Table Of ContentLandau’s theorem for slice regular
7
functions on the quaternionic unit ball
1
0
2
Cinzia Bisi
n
a Universit`adegli StudidiFerrara
J
Dipartimento di Matematica e Informatica
7
Via Machiavelli 35, I-44121 Ferrara, Italy
2
[email protected]
]
V Caterina Stoppato∗
C Universit`adegli StudidiFirenze
h. Dipartimento diMatematica e Informatica “U.Dini”
t Viale Morgagni 67/A, I-50134 Firenze, Italy
a
m [email protected]fi.it
[
1
Abstract
v
Along with the development of the theory of slice regular functions
2
1 overtherealalgebraofquaternionsHduringthelast decade,somenatu-
1 ralquestionsaroseaboutsliceregularfunctionsontheopenunitballBin
8 H. This work establishes several new results in this context. Along with
0 some useful estimates for slice regular self-maps of B fixing the origin, it
. establishestwovariantsofthequaternionicSchwarz–Picklemma,special-
1
0 ized to maps B → B that are not injective. These results allow a full
7 generalization toquaternionsoftwotheoremsprovenbyLandauforholo-
1 morphicself-mapsf ofthecomplexunitdiskwith f(0)=0. Landauhad
: computed, in terms of a := |f′(0)|, a radius ρ such that f is injective at
v leastinthedisk∆(0,ρ)andsuchthattheinclusionf(∆(0,ρ))⊇∆(0,ρ2)
i
X holds. Theanalogous resultprovenhereforsliceregularfunctionsB→B
r allows a new approach to the study of Bloch–Landau-type properties of
a slice regular functions B→H.
1 Introduction
The unit ball in the real algebra of quaternions H, namely
B:= q H: q <1 ,
{ ∈ | | }
isthesubjectofintensiveinvestigationwithinthetheoryofsliceregularquater-
nionic functions introduced in [19, 20]. The theory is based on the next defi-
nition, where the notation S := q H q2 = 1 is used for the sphere of
{ ∈ | − }
quaternionic imaginary units.
1
Definition1.1. LetΩbeadomain(anopenconnectedset)inHandletf: Ω
H. For all I S, let us use the notations L := R+IR, Ω := Ω L an→d
I I I
f := f . Th∈e function f is called (Cullen or) slice regular if, for ∩all I S,
thIe rest|rΩiIction f is holomorphic; that is, if, for all I S, f is differenti∈able
I I
and the function ∂ f: Ω H defined by ∈
I I
→
1 ∂ ∂
∂ f(x+yI):= +I f (x+Iy)
I I
2(cid:18)∂x ∂y(cid:19)
vanishes identically. Ifthisisthecase, thenasliceregularfunction∂ f :Ω H
c
→
can be defined by setting
1 ∂ ∂
∂ f(x+yI):= I f (x+yI)
c I
2(cid:18)∂x − ∂y(cid:19)
for I S, x,y R (such that x+Iy Ω). It is called the Cullen derivative of
∈ ∈ ∈
f.
For slice regular functions on the quaternionic unit ball B, the Schwarz
lemmaanditsboundaryversionwereprovenin[20,22]. Sliceregularanalogsof
theMo¨biustransformationsofBhavebeenintroducedandstudiedin[8,23,31],
leading to the generalization of the Schwarz–Pick lemma in [7]. Other results
concerning slice regular functions on B have been published in [2, 3, 11, 12, 14,
15, 17].
Within this rich panorama, the present work establishes the quaternionic
counterparts of the following results due to Landau [27, 28], which we quote in
the form of [26], 2.10.
§
Theorem 1.2 (Landau). For each a (0,1), let Φ denote the set of holomor-
a
∈
phic self-maps f of the complex unit disk ∆(0,1)such that f(0)=0, f (0) =a.
′
| |
Let
ρ:= inf r(f),
f∈Φa
r(f):=sup r (0,1) f is injective in ∆(0,r) .
{ ∈ | }
Then ρ= 1 √1 a2. Furthermore, for f Φ the equality r(f)=ρ holds if, and
− a− ∈ a
only if, there exists η ∂∆(0,1) such that f(z)=F(ηz)η 1 where
−
∈
a z
F(z):=z − .
1 az
−
Theorem 1.3 (Landau). For each a (0,1), let
∈
P := inf R(f),
f∈Φa
R(f):=sup r Ω ∆(0,1) s.t. 0 Ω and f :Ω ∆(0,r) is bijective .
r r r
{ |∃ ⊆ ∈ → }
Then P = ρ2 with ρ = 1 √1 a2. Furthermore, for f Φ it holds R(f) = ρ2
− a− ∈ a
if, and only if, there exists η ∂∆(0,1) such that f(z)=F(ηz)η 1.
−
∈
2
Besides their independent interest, Theorems 1.2 and 1.3 can be used to
prove one of the most celebrated results in complex function theory:
Theorem 1.4 (Bloch–Landau). Let f be a holomorphic function on a region
containing the closure of ∆(0,1) and suppose f(0) = 0,f (0) = 1. Then there
′
is a disk S ∆(0,1) on which f is injective and such that f(S) contains a disk
⊆
of radius b>1/72.
The reference [10] presents in Ch. XII 1 a proof of the Bloch–Landau
§
theorembasedonreducingtoboundedfunctionsandapplyingtothemavariant
of Theorem 1.3. The largest value of b for which Theorem 1.4 holds is known
as Bloch’s constant. As it is well-known, determining this constant is still a
challenging problem nowadays.
While the situation is considerably different in the case of several complex
variables[4,9,13,25],variantsofTheorem1.4holdinthetheoryofsliceregular
functions, see [12], and in other hypercomplex generalizations of the theory of
one complex variable: see [30]for the classof T-holomorphic functions overthe
bicomplexnumbers;and[29]forsquareintegrablemonogenic functionsoverthe
reduced quaternions. As we already mentioned, in the present work we estab-
lish perfect analogs of Theorems 1.2 and 1.3 for slice regular functions. Other
original results are proven along with them and a new version of Theorem 6
of [12] is obtained as an application. The paper is structured as follows.
In Section 2, we recall some preliminary material needed for our study, in-
cluding the algebraic structure of slice regular functions: a ring structure with
the usual addition + anda multiplication , as well as the existence of a multi-
∗
plicativeinversef whenf 0. Thisstructureisthebasisfortheconstruction
−∗
of regular M¨obius transforma6≡tions, namely, (q)u for q B,u ∂B, where
Mq0 0 ∈ ∈
Mq0(q):=(1−qq¯0)−∗∗(q−q0).
We recall some known results on the differential of a slice regular function and
wederiveacharacterizationoffunctionsthatarenotinjective. Thequaternionic
Schwarz–Picklemma is also recalled in detail.
Section 3 establishes two variants of the quaternionic Schwarz–Picklemma,
specializedtoself-mapsofthe quaternionicunitballthatarenotinjective. The
first one is:
Theorem 1.5. Let f : B B be a regular function, let q B and set v :=
0
→ ∈
f(q ). If f is not injective in any neighborhood of f 1(v) then there exists
0 −
p B such that
0
∈
(f(q) v) (1 v¯ f(q)) (q) (q).
| − ∗ − ∗ −∗|≤|Mq0 ∗Mp0 |
A second variant can be derived from the former:
Theorem 1.6. Let f: B B be a slice regular function, which, for some
r (0,1), is injective in B→(0,r) := q H : q < r but is not injective in
∈ { ∈ | | }
3
B(0,r ) for any r > r. Then there exists q ∂B(0,r) such that f is not
′ ′ 0
∈
injective in any neighborhood of f 1(f(q )) and
− 0
(f(q) f(q )) (1 f(q ) f(q)) (q) (q).
| − 0 ∗ − 0 ∗ −∗|≤|Mq0 ∗Mp0 |
for some p B(0,r). In particular, if f(0)=0 then f(q ) r2.
0 0
∈ | |≤
We add,inSection4, someusefulestimatesforsliceregularself-mapsofthe
unit ball fixing the origin:
Theorem 1.7. Let f : B B be a slice regular function with f(0) = 0. If
→
a:= ∂ f(0) belongs to (0,1) then
c
| |
a q q +a
q −| | f(q) q | | . (1)
| |1 aq ≤| |≤| |1+aq
− | | | |
for all q B. Furthermore, if there exists q B such that equality holds on
∈ ∈
the left-hand side or on the right-hand side, then f(q) = q (q) where is a
regular M¨obius transformation of B with (0) =a. M M
|M |
The aforementionedtheorems allow us to achieve, in Section 5, a full gener-
alization of Landau’s results:
Theorem 1.8. Let f : B B be a slice regular function with f(0) = 0. If
→
a := ∂ f(0) belongs to (0,1) and if we set ρ := 1 √1 a2 then the following
| c | − a−
properties hold.
1. The function f is injective at least in the ball B(0,ρ).
2. For all r (0,ρ), B 0,r a r f(B(0,r)) B 0,r a+r . As a conse-
∈ 1−ar ⊆ ⊆ 1+ar
quence, (cid:0) − (cid:1) (cid:0) (cid:1)
B(0,ρ2) f(B(0,ρ)).
⊆
3. The following are equivalent:
(a) B(0,ρ) is the largest ball centered at 0 where f is injective;
(b) there exists a point q ∂B(0,ρ) with f(q ) ∂B(0,ρ2);
0 0
∈ ∈
(c) f(q) = q (q) where is a regular M¨obius transformation of B
M M
(necessarily such that (0)=∂ f(0), whence (0) =a).
c
M |M |
In Section 6, as an application, we obtain a quaternionic Bloch–Landau-
type result in the spirit of [12]. Although finding a full-fledged quaternionic
generalization of Theorem 1.4 is still an open problem, the new approach used
here opens a new path towards such a generalization,which will be the subject
of future research.
4
2 Preliminary material
Letusrecallsomebasicmaterialonsliceregularfunctions,seeChapter1in[18].
We will henceforth use the adjective regular, for short, to refer to slice regular
functions.
Proposition 2.1. The regular functions on a Euclidean ball
B(0,R):= q H q <R
{ ∈ | | | }
warheicehxacocntlvyertgheeisnumBs(0o,Rf )t.hose power series Pn∈Nqnan (with {an}n∈N ⊂ H)
Between two regular functions on B(0,R), say
f(q):= qna , g(q):= qnb ,
n n
nXN nXN
∈ ∈
the regular product is defined as follows:
n
(f g)(q):= qn a b .
k n k
∗ nXN kX=0 −
∈
The regular conjugate of f is defined as
fc(q):= qna¯ .
n
nXN
∈
The next definition presents a larger class of domains that are of interest in
the theory of regular functions.
Definition 2.2. A domain Ω H is called a slicedomain if Ω =Ω L is an
I I
open connected subset of LI ∼=⊆C for all I ∈S, and Ω intersects the re∩al axis. A
slice domain Ω is termed symmetric if it is symmetric with respect to the real
axis, i.e., if for all x,y R,I S the inclusion x+Iy Ω implies x+yS Ω.
I
∈ ∈ ∈ ⊂
The definition of the multiplication (f,g) f g can be extended to the
7→ ∗
case of regular functions on a symmetric slice domain Ω, leading to the next
result. For more details, see 1.4 in [18].
§
Proposition 2.3. Let Ω be a symmetric slice domain. The set of regular func-
tions on Ω is a (noncommutative) ring with respect to + and .
∗
The operation f fc can also be consistently extended to the case of
7→
regular functions on a symmetric slice domain Ω. The additional operation of
symmetrization, defined by the formula
fs :=f fc =fc f
∗ ∗
allows to define the regular reciprocal of f as
f (q):=fs(q) 1fc(q)
−∗ −
5
andto provethe next results,where we use the notation := q Ω h(q)=
h
0 for the zero set of a regular h : Ω H. For details, wZe refe{r t∈he re|ader to
} →
Chapter 5 in [18].
Theorem 2.4. Let f be a regular function on a symmetric slice domain Ω and
suppose that f 0. Then f−∗ is regular in Ω fs, which is a symmetric slice
6≡ \Z
domain, and
f f =f f 1.
−∗ −∗
∗ ∗ ≡
Theorem 2.5. Let f,g be regular functions on a symmetric slice domain Ω.
Then
0 if f(q)=0,
(f g)(q)=
∗ (cid:26) f(q)g(f(q)−1qf(q)) otherwise.
If we set Tf(q):=fc(q)−1qfc(q) for all q Ω fs, then
∈ \Z
(f g)(q)=f(T (q)) 1g(T (q)), (2)
−∗ f − f
∗
for all q Ω fs. For all x,y R with x+yS Ω fs Ω fc, thefunction
T maps∈x+\yZS to itself (in pa∈rticular, T (x)=⊂x f\oZr all⊂x \RZ). Furthermore,
f f
∈
Tf is a diffeomorphism from Ω fs onto itself, with inverse Tfc.
\Z
Let us now recall some material on the zeros of regular quaternionic func-
tions,seeChapter3in[18]. Webeginwitharesultthatisfolkloreinthetheory
of quaternionic polynomials. For all q = x +Iy with x ,y R,I S, we
0 0 0 0 0
∈ ∈
will use the notation
S :=x +y S.
q0 0 0
Theorem 2.6. Let q ,q H and f(q)=(q q ) (q q ). If q S then f
has two zeros in H, n0am1el∈y q and (q q¯ )q−(q0 ∗q¯ )−11 S .1N6∈owqs0uppose,
0 0− 1 1 0− 1 − ∈ q1
instead, that q S . If q = q¯ then f only vanishes at q , while if q = q¯
1 ∈ q0 1 6 0 0 1 0
then the zero set of f is S .
q0
As in the case of a holomorphic complex function, the zeros of a regular
quaternionic function can be factored out. The relation between the factoriza-
tion and the zero set is, however, subtler than in the complex case because of
the previous theorem.
Theorem 2.7. Let f 0 be a regular function on a symmetric slice domain Ω
and let x+yS Ω. T6≡here exist m N = 0,1,2,... and a regular function
g :Ω H, not⊂identically zero in x+∈yS, su{ch that }
→
f(q)=[(q x)2+y2]mg(q)
−
If g has a zero p x+yS, then such a zero is unique and there exist n
1
N,p ,...,p x+y∈S (with p =p¯ for all l 1,...,n 1 ) such that ∈
1 n l l+1
∈ 6 ∈{ − }
g(q)=(q p ) ... (q p ) h(q)
1 n
− ∗ ∗ − ∗
for some regular function h:Ω H that does not have zeros in x+yS.
→
6
This motivates the next definition.
Definition 2.8. In the situation of Theorem 2.7, f is said to have spherical
multiplicity 2m at x+yS and isolated multiplicity n at p . Finally, the total
1
multiplicity of x+yS for f is defined as the sum 2m+n.
Let us now recall a definition originally given in [24] and a few results
from [16], which concern the real differential of a regular function.
Definition 2.9. Let f be a regular function on a symmetric slice domain Ω,
and let q = x +Iy Ω with x ,y R,I S. If y = 0 then the spherical
0 0 0 0 0 0
∈ ∈ ∈ 6
derivative of f at q is defined by the formula
0
∂ f(q ):=(q q¯ ) 1(f(q ) f(q¯ )).
s 0 0 0 − 0 0
− −
Proposition 2.10. Let f be a regular function on a symmetric slice domain
Ω, and let q = x + Iy Ω with x ,y R,I S. If y = 0 then the
0 0 0 0 0 0
∈ ∈ ∈
real differential of f at q acts by right multiplication by the Cullen derivative
0
∂ f(q ) on the entire tangent space T Ω H. If, on the other hand, y = 0
ancd if0 we split such space as H = Lq0 L≃ then the real differential ac0ts6 on
I ⊕ ⊥I
L by right multiplication by ∂ f(q ) and on L by right multiplication by the
I c 0 ⊥I
spherical derivative ∂ f(q ).
s 0
The next result characterizes the singular set N of a regular function f,
f
that is, the set of points where the real differential of f is not invertible.
Proposition 2.11. Let f be a regular function on a symmetric slice domain Ω,
and let q Ω. The real differential of f at q is not invertible if, and only if,
0 0
there exist∈q S and a regular function g :Ω H such that
0 ∈ q0 →
e f(q)=f(q )+(q q ) (q q ) g(q); (3)
0 0 0
− ∗ − ∗
that is, if, and only if, f −f(q0) has total multipelicity n ≥ 2 at Sq0. We can
distinguish the following special cases:
equality (3) holds with q = q¯ if, and only if, the spherical derivative
0 0
•
∂ f(q ) vanishes;
s 0
e
equality (3)holdswithq =q if, andonlyif, theCullenderivative ∂ f(q )
0 0 c 0
•
vanishes.
e
The following theorem asserts that the total multiplicity n of f f(q ) at
0
−
S is constant in a neighborhood of S .
q0 q0
Theorem 2.12. Let Ω be a symmetric slice domain and let f : Ω H be
→
a non-constant regular function. Then its singular set N has empty interior.
f
Moreover, for a fixed q N , let n>1 be the total multiplicity of f f(q ) at
0 f 0
∈ −
S . Then there exist a neighborhood U of q and a neighborhood T of S such
q0 0 q0
that, for all q U, the sum of the total multiplicities of the zeros of f f(q )
1 1
∈ −
in T equals n; in particular, for all q U N the preimage of f(q ) includes
1 f 1
∈ \
at least two distinct points of T.
7
Corollary 2.13. Let Ω be a symmetric slice domain and let f : Ω H be a
→
regular function. If f is injective, then its singular set N is empty.
f
For the purposes of our present work, we add the following remark.
Proposition 2.14. Let Ω be a symmetric slice domain and let f :Ω H be a
→
regular function. For each value v of f, the following are equivalent:
f is not injective in any neighborhood of f 1(v);
−
•
there exist q ,q Ω and a regular g :Ω H such that
0 1
• ∈ →
f(q)=v+(q q ) (q q ) g(g). (4)
0 1
− ∗ − ∗
Proof. Suppose
f(q)=v+(q q ) (q q ) g(g),
0 1
− ∗ − ∗
so that in particular f(q ) = v. If q S then f v vanishes at some
0 1 6∈ q0 −
q S . As a consequence, f(q ) = v = f(q ) and f is not injective on any
1 ∈ q1 1 0
neighborhoodoff 1(v) q ,q . If,ontheotherhand,q S thenq N
bey Proposition 2.1−1. In⊇su{ch0a 1ce}ase, f is not injective in a1n∈y nqe0ighborh0o∈od off
f 1(v) q by Theorem 2.12. e
− 0
∋
Now let us prove the converse implication. If v is the value of f at q then
0
there exists h : Ω H such that f(q) = v +(q q ) h(q). If h does not
0
→ − ∗
admit any zero q Ω then f does not take the value v at any other point
1
∈
of Ω other than q and q N . As a consequence, f 1(v) = q and f is
0 0 f − 0
6∈ { }
a local diffeomorphism near q . A fortiori, f is injective in a neighborhood of
0
f 1(v).
−
Weconcludeouroverviewofpreliminarymaterialwithafewresultsconcern-
ing the unit ball B := q H q < 1 . The work [31] introduced the regular
M¨obius transformation{s, ∈name|ly| |the f}unctions q (q)u with u ∂B,
q B and 7→ Mq0 ∈
0
∈
(q):=(q q ) (1 q¯ q) =(q q ) (1 qq¯) =(1 qq¯ ) (q q ).
Mq0 − 0 ∗ − 0∗ −∗ − 0 ∗ − 0 −∗ − 0 −∗∗ − 0
These transformations are the only bijective self-maps of B that are regular. In
[7], the following result has been proven.
Theorem 2.15 (Schwarz-Pick lemma). Let f : B B be a regular function
and let q B. Then in B: →
0
∈
(f(q) f(q )) (1 f(q ) f(q)) (q). (5)
| − 0 ∗ − 0 ∗ −∗| ≤ |Mq0 |
Moreover,
1
∂ f (1 f(q ) f(q)) (6)
| c ∗ − 0 ∗ −∗||q0 ≤ 1 q0 2
−| |
∂ f(q ) 1
s 0
| | (7)
1 fs(q ) ≤ 1 q 2
0 0
| − | | − |
8
If f is a regular M¨obius transformation of B, then equality holds in the previous
formulas. Else, all the aforementioned inequalities are strict (except for the first
one at q , which reduces to 0 0).
0
≤
The proof was based on the following lemmas, proven in [8] and in [7],
respectively.
Lemma 2.16. If f : B B is a regular function then for all q B, the
0
→ ∈
function f(q):=(f(q) f(q )) (1 f(q ) f(q)) is a regular function from
0 0 −∗
− ∗ − ∗
B to itself with f(q )=0.
e 0
Lemma 2.17. Ief f :B B is a regular function having a zero at q B, then
0
f is a regular fun→ction from B to itself. ∈
M−q0∗∗
The proof of Lemma 2.17 used the next result (see 7.1 in [18]), which will
§
be thoroughly used in the present work.
Theorem 2.18 (Maximum Modulus Principle). Let Ω H be a slice domain
and let f: Ω H be regular. If f has a relative maximu⊆m at p Ω, then f is
→ | | ∈
constant. As a consequence, if Ω is bounded and if, for all q ∂Ω,
0
∈
limsup f(q) M
| |≤
Ω q q0
∋ →
then f M in Ω and the inequality is strict unless f is constant.
| |≤
Finally, the following lemma, proven in [7] as a further tool for the proof of
Theorem 2.15, will also be useful later in this paper.
Lemma 2.19. Let f,g,h : B = B(0,R) H be regular functions. If f g
→ | | ≤ | |
then h f h g . Moreover, if f < g then h f < h g in B ,
h
| ∗ | ≤ | ∗ | | | | | | ∗ | | ∗ | \Z
where we recall that := q B h(q)=0 .
h
Z { ∈ | }
3 A generalized Schwarz-Pick lemma
Our first step towards a quaternionic version of Landau’s results is a special
variant of Theorem 2.15, concerned with self-maps of B that are not injective.
We will start with the next theorem and then achieve the result which we will
apply later in the paper.
Theorem 3.1. Let f : B B be a regular function, let q B and set v :=
0
→ ∈
f(q ). If f is not injective in any neighborhood of f 1(v) then there exists
0 −
p B such that
0
∈
(f(q) v) (1 v¯ f(q)) (q) (q).
| − ∗ − ∗ −∗|≤|Mq0 ∗Mp0 |
Namely, if q ,q B are such that f(q) = v+(q q ) (q q ) g(g) holds
0 1 0 1
for some regular∈g : B H, then the previous −inequa∗lity−holds∗with p :=
0
→
(1 q q ) 1q (1 q q ) S .
− 1 0 − 1 − 1 0 ∈ q1
9
Proof. Iff isnotinjectiveinanyneighborhoodoff 1(v)thenProposition2.14
−
applies. Let q ,q B be such that
0 1
∈
f(q)=v+(q q ) (q q ) g(g)
0 1
− ∗ − ∗
for some regular g :B H. Now,
→
f(q):=(f(q) v) (1 v¯ f(q))
−∗
− ∗ − ∗
e =(q q0) (q q1) g(g) (1 v¯ f(q))−∗
− ∗ − ∗ ∗ − ∗
isaself-mapofBwithazeroatq . Inparticular,byLemma2.17,h:= f
is a regular function from B to it0self. Moreover, M−q0∗∗
e
h(q)=(1 qq¯0) (q q1) g(g) (1 v¯ f(q))−∗
− ∗ − ∗ ∗ − ∗
vanishes at p := (1 q q ) 1q (1 q q ) S by Theorem 2.5. By applying
0 − 1 0 − 1 − 1 0 ∈ q1
again Lemma 2.17 we find that
h= f
M−p0∗∗ M−p0∗∗M−q0∗∗
is again a regular function from B to itself. By Lemmea 2.19 we conclude that
f
| |≤|Mq0 ∗Mp0|
in B, as desired. e
Thepreviousresultwasalreadyknowninthespecialcaseswhentheviolation
ofinjectivity iscausedbythe vanishingoftheCullenorthe sphericalderivative
at q : see Theorems 5.2 and 5.4 in [7]. In such cases, the point p appearing in
0 0
the statement coincides with q or q¯ , respectively.
0 0
WenowexploitTheorem3.1andturnitintoaresultthatwillbeparticularly
useful in the sequel.
Theorem 3.2. Let f: B B be a regular function, which, for some r (0,1),
→ ∈
is injective in B(0,r) but is not injective in B(0,r ) for any r > r. Then
′ ′
there exists q ∂B(0,r) such that f is not injective in any neighborhood of
0
∈
f 1(f(q )) and
− 0
(f(q) f(q )) (1 f(q ) f(q)) (q) (q).
| − 0 ∗ − 0 ∗ −∗|≤|Mq0 ∗Mp0 |
for some p B(0,r). In particular, if f(0)=0 then f(q ) q p r2.
0 0 0 0
∈ | |≤| || |≤
Proof. For each n 1, since f is not injective in B(0,r+1/n), there exist two
≥
distinctpointsp ,q inB(0,r+1/n)wheref takesthesamevaluev . Sincewe
n n n
supposedf tobeinjectiveinB(0,r),onlyoneofthetwopoints,sayp ,maybe
n
included in B(0,r) while q must have r q <r+1/n. Therefore, q r
n n n
≤| | | |→
as n + and, up to refining the sequence, q q for some q ∂B(0,r).
n 0 0
Up to→fur∞ther refinements, p p for some p →B(0,r) and v ∈ v B.
n n
→ ∈ → ∈
Clearly, f(q )=v =f(p). This immediately proves that f is not injective near
0
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