Advances in Pure Mathematics, 2011, 1, 305-314
doi:10.4236/apm.2011.16056 Published Online November 2011 (http://www.SciRP.org/journal/apm)
Copyright © 2011 SciRes. APM
Uniform Convergence of Extremal Polynomials When
Domains Have Corners and Special Cusps on the
Boundary
Cem Koşar1, Mehmet Kucukaslan2, Fahreddin G. Abdullayev2
1Department of Mat hem at i cs, Facult y of S cience and L iterat ur e , Bitlis Eren University, Bitlis, Turkey
2Department of Mat hem at i cs, Facult y of S cience and L iterat ur e , Mersin University, Mersin, Turkey
E-mail: ckosar77@gmail.com, {mkucukaslan, fabdul}@mersin.edu.tr
Received May 26, 2011; revised September 13, 2011; accepted Septemb er 22, 2011
Abstract
We study the approximation properties of the extremal polynomials in Ap-norm and C-norm. We prove esti-
mates for the rate of such convergence of the sequence of the extremal polynomials on domains with corners
and special cusps.
Keywords: Uniform Approximation, Extremal Polynomials, Conformal Mapping, p-Bieberbach
Polynomials
1. Introduction and Main Results
1.1. Statement of Problem and Some Definitions
Let be a finite domain bounded by Jordan curve
and 0
G
G:=LG
be an arbitrary fixed point. Let
be conformal mapping of onto the disk

wz

=G
0
0
0, :=Dr

0=0,
: <wwr

0=1

with normalization
, where 0 is a conformal
radius of G with respect to

r
0
and let 1
:=
be
an inverse mapping.
Let We denote by
>0.p

1
p
A
G the set of
functions

f
z
=0,f analytic in and normalized by
such that
G

0
f

0=1


1
1:=d< ,
p
p
'z
AG
pG
ffz




where d
z
is a two-dimensional Lebesque measure on
.G
We denote by n of all algebraic polynomials
satisfying
n
n

,deg
n
Pz P

0=0,
n
P

0=1.
n
P
Let us consider following extremal problem:

1:
nnn
AG
p
PP

min (1.1)
Using a method similar to the one given in ([1], p.
137), it is seen that there exists an extremal polynomial
n
Pz
furnishing to the problem (1.1), these polyno-
mials
n
Pz
are determined uniquely in case of
([1], p. 142). This extremal problem was first considered
by Kucukaslan M. and Abdullayev F.G. and they were
called the p-Bieberbach polynomial of degree for the
pair
>1p
n
0
,G
in [2], and denoted by

,.
np
Bz
The main goal of this paper is to investigate the
approximation rate of convergence to the
function

,np
Bz
in uniform norm for some domains when
has some certain singularity, i.e.


,,
()
:= max,
np np
CG zG
c
BzBz
n

 (1.2)
where
:=,> 0,pG

the constant is independent
of
c
.n
In case of the solution of (1.1) coincides with
the well known Bieberbach polynomial
=2,p
-nth
z
,2nn
(see, for example, [3,4] ). The appro-
ximation properties in the uniform norm of
Bz B
n
Bz on
G first was observed by Keldysh in 1939 [3] for the
domains with sufficiently smooth boundary. A consi-
derable progress in this area has been achieved by
Mergelyan [5], Suetin [4], Simonenko [6], Andrievskii
[7,8] Gaier [9,10], Abdullayev [11-13], Israfilov [14,15]
and the others.
In this paper, we are going to consider the case
the problem in (1.2). First, we will investigate the
approximation rate of to the function
>1p

,np
Bz
in
C. KOŞAR ET AL.
306
1-
p
A
norm and using well known Simonenko and
Andrievskii method (see, for example, [6,7]), the appro-
ximation rate of to the function

,np
Bz
in uniform
norm will be obtained.
Now we need some definitions:
Definition 1.1. ([16], p. 97)The Jordan arc or a curve
is called a quasiconformal () arc or curve
if there is a quasiconformal mapping
L-K1K
-K
f
of a region
containing such that
D L

f
L is line segment or
circle.
Let

F
L denote the set of all sense-preserving
plane homeomorphisms
f
of regions such
that
DL

f
L is a line segment or circle and let

=inf
 
:

,
K
LKffFL
where

K
f
.
is the maximal dilatation of such a map-
ping
f
Then is L-
K
quasiconformal if and only if
If is a

<.KL L-
K
quasiconformal, then

.
K
LK
D gives the global definition of a -
K
qua-
siconformal arc or curve consequently. This definition is
common in the literature. At the same time, we can
consider the domain as the neighborhood of the
curve In this case, Definition 1.1 will be called local
definition of quasiconformal arc or curve. Through this
work we consider the local definition. The local defi-
nition has an advantage in determining the coefficients of
quasiconformality for some simple arcs or curves.
D

=,zzs
L
.L
Let us denote
0,
s
mesL natural repre-
sentation of
:= .LG
Definition 1.2. ([4]) We say that GC
if G
has
a continuous tangent
 
:=
s
zs

,GC
for every points
.
zs
Corollary 1.1. ([17]) If
then =1K
for
all >0.
Definition 1.3. ([12]) We say that
if is expressed
as a union of a finite number of

;, 0
 
0<<2,
,GC :=L
-CG
arcs, connecting at
the points 0, 1 such that is locally smooth at
0 and in the local coordinate system
,,,
m
zzz L
,z
x
y with origin
at ,
j
z the following conditions are satisfied: 1j,m
,
1) For every
j
z the domain has 1jp ,mG
π,
j
0< <
j2,
exterior angle at the corner ,
j
z

=: .
j
min
2) For every ,
j
z in 1,pj m
,
x
y coor-
dinate system with origin at
j
z we have

11


12 1
=:,0,zxiyycx xG 

cx

,
21
=: ,0zxiyyxxG
C
>0
for some constants
12
<< <cc 
 ,,
i
=1,2.i
It is clear from the definition that each domain
;
GC
may have exterior π
j
angles,
0< <2
j
, at the points
j
z, , and interior
zero angles at which the boundary arcs are touching with
1jp
1-x
speed at the points
j
z, . If
1pj m=0
then the domain does not has interior zero angles
and
G
,0< 2GC
<
, i.e.,

.;0CC

If =1
then the domain has piecewise smooth
boundary with only interior zero angles. We denote the
class of domains by
G
.1;C
1.2. New Results
We introduce the following notation. For any
and
1<< 2p
0< <2
we set

,,,
j
pp
j
 
and
,
jp

as follows:

 
11
0 <,
2
42 12
,:=, <,
222 23
12
1, < 2,
23
p
 

 

,
3
pp
= 1,
=2,
= 3,
j
pj
pj
j



2
2
8
pp

1,
42 4
33
,
42 4
,:=
23
,
4
1
142, .
2
ppp j
j
ppj
pp p



 
= 1,
=2,
= 3,
=4
j
j

2,=
2
11
, =2,
22
,:=1, =3,
12 1
,=
12
j
j
p
j
p
pj
p
j
p









1,
4.
Throughout this paper, 12 are positive and
12
,,,ccc
,, ,,

sufficiently small positive constants which
in general depend on .
G
Theorem 1.2. Let and assume that 1< <2p
;
GC
for some 1
0< 2
and
11
min,, ,pp0< .

Then the Bieberbach polynomial satisfies
-p

,np
Bz

,1np CG
Bcn

with

1
2
0<<22.
p
p
 
 for each
Copyright © 2011 SciRes. APM
C. KOŞAR ET AL.307
Theorem 1.3. Let

62 <<2
6p
and assume that
;GC
for some 1
<
23
2
.
and


22
0<min,,,pp


,np
Bz
Then the Bie-
berbach polynomial satisfies
-p

,2np CG
Bcn

for each
with

2
2
0<<2 2.
p
p
 

Theorem 1.4. Let 3<<2
2p and assume that
;GC
for some 2<<2
3
and

3
23
0<min ,,
4
p
p




.
Then the Bieberbach polynomial satisfies
-p

,np
Bz

,3np CG
Bcn

for each
with

3
2
0<<2 2.p
p
 

For 0<<1
we obtain



1
1
max,,=1, 2, 3=,=
ipj p
 
and so, we
have the following theorem.
Theorem 1.5. Let



2
2
14 2
max1,<< 2
222
p

 







and assume that
;GC
for some 2<<1
9
and
4
1<,
p
.

Then the Bieberbach poly-
-p
nomial satisfies

,np
Bz

,4
ln
np CG
Bcnn

for each
with

4
2
0<<2 2.p
p
 

Analogously result can be written for case 1<2.
Theorem 1.6. Let 2
2< <21
p
and assume that
;GC
for some 0<<2
, and 0.
Then,
for any and arbitrary small
3n>0
.

,5np CG
Bcn

for each
Corollary 1.7. Let and assume that
=2p
;GC
for some 0< <2
and
11
0<min,
222
.
with 1
0< <,
p
if0< <1
and

12
0<<11 ,
pp


 

 if1<2.

Then the Bieberbach polyno-
mial
,np
Bz satisfies

,26 log
nCG
Bcnn

for each
with 1
0<<min,2 .
22



Remark 1.1. 1) Theorems 1.2-1.2, extend the cor-
responding results in [3-5, 10,12,13] to case 2p
.
2) Corollary 1.2 is extending the one result [10] to
domains bounded by a piecewise smooth curve with
interior zero angles and in =0
coincides with the
corresponding result of this work.
2. Some Auxiliary Results
The notation “”, we mean that 1 for a
constant , which doesn’t depend on and b. The
relation “” indicates that c where
are independent of and
ab
b
acb
a
,bacb
1
c
a23
23
,cc a.b
Let be finite domain bounded by Jordan
curve and let
G
L
=wz () be the con-
formal mapping of

ˆ
=w
z
:= extG
G onto
ˆ=:>1
ww
:<1,ww normalized by
=, >0
 (

00
ˆˆ
=0, >0
 
).
The level curve (exterior or interior) can be defined for
as
>0t
 
1
ˆ
:=:=,if<1;=,if> 1,.
t
Lzzttztt L
L
Let us denote t
and := ,
tt
GintL :=
t
extL
,:=inf :dzLz L

.
Let be a
L-
K
quasiconformal curve and .D
Then the region can be chosen to be the region
0
D
0
\,
R
r
G
,,
G
ˆ
for a certain number 0
1< depending
on
2R
f
and 1
0
R
0
=r
. In this case, it is known that
the function
 
1
1
.= .ff

2
is a -
K
quasi-
conformal reflection across as shown in ([18], p. 28)
by analogously in ([19], p. 75), that is, is a
L

.
2-
K
quasiconformal mapping leaving points on fixed and
L
satisfying the conditions \
R
GG G



0
\
r
G,
0
\
rR
GGG G
\ for some
0
1< <,RR
0<<1.rr
By using the facts in ([16], p. 97) and ([19], p. 76, [20], p.
26) we can find a
-CKquasiconformal reflection
.
across such fies the following L that it satis
Copyright © 2011 SciRes. APM
C. KOŞAR ET AL.
308

111
22
1
,,<<,zzzzzLz

1
1, << ,
1
,<, ,>,
z
z
z
zz
z
zz
z
 
 


(2.1)
and Jacobian

2
2
=z
z
J
of

.
satisfied
by means of thef a
qu
D
Lemma 2.1. ([18]) Let be a
1.J
Therefore, exn theorem otensio
asiconformal mapping, without loss of generality we
may assume that
 
=, .zzz

L-
K
quasiconformal
curve;

123 111
0
;,:, ,
R
LzzG zzzcdzL z
ˆ
=
j
j
wz


231210
,: ,, =
=1,2,3.
jj
orzzGzzzc dzLwz
j
 
Then,
ents 1) The statem12 13
zz zz and
12 13
w ww are are
w equivalent. So
12 13
zz zz and 1213
ww ww
2) If 12 13
zz,zz then
22
1313 13
1212 12
K
K
wwzz ww
wwzz ww



and, consequently, for any

33
00
R
zLzL

2
121
K
ww z
2
212,
K
z ww
where and are fixed cotants.
al
cu
0
1<< 2R
ma 2.2. ([2
1
00
=R
t :=L
ns
Lem 1]) LeG be a quasiconform
rve. Then, For every zL
0
there exists an arc

0,z

in G joining
to z with following
p:
1)

roperties
,dL

z
for every
If
0,z

2)
12
,

is joining the subarc of

,z

0
1
to 2,
then

12 1
,es 2
m
 
very pair
2 0
,,.z
 
([7] a -Kqua-
for e
) Let be
si
e aritrary Jan domain and
le arc except

Lemma 2.3.
1
:=LG
mes
conformal curve. Then

mes
, fvery
rectifiable arc .G
Let G bbord

or e
0
:=,, , zL

  an rectifiab
0L which satisfies the
following conditions:
1) mes

0
z
for one of its endpoints of its z
12
,
1 2
,

 for all 12
,.

ng function 2) There exists a monotone increasi
g
t
suhat ch t
0
,dLg z

for all .

Lemma 2.4. ([22]) Let

;GC
for some
0< <2,
0
and ofunc-
tion
n the arc a measurable
f
n such that
b
e give
1
2
0
zf

for
all
.en for all z
. Th



21
12
2
21
22
ln1<<2,
d2
2<2 1.
p
AG
p
p
f
zp

p

. Approximation in the 3

-
1
p
A
GNorm
Suppose that
;GC
f0<<2or some
and
0
=z
is giveof simput n. For the sake
, we assume
licity, but witho
=1,z loss of generality that =2,m 1
21;
1,1 G
and let the local coordinate axes
be parallel to OX and OY in the coordinate system;
,z

2:=:,0 .LzzLImz
Then
0
z is taken a arbitrary point on 2
L (or on 1
L
We recall that the domain
1:=Lz
subject t
:
o
, 0
LImz
s an
hosen directio the cn).
;GC
has exteri r
π
o
0< <2
non zero angle and 1
x
d of the po
type interior
zero angle in the neighborhooints 1=1z
2ectively.
We can say that the function

ˆ
=wz
domain
and =1,z resp
for the
;C
G
satisfies the escribed
in
conditions d
int 1
z
Lemma 2 in the neighborhood of po1=
. So,
we can e Lemma 2
 
asily get from


 
1
ˆ
1,z
(3.1)
for all
22
1
ˆˆ
,1; 1dzL zz


1
:=:1 >.zM zGz

On the other hand, using properties of the function
z
ˆ
=w
in the neigbourhood of the point 1=1z
(see, [7,23]) we obtain

1
ˆˆ
1ln 1zz



(3.2)
Because each is a
,=1,2
j
Lj

1-
i
quasi-
.
j mu
conformal arc, st be
re
the quasiconformal
flection across .
j
Lsider the Let us con curve

1
112
11
2
:=: =1;
3
cc
zxiyyx

=




1
212
12
2
:==: =1;
3
cc
zxiyy x





1
23
:==:=1 ;zxiyycx

2
24
:==:=1;zxiyycx
Copyright © 2011 SciRes. APM
C. KOŞAR ET AL.309
for some where a constnts and
from
mma 2.2 that
43
<<<,cc

Definition 1.3.
It is easy to check from Le
a1
c
2
c

121 2
,,
j
mes i

fo
Let fficiently large ntural number.
For arbitrary
 
r all 12
,, ,=1,2.
i
jij
 

=NNR su
0
>nN and
a
0< <1,
let us choose
1
cn
0
=Rr
such
=1, 2 such
that and points
00
<<rRR
, ,
jjthat they are in the intersection of
i
zi
R
L and i
j
ts divide These poin
R
Lparts:
 
11312
21 11
:=,, :=,,
RR RR
LzzLLzz
into four
 
22
421
12 22
:= , :=,
RR RR
LL
z LLzz
1
2
,
L
z
are traversed in the positive direction (counterclockwise).
4132
:=
R
RRR
L LLLL
R
and are subarcs of

i
jR
i
j
joining points 1,1 with i
j
z. Lenote t us de
 
1112 22
1221
:= ,
RR R
RL RRLR



:=
R
Uint and :=\.
R
UUG
We can extend the function to the
following way

z
Uin



2
,,
():=
zzG
zr
(3.3)
0,,
=1,2.
R
i
zUi
z

Then,
 


,
0, ,
=,,=
ziR
iz
zG
zzzzUi
 
1,2.
(3.4)
From the Cauchy-Pompeiu formula ([16], p. 148), we
get
  
11
=d d,.
2ππ
zzG
iz z






U
RR

Then, using the above notations we obtain
 




2
,=1
1
=d
2πLR
iz




12
3
4
or,
=1 ,
1 .
R
R
R
R
LL
fL
L
 
 



Since the first part of right hand side in (3.5) is
analytic function in G, there exists a polynomial
1,
n
Pz
where
1
deg 1
n
Pzn
 ([24], p. 142), such
that

 
1
2
1d<
2πLR
n
fc
Pz
in
z
.
(3.6)
Let Then, and from
(3

1
0
:=( )d .
z
nn
QzPtt

0=0
n
Q
.5) and (3.6) we have
  


1
1
d
2π
1
d,
π
iR
j
j
ij
UR
f
z
iz
z

 



(3.5)
where



2
2
,=1
2
1
2π
1
d
π
nij iR
j
UR
c
znz
z
1
d
j
Q
z
 






an us take integrals ovef the
each de we obtain
d letr G opower of
si
-pth
 






2
2
,1
2
d
1
1dd
dd.
ij
p
nz
G
p
j
pij
GR
p
z
nz
z
 
z
GU
R
zQz
 

 
 
(3.7)
n-Zygmund inequality ([19], p. 98),
we obtain


From the Caldero

 

2ddd, =1,
UR
p
p
zi
GU
R
i
z

  
 
2.
So, (3.7) and (3.8) give us
(3.8)








2
2
,=1
() 1
1d
d.
p
j
p
np
AG
pij
p
i
UR
Qnz
 

 




(3.9)
iR
jAG
p
Copyright © 2011 SciRes. APM
C. KOŞAR ET AL.
Copyright © 2011 SciRes. APM
310
Let us consider two case of in the last double
integral in (3.9) as: an
If then, using Hölder Inequality ([25
10
p
d p1< <2p2.
1< <2,p ], p.
5) we obtain





 






1
22
2d
p
1
22
2
1
22
dd
dd
.
p
p
ii
UUU
RRR
pp
UU
iR iR
p
p
iR iR
mesUmesU



 
 
 










 
(3.10)
Thus, (3.9) and (3.10) give us
 







 








2
2
,=1
1
22
1
1d
,1<<2, =1,2,
,2
p
j
p
np
AG
pijiR
jAG
p
pp
iR iR
p
i
UR
Qnz
mesUmesUp i
dp


 





(3.11)
, =1,2.
i
Now, we need some technical estimates to attack the
problem in (3.11).
Lemma 3.1. Let

;GC
for some 0< <2,
0.
Then for all >0



 
12 1
112
2
12 1
12
221
21
2
1
ln,1<2,=1,
12
, 2<<2,=1,
11
d
12,=2,
1,
p
p
j
p
p
npj
n
pj
n
j
n
n


 



























1
2
ln
iR
jAG
p
zn

 2, 1<p
2
2<<2,=2.
1
pj
Proof. Let us choose

:=1 ,
j
f


=1,2j, in Lemma 2.4, we obtain



 
21
1
2
2
,,
21
2
2
,
ln,1 <2,
1
d2
, 2<<2,
1
p
jij ij
p
iR
jij
AG
p
p
zp
 


(3.12)
here On the other hand, according
1, we have
1
,.
ii
j
dz Ln
all >

,:= .
i
ij j
mes R
w
to
Then, from (2.1), (3.1) and (3.2), for
Lemma 2.1 and Corollary 1.0, we get
C. KOŞAR ET AL.311



1
1
1
1
,1
1,=1
,, =1
1.
1
,, =2,=
ii
j
ij
ij jii
j
j
dzL jn
zdz Ljj
n



 





2
(3.13)
Then, the following inequalities are obtained from (3.12) and (3.13)
 

 
12 1
112
2
2
ln << 2,
1d
np
z
 


11
12
1
1,1
12
,2 <2.
1
p
iRAG
p
n
p
n

















2
p

(3.14)
and
 



2
221
2
1
iRp
z




21
12
21
ln,1 <<2,
1d
2
,2 <2.
1
p
AG
p
np
n
p
n
 










(3.15)
Combining (3.14) and (3.15) the proof is completed.
Now, for sufficiently small
01
0<< ,
12
33
VV where
we are going
to use following notations:
0
1212
22
:=
R
UVVVV

j
11
1:=
jR
V
 
0
1,:>0, =1,2UDzImzj
 



2
0
:=1,:<0, =1,2
j
jR
VUDzImzj
 


1
30
:=:>0\1,1,,
R
VUzImz DD






2
30
:=:< 0\1,1,.
R
VUzImzDD

0



Lemma 3.2. ([26], p. 10) Let

;GC
for some
0< <2,
0.
ll >0 Then for a




11
13
11
,
i
ii
mes V ,
i
mes V
nn


 





and


2
2
1, =1,2
i
i
mesVi
n





where 1
:= min,.
22



Lemma 3.3. Let

;GC
for some 0< <2
,
0
. Then for all >0

min 2
1=
R
m Ui


,1
,
1, 2.
i
es n


Proof. We have
3
i
Now, let us estimate

 
2
12
=1
=
ii
iR i
i
UmesVmes VV





1
i
i
s V
by choosing me
=1, >0K
and from Lemma 2.1 we have


1
1
2
1
31
1
,.
iii
i
mesVdzLn






 

1
1
2
1
11
1
,
iii
i
mesVdzLn






On the other hand if we use
but except the conformal mapping
the method of Gaier in [9]
in method, then
we obtain


min 2,1
2
1.
i
mes Vn



After then using the second and third inequality in the
we obtain the desireroof.
Lemma 3.4. Let
first oned p

;GC
for some 0< <2
,
0
. Then for all >0




12 1
1
d,
UR
p
p
ii
n







=1,2
where 1
2<2
1
p


=min ,2.

and
Proof. Let ,zG z
the point of nearest
to
:=LG
.z Then we have


11
,zzzz zz




and from ([18], Lemma 3)
Copyright © 2011 SciRes. APM
C. KOŞAR ET AL.
312
 


1
00
,1.
oo
wwww
dwL
wrwrwrwrw









After this estimation, we get
















12
1
21 121
00
1
()
1
d,
UU
RiR
i'
U
iR
p
pp
w
rw n


2
d
p
w
dd
pp
wrr
U
iR

  
  






 



wh


 




ere

:zzL

:= minr
and
iR
1
<2 .
1
2p
By using Lemma 3.1, 3.2, 3.3, 3.4 and (3.11) we get
the following results. We need this notation:






12 1
112
2
2
2
1
11
22
2
1
3
1
ln,max,,=1, 2,3.
11
, <,, 0 <.
2
,:=
11
, <,, <.
23
12
, <,, <<2.
3
p
i
p
n
p
p
np
n
p
n
p
p
n
p
n




 

 
 



















2
i
Lemma 3.5. Let and assume that
1< 2p

;GC
for some 0<<2
, 0.
Then, for
and arbitrary smany 3nall >0


1
,,.
np n
AG
p
Bp

(3.16)
Lemma 3.6. Let 2
2< <21
p
and assume that

;GC
 
1
1
,12 1
1,0<<1,
.
11<2.
p
np p
AG
pp
n
B
n







(3.17)
Proof. The proof of Lemma 3.5-3.6 is similar. So, we
give them together. From (3.11) and Lemma 3.1-3.4 we
obtain
for some 0<<2
, 0.
Then, for
arbitrary smany 3n andall >0

12 1
112
2
12 1
12
21
12
2
ln n
21
2
1
ln1 <<2,= 1,
1
12
2<2 ,=1,
1
11 <<2,=2,
1
p
nAG
pp
p
p
npj
n
Qnpj
n
pj
n
n



































2

2
1< 2,
211
2<2 ,=2,,2 <2,
11
p
p
p
p
pj p
n
12 1
,1 <
p
n

 
(3.18)






 




Copyright © 2011 SciRes. APM
C. KOŞAR ET AL.313
where 1
:= min,
22




,

:=min2;1,:=min1, 2
 

Case 1. Let In this case from (3.18) we get
1< <2.p


2
12 1
112 2
211
ln
,.
p
pp
nAG
p
n
Qn
nn
p








 


 
 
Case 2. Let

21
2<min2 ;2,:=min1,2
11
p










12 1
21
2
11
p
pp
n
Q

1
12 1
1, 0<< 1,
1,1 <2.
AG
p
p
p
p
nn
n
n
 

 


 

 






Now, let us consider the polynomial
.
In this case from (3.18) we have

00
:= 1.
nn n
Qz QzQz


 

nn
Q
It is clear
that

00
=0, =1
nn
Qz Qz

. So, we have


1
1
12 1
1, 1 <<2,
12
, 0<<1,2<2,
1
12
,1<2,2<2 .
1
p
nAG
p
p
p
p
n
Qp
n
p
n


 







 


So, if we consider the extremal property of the
polynomials then we obtain (3.16) and (3.1)
respectively.
To prove Theorem 1.2-1.6 and Corollary 1.7, we use a
sio the one of Andrievskii and Simonenko
employed in the proofs of analogous theorems for
(see [6,8,10]).
simply connected
4.1) in (3.19) to prove Theorems 1.2-1.6.
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B
7
milar method t
=2p
Lemma 3.7. [2] Let Gbe a
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
1
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p
Bn
(0,1),
=2,3,,n
for each and
1
() ()
1,> 2,
log,= 2,
,0<<2,>
nn
CGA G
p
p
PP np
np
0
r all polynomials of degfo

Pz
nn
Pn
and
normalized with

0=0.
n
P
Then,
,()
np CG .Bn
(3.19)
It is enough replacing
by
in (3.16) and (3.17)
rstively and
e pec

2
=22p
p

 in ([22], Corollary
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