THE SET OF INFINITE VALENCE VALUES OF AN ANALYTIC FUNCTION Julian Gevirtz 2005 North Winthrop Road Muncie, Indiana USA
[email protected] ABSTRACT It is shown (Theorem A and its corollary) that if 1 is any nonconstant nonunivalent analytic function on a half-plane ‡ and if H is either a half-plane or a smoothly bounded Jordan domain, then there is a function 0 on H for which 0 w ÐHÑ § 1w Ð‡Ñ such that for any neighborhood Y of any point of 0 Ð`HÑ the set of values A − Y which 0 assumes infinitely many times in H has Hausdorff dimension ". From this it follows (Theorem C) that in the Becker univalence criteria for the disc and upper" " half plane (l0 ww ÐDÑÎ0 w ÐDÑl Ÿ "lDl and l0 ww ÐDÑÎ0 w ÐDÑl Ÿ #eÖD× , respectively) if the " in # the numerator is replaced by any larger number, then there are functions 0 satisfying the resulting bounds the set of whose infinitely assumed values has this same dimension " property. 1. Introduction. In what follows we study the set of values of infinite valence of an analytic function 0 on a domain H, that is, the set \ Ð0 Ñ of points A − 0 ÐHÑ for which 0 " ÐAÑ is infinite. We will limit the discussion to H − ·, the class consisting of the upper halfplane ‡ together with all smoothly bounded Jordan domains. The word "smooth" is used here somewhat vaguely, G _ being both clearly enough and clearly far too much. It is immediate that if H is a bounded domain and 0 has a continuous extension to H, then \ Ð0 Ñ § 0 Ð`HÑ since in this case, if 0 " ÐAÑ is infinite then 0 " ÐAÑ has an accumulation point A! − `H. Let VÐ7ß Q Ñ œ ÖD À 7 lDl Q ×, ?Ð+ß − Ò"ß R #Ó there are at most two integers between 1 and R " whose distance from 5 is >, so that there are at most two siblings s of k for which the distance between -ÐkÑ and -ÐsÑ is 6" > #R , as follows from (8) of the - (l). For such s the minimum distance between points of 6"
6"
?Ð-ÐkÑß !# # 6" Ñ and N ÐsÑ is ( R> !)# 6" (> %) #R #> #R , since % ". From this and the fact that -" ÐN ÐsÑÑ œ !# 6" it follows that for D − ?Ð-ÐkÑß !# 6" Ñ
7
.> "' .> ! 6" (#R ) lP w Ð]s ß D Ñl œ 1" l'N ÐsÑ ÐD>Ñ œ # l Ÿ 1 N ÐsÑ lÐD>Ñ# l Ÿ 1 # (>#6" )# #
%! R # 1 ># #6" ,
for D − ?Ð-ÐkÑß !# # 6" Ñ. Thus, l ! P w (]s ß DÑl Ÿ s−f ÐkÑ
_
as claimed since ! >"# œ >œ"
1# '
# %! R # 1 #6"
R#
!
>œ"
" >#
_
! "# >
%! R # #3"
)! R # #3" ,
>œ"
# .è
Lemma 3. Let PÐkÑ œ 6 and T ÐkÑ be as in (6). Then, lP w (\TÐkÑ ß DÑl
% , !"" #6"
D − ?Ð-ÐkÑß !"% " # 6" Ñ.
Proof. Clearly, from (2) we have (10)
" " Pw ÐMß D Ñ œ 13 Ö D" D" D""" D" " × œ "
so that, since ""
" "!! ,
# #3 Ð"" "ÑÐD "" Ñ 1 Ð"D # ÑÐ""# D # Ñ ,
we have for lDl Ÿ "# "" that
lPw ÐMß D Ñl Ÿ 1# Ð
Ð""" ÑÐ"" "% ""# Ñ Ñ Ð" "% ""# ÑÐ $% ""# Ñ
" "% "" ) $1 Ð" "% ""# Ñ""
Ÿ
" "" .
- Ða3 Ñ Since by (3) of §3.1, PÐM Ða3 Ñß DÑ œ PÐ !# # 6" M - Ða3 Ñß DÑ œ PÐM ß D ! 6" Ñ, it follows that # #
lPw ÐM Ða3 Ñß D Ñl Ÿ
2 !"" #3" ,
D − ?Ð-Ða3 Ñß !"% " # 3" Ñ, " Ÿ 3 Ÿ 6.
From this together with nesting of the disks ?Ð-Ða3 Ñß !"% " # 3" Ñ established in Lemma 1, and the fact that # "# we conclude that 6
6
lP w (\TÐkÑ ß DÑl œ l ! P w (MÐa3 Ñß DÑl Ÿ ! !""2#3" 3œ"
3œ"
_ 2 !#3 6" !"" # 3œ!
% !"" #6" ,
for D − ?Ð-ÐkÑß !"% " # 6" Ñ as claimed.è Lemma 4. Let PÐkÑ œ 6. For each node m − VÐkÑ œ ^6 ÏÖk×, let ]m be a measurable subset of N ÐmÑ, then l ! PÐ]m ß DÑl %%, D − ‡ ?Ð-ÐkÑß !# # 6" Ñ. m−V ÐkÑ
8
Proof. As in (6) let TÐkÑ œ Öa6 œ k, a6" ß á ß a" ×. For " Ÿ 3 Ÿ 6 ", _3 be the set of descendants of the siblings of a3 of length 6 (that is, _3 œ WÐa3 Ñ ^6 ) and let _6 œ f ÐkÑ, the set of siblings of k. Clearly, 6
VÐkÑ œ _3 , 3œ"
so that 6
l ! PÐ]m ß DÑl œ l! ! PÐ]m ß DÑl. 3œ" m−_3
m−V ÐkÑ
We estimate each of the inner sums 53 œ ! PÐ]m ß DÑ. For each of the R # siblings m−_3
b" ß á ß bR# of a3 there are ÐR "Ñ63 R 63 summands in 53 . Let the last element of the sequence a3 be 5 . As pointed out in the proof of Lemma 2, for each integer > − Ò"ß R #Ó there are at most 2 integers 4 between " and R ", at distance > from 5 , 3" so that for such 4 the distance from -Ðb4 Ñ and -Ða3 Ñ is > #R since for nodes of length 3 the 3" distances between the - 's corresponding to siblings are multiples of #R . For any descendant m − VÐkÑ of such a b4 the minimum distance between points of 3" 3" ?Ð-Ða3 Ñß !# # 3" Ñ and N ÐmÑ is (> !) #R ( R> %)# 3" #> #R , since % #" . Grouping the (R "Ñ63 summands 53 according as they are descendants of b" ß á ß bR# and taking into account that the linear measure of ]m for each of these descendants m is at most !# 6" R#
l ! PÐ]m ß DÑl Ÿ 1# R 63 !# 6" !
>œ"
m−_3
R#
œ 14 !R 6"3 # 63 !
>œ"
" >
#R >#3"
#!R 63 # 63 R logR œ #!ÐR # Ñ63 R logR ,
for D − ‡ ?Ð-Ða3 Ñß !# # 3" Ñ. But !R logR %, and R # œ "# R !#" 6
l ! PÐ]m ß DÑl #% D m−V ÐkÑ
3œ" #
"
63
" #
and therefore,
%% ,
as asserted.è Lemma 5. Let PÐkÑ œ 6 and let ] § -ÐkÑ !% # 6" F have measure at most # 6 . Then lPÐ] ß DÑl % for D − -ÐkÑ !# # 6" Ї Ð?Ï?Ð"" ÑÑ. Proof. For convenience we abbreviate !# # 6" by !w and -ÐkÑ by - . Let [ œ ]!w . From D! 6" D(3) in §3.1, PÐ] ß DÑ œ PÐ[ ß !w Ñ. But D − -ÐkÑ % # F is equivalent to !w − #" F , so that [ § "# F . Also, -" Ð[ Ñ œ !"w -" Ð] Ñ # 6 Î !# # 6" œ !# # œ #" , so that from the
9
definition of #" , lPÐ[ ß D!w Ñl % ! 6" D − -ÐkÑ # # Ї Ð?Ï?Ð"" ÑÑ.è
for
D!w
− ‡ Ð?Ï?Ð"" Ñ,
i.e.,
for
For convenience we define ^w ÐkÑ œ Öl À PÐlÑ Ÿ PÐkÑ×. We will now show by a simple induction on PÐkÑ that we can choose the points BÐkÑ, k − ^ in such a way that when the -ÐkÑ, N ÐkÑ, MÐkÑ, FÐkÑ aredefined by (7) and (8), then we have (11) (12) (13)
PÐ\^w ÐkÑ ß BÐkÑÑ œ !ß BÐk) − %%!# F - ÐkÑ, i.e., lBÐkÑ -ÐkÑl %%!"" # PÐkÑ" lPÐ\^w ÐkÑÏÖk× ß DÑl (%, D − ‡ ?Ð-Ðk Ñß !# # PÐkÑ" Ñ. PÐkÑ"
To begin the induction let 6 œ ", so that k œ Ð4Ñ, for some 4 − Ò"ß R "Ó. It follows from (7) and Lemma 4 with 6 œ " and ]m œ MÐm) § N ÐmÑ, for each m − ^w ÐkÑÏÖk×, that (14)
PÐ\^w ÐkÑÏÖk× ß DÑ %%, D − ‡ ?Ð-ÐkÑß !# Ñ.
Now, as indicated in the comments immediately following the definition (1) of PÐ\ß DÑ, PÐ\^" ß DÑ is pure imaginary on each of the gaps FÐkÑ œ !# F -ÐkÑ œ Ð R4 !# "" ß R4 !# "" Ñ. It follows from the formula (10) for Pw ÐMß DÑ in the proof of Lemma 3 that (15)
eÖPw ÐMß D Ñ×
" #"" ,
D − F.
kÑ " w Since P ÐMÐkÑß D Ñ œ P Ð !# M - ÐkÑß DÑ œ PÐMß D-Ð !Î# Ñ, eÖP ÐMÐkÑß D Ñ× !"" on FÐkÑ. Since PÐMÐkÑß -ÐkÑÑ œ !, it follows from (14) and the intermediate value theorem that there indeed exists an BÐkÑ within %%!"" of -ÐkÑ for which (11) holds, so that condition (12) is satisfied. Finally, (13) follows immediately from (14). Now assume that 6 " and that we have defined BÐkÑ for all k of length at most 6 in such a way that (11)-(13) hold. Let PÐkÑ œ 6, let 4 be an integer in Ò"ß R "Ó and let kw denote the sequence of length 6 " consisting of k followed by 4. The objects - (kw ), MÐkw Ñ, N (kw ) and F (kw ) are as defined in (8). Let T ÐkÑ œ Öa6 œ k, a6" ß á ß a" ×, the sequence of k together with its ancestors, as in (6). It is clear that ?Ð-Ða3 Ñß !# # 3" Ñ ¨ ?Ð-Ða3" Ñß !# # 3 Ñ, " Ÿ 3 Ÿ 6 " so that from Lemma 2 it follows that 6
" lPw Ð\^w ÐkÑÏTÐkÑ ß DÑ| )!R # ! #3" , D − ?Ð-ÐkÑß !# # 6" Ñ, 3œ"
so that,
10
6
" lP Ð\^w ÐkÑÏTÐkÑ ß D" Ñ P Ð\^w ÐkÑÏTÐkÑ ß D# Ñ| )!R # lD" D# l! #3" ,
(16)
3œ"
for D" ß D# − ‡ ?Ð-ÐkÑß !# # 6" Ñ. But lBÐkÑ -ÐkÑl %%!"" # 6" , so that ?ÐBÐkÑß # 6 Ñ § ?Ð-ÐkÑß %%!"" # 6" # 6 Ñ § ?Ð-ÐkÑß !"% " # 6" Ñ,
(17)
since %%!"" # 6" # 6 œ Ð%%!"" # Ñ# 6" œ Ð%%!"" # Ñ# 6" œ Ð%%!"" œ !Ð%%"" %"# " Ñ# 6" œ *# %!"" # 6" !"% " # 6" . Thus, (16) is !" ! D" ß D# − ‡ ?ÐBÐkÑß # 6 Ñ, since % " # . By Lemma 3 we have lP w (\TÐkÑ ß DÑl so that (18)
% !"" #6" ,
" #
for
%lD" D# l !"" #6" ,
In light of (17), (18) is valid for D" , and (18) it therefore follows that for
lP Ð\^w ÐkÑ ß DÑ P Ð\^w ÐkÑ ß BÐkÑÑ| But since # œ "# #" !
valid
D − ?Ð-ÐkÑß !"% " # 6" Ñ,
lP Ð\TÐkÑ ß D" Ñ P Ð\TÐkÑ ß D# Ñ|
for D" , D# − ‡ ?Ð-ÐkÑ, !"% " # 6" Ñ. D# − ‡ ?ÐBÐkÑß # 6 Ñ. From (16) 6 D − ‡ ?ÐBÐkÑß # Ñ
!#" 6" # Ñ#
%# !""
6
" )!R # # 6 ! #3" . 3œ"
and # %"" , 6
_
" )!R # # 6 ! #3" )!R # # ! # 3 "'!R # # œ )Ð! R Ñ# #" 3œ"
% )Š Ð"log RÑ ‹
œ Also
%# !""
#
3œ!
#
% #" )Š Ð"log RÑ ‹ %"" % .
œ # "#"" #%. Thus by (11), lPÐ\^w ÐkÑ ß DÑl $%, D − ‡ ?ÐBÐkÑß # 6 Ñ.
(19)
All of the -ÐlÑ, N ÐlÑ, MÐlÑ with PÐlÑ œ 6 " are defined by (8) so that we can address (13) before defining BÐlÑ and verifying (11) and (12). By Lemma 4 (20)
l ! PÐMÐlÑß DÑl %%, D − ‡ ?Ð-Ðkw Ñß !# # 6 Ñ, l−V Ðk w Ñ 6
so that, since ?Ð-Ðkw Ñß !## Ñ § ?ÐBÐkÑß # 6 Ñ, from (19) and (20) we conclude that (21)
l
!
l−^w Ðk w ÑÏÖk w ×
PÐMÐlÑß DÑl (%, D − ‡ ?Ð-Ðkw Ñß !# # 6 Ñ,
11
which establishes (13) for 6 ". We now show that BÐlÑ can be defined and in a way that makes (11) and (12) valid. This, however, follows exactly as in the case 6 œ ". Since kÑ " w w w P ÐMÐkw Ñß D Ñ œ P Ð !# # 6 M - ÐkÑß DÑ œ PÐMß D-Ð !#6 Î#Ñ Ñ, eÖP ÐMÐk Ñß D Ñ× !#6 "" on FÐk Ñ by (15), it follows from (21) and the intermediate value theorem that there is an BÐkw Ñ within (%!"" # 6 of -Ðkw Ñ for which PÐ\^w Ðkw Ñ , BÐkw Ñ) œ !, which establishes (11) and (12). Hence by induction we have defined ÖBÐkÑ À k − ^× in such a way that (11), (12) and (13) hold for all k − ^. It is clear from Lemma 4 that if a ÐkÑ œ Öl À PÐlÑ PÐkÑ, l  WÐkÑ×, then for all k − ^ (22)
lPÐ\a ÐkÑ ß DÑl %%, D − ‡ ?Ð-ÐkÑß !# # 6" Ñ,
and follows from Lemma 5 that lP(\WÐkÑ ß DÑl %, D − -ÐkÑ ‡ !# # 6" Ð?Ï?Ð"" ÑÑ Putting this together with (13) and (22) we conclude that for (23)
lPÐ\^ÏÖk× ß DÑl "#%, D − -ÐkÑ !# # 6" Ї Ð?Ï?Ð"" ÑÑÑ, k − ^. We define the fundamental sets \ ÐRÑ œ \^ œ MÐkÑ I
ÐRÑ
_
k −^
œ Ð N ÐlÑÑ. 8œ" PÐlÑœ8
3.3. Dimension of I ÐR Ñ . The set I œ I ÐRÑ is very similar to the set described §3.1 except that we are using the BÐkÑ instead of the - ÐkÑ, so that things are slightly "off center," but this does not affect the calculation. Let ÖY3 × be a covering of I . We can assume that all these Y3 are intervals since we are only interested in !lY3 l= , where here lY l denotes the diameter of 3
Y . Furthermore, since we can enlarge each Y3 in such a way that !lY3 l= is increased by 3
an arbitrarily small amount, we can assume that all of the Y3 are open and therefore, since I is compact, we can also assume that ÖY3 × is a finite covering. In light of the definition of =-dimensional Hausdorff measure we may also assume that lY3 l R" !. For each 3 let 8Ð3Ñ " be the integer 8 for which
12
# 8Ð3Ñ Ð R" !Ñ Ÿ lY3 l # 8Ð3Ñ" Ð R" !Ñ.
(24)
The minimum distance between the intervals which make up I6 œ
N ÐlÑ is
PÐlÑœ6
# 6" Ð R" !Ñ, so that Y3 can intersect at most one of the N ÐlÑ that make up I8Ð3Ñ , and therefore, more generally, for 7 !, Y3 can intersect at most ÐR "Ñ7 of the closed intervals which make up I8Ð3Ñ7 since I8Ð3Ñ7 § I8Ð3Ñ . In other words, if 4 8Ð3Ñ, then Y3 intersects at most ÐR "Ñ48Ð3Ñ of the intervals in I4 . Now fix 4 so large that # 4" Ð R" !Ñ Ÿ lY3 l for all 3. Since ÖY3 × is a covering of I , for each interval N ÐlÑ making up I4 , some Y3 must touch N ÐlÑ. Thus, ÐR "Ñ4 œ #of intervals in I4 Ÿ !ÐR "Ñ48Ð3Ñ .
(25)
3
We express R " as Ð #" Ñ= , where = œ
logÐR"Ñ logÐ"Î#Ñ
!. From (24) we have that
R = = ÐR "Ñ8Ð3Ñ œ (# 8Ð3Ñ Ñ= Ÿ Ð "R ! Ñ lY3 l ,
so that from (25) it follows that R =! ÐR "Ñ4 Ÿ !ÐR "Ñ48Ð3Ñ œ ÐR "Ñ4 Ð "R lY3 l= , !Ñ 3
3
! = and therefore that !lY3 l= Ð "R R Ñ !.
dimÐIÑ = œ (26)
3 logÐR"Ñ logÐ"Î#Ñ .
Since
" #
œ
#RÐ"log RÑ , %#1
Thus L Ð=Ñ ÐIÑ !, which means that we have that logÐR "Ñ
dimÐI ÐRÑ Ñ .ÐR Ñ œ log R logÐ"log R ÑlogÐ#Î%# Ñ , "
which tends to " as R p_. 3.4. A Normalization. The following lemma allows us to introduce some simplifying conditions. Lemma 6. Let 1! be nonunivalent in ‡ with nonvanishing derivative there. Then there is D a function 3! which maps ‡ into itself such that 1ÐDÑ œ '! 1w! Ð3Ð' ÑÑ . ' satisfies (Gi) 1w is analytic and nonvanishing on ‡, (Gii) 1w ÐDÑ is analytic at _ and 1w Ð_Ñ œ ,! Á !, (Giii) 1ÐD! Ñ œ 1Ð!Ñ œ ! where D! − ‡ ` ?Ð "# Ñ. Proof. For < ", let ;< map ‡ one-to-one onto ?Ð
" w #l,w +w l Ð0 Ð#l,
+w lD +w Ñ 0 Ð+w ÑÑ,
w
w w so that 1Ð!Ñ œ 1Ð #l,, w+ +w l Ñ œ !. Also, 1 ÐDÑ is clearly of the form 1! Ð3! ÐDÑÑ with 3 ! Ð ‡ Ñ § ‡ .è
Henceforth we shall work with a function 1 which has the properties given in the conclusion of this lemma. 3.5. KÐ\ ÐR Ñ ß DÑ is bi-Lipschitz on I ÐR Ñ . We begin with the following
14
Lemma 7. Let ,, (, X !, let R $ be an integer and let \ § Ò!ß ,Ó satisfy , , -" Ð\ [ Ñ Ÿ RÐ"(log RÑ for all intervals [ of length R . Then lPÐ\ß B
X ,( R 3Ñl
Ÿ 1" Ð X$ #(Ñ, ! Ÿ B Ÿ ,.
Proof. Let B − Ò!ß ,Ó. For l5l Ÿ R , let N5 be the interval B R, Ò5 "# ß 5 "# Ó. The " union of these intervals of length R, covers Ò!ß ,Ó. Let D œ B XR,( 3. Since lD>l Ÿ XR,( , for > − ‘, it follows from the hypothesis that ! lPÐ\ N5 ß DÑl Ÿ
l5lŸ"
. For l5l #, distÐDß N5 Ñ œ R, Ðl5l "# Ñ Thus, it follows from the hypothesis that ! lPÐ\ N5 ß DÑl Ÿ
#Ÿl5lŸR
(, 1 RÐ"log RÑ
$( , 1 RÐ"log RÑ
,Ðl5l"Ñ , R
!
#Ÿl5lŸR
†
R X ,(
$ 1X .
" lD>l
so that for > − N5 ,
R ,Ðl5l"Ñ
œ
#( 1 Ð" log RÑ
R"
!
5œ"
" 5
Ÿ
R ,Ðl5l"Ñ .
#( 1 .
The desired conclusion follows immediately from these two bounds.è We next use Lemma 7 to establish that the mapping KÐ\ ÐRÑ ß DÑ is bi-Lipschitz on I and, as will be significant when in §3.7 it comes to constructing an 0 for which \ Ð0 Ñ has dimension " everywhere on 0 Ð` ‡Ñ, that the same upper and lower bounds will hold even if we make (small) alterations in \ ÐRÑ . Let Q be an upper bound for l1w ÐDÑ| on ‡. From properties of L!" and 1 it follows that there is a $ ! such that l, l #% l1w ÐL!" ÐAÑÑ ,! l %! for lAl $ . Let X œ 1$ and let ( be so small that ÐRÑ
" $ 1ÐX
#(Ñ $% ,
l,! l X ( minÖ"ß %Q × and (
l,! l )Q ,
that is, 1$ l,! l l,! l ( minÖ 1$ #% ß *'Q ß )Q ×
(27)
Note that $ , ( and X depend solely on 1. If ] § ‘ is a measurable set for which (28)
lPÐ] ÏÒ!ß ,Óß DÑl $% , for D − ‡ ?Ð #, ß #, Ñ
and the set \ œ ] Ò!ß ,Ó satisfies the hypothesis of the preceding lemma, then lPÐ] ß DÑl $# , for D − J œ Ò R, ß ÐR"Ñ, Ó R so that with KÐDÑ œ KÐ] ß DÑ
15
X ,( R 3,
lKÐÐ4 X (3Ñ R, Ñ KÐÐ5 X (3Ñ R, Ñl $% l,! ll4 5l R, . (Note that we have used the fact that J § ‡ ?Ð #, ß #, Ñ, which follows by a trivial calculation since we have stipulated that X ( ".) But then lKÐ4 R, Ñ KÐ5 R, Ñl $% l,! ll4 5l R, #Q XR,( œ Ð %$ l,! ll4 5l #Q X (Ñ R, l, l Ð $% l,! ll4 5l #! Ñ R, "% l,! ll4 5l R, . Since (
l,! l )Q
we have (with N œ Ð "ß "Ñ, as defined in §3.2) that
distÐKÐ4 R,
(, (, (, , " , #R N Ñß KÐ5 R #R N ÑÑ % l,! ll4 5l R #Q #R (, (, ,)! distÐ4 R, #R N ß 5 R, #R N Ñ.
,! ) l4
5l R,
If we let , œ "" and % Ÿ #(1 (so that "#% $% which allows us to apply (23) and therefore have (28) hold with , œ "" ) we immediately conclude that KÐ\ ÐRÑ ß DÑ is bi-Lipschitz on I ÐRÑ , that is, that for R $ and with the indicated values of % and ,, (29)
,! ) lB"
B# l Ÿ lKÐ\ ÐRÑ ß B" Ñ KÐ\ ÐRÑ ß B# Ñl Ÿ Q lB" B# l, B" , B# − I ÐRÑ .
It therefore follows from the preservation of Hausdorff dimension under bi-Lipschitz mappings and the lower bound for dimÐI ÐRÑ Ñ obtained in §3.3 that logÐR "Ñ
(30) dimÐKÐ\ ÐRÑ ß I ÐRÑ ÑÑ œ dimÐI ÐRÑ Ñ .ÐR Ñ œ log R logÐ"log R ÑlogÐ#Î%# Ñ . " We will make use of the following observation in establishing that for the functions of the conclusion of Theorem A the set \ Ð0 Ñ has dimension 1 locally in 0 Ð`HÑ. Observation 1. It is clear form the foregoing considerations that for any fixed R $ every open interval Y of ‘ has a subinterval Y w such that (29), with the constant ,)! replaced by any smaller number, and consequently (30), also hold for any ] for which \ ÐRÑ ?] § Y w . 3.6. \ ÐKÐR Ñ Ñ ¨ KÐ\ ÐR Ñ ß I ÐR Ñ Ñ. Before beginning we note that the following weak version of Theorem A follows immediately from this inclusion together with (30): For any 5 " there is a mapping D 0 ÐDÑ œ '! 1w! Ð3Ð' ÑÑ . ' for which dimÐ\ Ð0 ÑÑ 5. For ! - " let HÐ-Ñ œ Ї -3Ñ ?, for ! Ÿ " " let MÐ" Ñ œ N ÏÒ " ß " Ó, L" ÐDÑ œ PÐMÐ" Ñß DÑ and 1" ÐDÑ œ KÐL" ß DÑ, so that in particular 1! œ 1. In addition, we denote by h" the class of functions ? on ‡ for which ? L" maps ‡ into ’, and we denote by m?mE the sup-norm of ? on the set E § ‡. By the normalization conditions (Gi) and (Gii) of Lemma 6 there are positive constants Q " and 7 for which
16
1w Ð‡Ñ § VÐ7ß Q Ñ, so that Kw ÐL" ?ß DÑ satisfies the same inclusion for all ? − h" . Clearly, there exist "# , 0 − Ð!ß "Ñ which depend solely on 1 such that for all " − Ò!ß "# Ó we have 1" ÐD" Ñ œ 1" Ð!Ñ œ ! for some D" − ‡ with ?ÐD" ß 0Ñ § HÐ0Ñ. It is a simply verified general fact that there exist positive constants E and F which depend only on 7 and Q such that if 0 w Ð?Ñ § VÐ7ß Q Ñ then 0 is univalent on ?ÐEÑ and 0 Ð?ÐEÑÑ ¨ ?Ð0 Ð!Ñß FÑ. Thus, there are positive constants 3 #0 and
