
∞ ∈
w f(a) = lim f(q ) f(a) liminf f(q) f(a) =s>0.
n
| − | n + | − |≥ q ∂Ω | − |
→ ∞ →
In other words, the boundary ∂f(Ω) of the open set f(Ω) lies outside of the ball
B f(a),s . Consequently, f(Ω) must contain the ball B(f(a),s). (cid:3)
(cid:0) (cid:1)
Remark 3.2. From the proof of Proposition 3.1 above, the result holds naturally
underthe conditionsas describedinProposition3.1forgeneralsetting Rn, instead
of quaternions H=R4.
∼
The so-called Apollonius circle also can be generalized trivially to the quater-
nionic setting.
Lemma 3.3 (Apollonius). Let a,b H, and R t = 1. Then the set q H :
∈ ∋ 6 { ∈
q a tq b is the closed ball B(c,r) with center and radius given by
| − |≤ | − |}
a t2b a b
c= − , r = | − |t.
1 t2 1 t2
− −
Let (B,B) be the class of regular function f (B) with values in B. A
typicalSexRample of (B,B) is a regular Mo¨bius tra∈nsSfoRrmation of B onto B given
SR
by
f(q)=(1 qu) (q u)v
−∗
− ∗ −
with u B and v ∂B (cf. [16, Corollary 9.17]).
∈ ∈
Proposition 3.4 (Lindel¨of). Let f (B,B). Then for all q B, the following
∈SR ∈
inequalities hold:
1 q 2 q 1 f(0)2
(3.2) f(q) −| | f(0) | | −| | ;
(cid:12) − 1 q 2 f(0)2 (cid:12)≤ 1 (cid:0) q 2 f(0)2(cid:1)
(cid:12) −| | | | (cid:12) −| | | |
(cid:12) (cid:12)
f(0) q q + f(0)
(3.3) | |−| | f(q) | | | | ;
1 q f(0) ≤| |≤ 1+ q f(0)
−| || | | || |
q 1 f(0)2
(3.4) f(q) f(0) | | −| | .
− ≤ 1(cid:0) q f(0) (cid:1)
(cid:12) (cid:12) −| || |
(cid:12) (cid:12)
Equality holds for one of inequalities in (3.2) (3.4) at some point q B 0
0
if and only if f is a regular M¨obius transformati−on of B onto B. ∈ \{ }
ON TWO BLOCH TYPE THEOREMS FOR QUATERNIONIC SLICE REGULAR FUNCTIONS9
Proof. The Schwarz-Pick theorem (see [4]) gives, for all q B,
∈
(3.5) 1 f(q)f(0) −∗ f(q) f(0) q ,
− ∗ − ≤| |
(cid:12)(cid:0) (cid:1) (cid:0) (cid:1)(cid:12)
which together with P(cid:12) roposition 2.9 implies that (cid:12)
f T (q) f(0)
(cid:12) ◦ 1−ff(0) − (cid:12) q .
(cid:12)1 f T (q)f(0)(cid:12) ≤| |
− ◦ 1 ff(0)
(cid:12) − (cid:12)
According to Lemma 3.3,(cid:12)the preceding inequality(cid:12)is equivalent to
1 q 2 q 1 f(0)2
f T (q) −| | f(0) | | −| | .
(cid:12) ◦ 1−ff(0) − 1 q 2 f(0)2 (cid:12)≤ 1(cid:0) q 2 f(0)2(cid:1)
(cid:12) −| | | | (cid:12) −| | | |
Since f(B) (cid:12)B, it follows from Proposition 2.9 th(cid:12)at T is a homeomorphism
⊂ 1 ff(0)
−
with inverse T . Replacing q by T (q) in the preceding inequality
1 f(0) fc 1 f(0) fc
gives that − ∗ − ∗
1 q 2 q 1 f(0)2
f(q) −| | f(0) | | −| | , q B.
(cid:12) − 1 q 2 f(0)2 (cid:12)≤ 1 (cid:0) q 2 f(0)2(cid:1) ∀ ∈
(cid:12) −| | | | (cid:12) −| | | |
If equalit(cid:12)y achieves at some point 0(cid:12)=q B in the last inequality, then it also
0
6 ∈
achieves at point 0 = q = T (q ) in inequality (3.5). It thus follows from
0 1 f(0) fc 0
the Schwarz-Pickth6eorem tha−t f is∗a regular Mo¨bius transformations of B onto B.
Inequalities in(3.3)eand(3.4)followfrom(3.2)aswellasthe triangleinequality.
The proof is complete. (cid:3)
For each p = x+yI H R with x,y R and I S, we denote by S the
p
∈ \ ∈ ∈
2-dimensional sphere given by
S := x+yJ :J S .
p
∈
(cid:8) (cid:9)
The next result is a crucial step towards Theorem 1.4.
Lemma 3.5 (Landau). Let f (B,B) with f(0)=0 and f (0)=α>0. Then
′
∈SR
(a) f is injective on B(0,r ), where
0
α
r = ;
0
1+√1 α2
−
(b) for each positive number r r , f 0,B(r) contains the ball B 0,R(r) ,
0
≤
where (cid:0) (cid:1) (cid:0) (cid:1)
α r
R(r)=r − rr .
0
1 αr ≥
−
Proof. Clearly, α (0,1]. If α = 1, then f(q) = q for all q B and the results
∈ ∈
holdtrivially. Now we assumethat α (0,1). To prove(a), it suffices to showthat
∈
if there exist two distinct points mapped by f to one common point, then one of
these two points must lie outside of the open ball B(r ). We proceed as follows.
0
Suppose that there exists one point q B such that ρ := q >0 and f(q )=0.
0 0 0 0
∈ | |
From the maximum principle (see cf. [16, Theorem 7.1]) and α (0,1), it follows
thatthe function g(q):=q 1f(q)is a regularself-mapping ofB. ∈Applying the first
−
inequality in (3.3) to g shows that the following inequality
α q
(3.6) f(q) q −| |
| |≥| |1 αq
− | |
10 Z.XUANDX.WANG
holds for every q B. Now it follows from inequality (3.6) with q =q that
0
∈
(3.7) ρ α>r .
0 0
≥
Now suppose that there exist two distinct points q ,q B such that 0< q
1 2 1
∈ | |≤
q =:ρ<1 and f(q )=f(q )=:w =0. We claim that
2 1 2 0
| | 6
(3.8) w ρ2.
0
| |≤
Set
ψ(q)= f(q) w0) 1 f(q)w0 −∗.
− ∗ −
We must consider the followin(cid:0)g two cases. T(cid:0)he first case(cid:1)is that q ,q lie in a same
1 2
2-dimensionalsphere, i.e. S =S . Inthis case,the sphere S =S is contained
q1 q2 q1 q2
in the zero set of ψ, and hence the function
1
h(q):= (1 qq ) (q q ) s − ψ(q)
(cid:16)(cid:0) − 1 −∗∗ − 1 (cid:1) (cid:17)
is regular on B and is bounded in modulus by one, in virtue of the maximum
principle (see cf. [16, Theorem 7.1]). In particular, h(0) = w /q 2 1, and
0 1
| | | | | | ≤
thus
w q 2 =ρ2
0 1
| |≤| |
as claimed.
The second case is that q ,q lie in two different 2-dimensional spheres, i.e.
1 2
S S = . Set
q1 ∩ q2 ∅
ϕ(q)=(q q ) (1 qq ) , and φ(q)=(q q ) (1 qq ) ,
− 1 ∗ − 1 −∗ − 2 ∗ − 2 −∗
where
e e
q2 =Tϕc(q2).
Then ϕ φ has precisely two zeros q and q . Reasoning as before, (ϕ φ) ψ is
also a re∗gular function on B with vea1lues in2B. In particular, ∗ −∗∗
w /q q = (ϕ φ) ψ(0) 1,
0 1 2 −∗
| | | || | ∗ ∗ ≤
(cid:12) (cid:12)
implying that e (cid:12) (cid:12)
w q q = q q ρ2
0 1 2 1 2
| |≤| || | | || |≤
as desired.
e
Now applying inequality (3.6) with q = q once more, together with inequality
2
(3.8) implies that
α ρ
− ρ,
1 αρ ≤
−
which is equivalent to
α
(3.9) ρ r = .
0
≥ 1+√1 α2
−
Now from inequalities (3.7) and (3.9) we conclude that f is injective. This
|B(0,r0)
completes the proof of (a).
From(a) andthe open mapping theorem(see [16, Theorem7.7]), it followsthat
f :B(0,r ) f B(0,r )
|B(0,r0) 0 → 0
(cid:0) (cid:1)
is a homeomorphism. Thus for an arbitrary fixed r (0,r ], we easily deduce
0
∈
from inequality (3.6) that the boundary ∂f B(0,r) of the open set f B(0,r)
(containing the origin 0) lies outside of the b(cid:0)all B R(cid:1)(0,r) and (b) imm(cid:0)ediately(cid:1)
follows. (cid:0) (cid:1) (cid:3)