Massiveness of Exceptional Sets of Multi-Term Asymptotic Representations of Subharmonic Functions in the Plane
It was investigated the massiveness of exceptional sets that arise in multiterm asymptotic representations of subharmonic functions.
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Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
2006
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| Назва видання: | Журнал математической физики, анализа, геометрии |
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| Цитувати: | Massiveness of Exceptional Sets of Multi-Term Asymptotic Representations of Subharmonic Functions in the Plane / P. Agranovich // Журнал математической физики, анализа, геометрии. — 2006. — Т. 2, № 2. — С. 119-129. — Бібліогр.: 9 назв. — англ. |
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irk-123456789-1065862016-10-01T03:02:13Z Massiveness of Exceptional Sets of Multi-Term Asymptotic Representations of Subharmonic Functions in the Plane Agranovich, P. It was investigated the massiveness of exceptional sets that arise in multiterm asymptotic representations of subharmonic functions. 2006 Article Massiveness of Exceptional Sets of Multi-Term Asymptotic Representations of Subharmonic Functions in the Plane / P. Agranovich // Журнал математической физики, анализа, геометрии. — 2006. — Т. 2, № 2. — С. 119-129. — Бібліогр.: 9 назв. — англ. 1812-9471 http://dspace.nbuv.gov.ua/handle/123456789/106586 en Журнал математической физики, анализа, геометрии Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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It was investigated the massiveness of exceptional sets that arise in multiterm asymptotic representations of subharmonic functions. |
| format |
Article |
| author |
Agranovich, P. |
| spellingShingle |
Agranovich, P. Massiveness of Exceptional Sets of Multi-Term Asymptotic Representations of Subharmonic Functions in the Plane Журнал математической физики, анализа, геометрии |
| author_facet |
Agranovich, P. |
| author_sort |
Agranovich, P. |
| title |
Massiveness of Exceptional Sets of Multi-Term Asymptotic Representations of Subharmonic Functions in the Plane |
| title_short |
Massiveness of Exceptional Sets of Multi-Term Asymptotic Representations of Subharmonic Functions in the Plane |
| title_full |
Massiveness of Exceptional Sets of Multi-Term Asymptotic Representations of Subharmonic Functions in the Plane |
| title_fullStr |
Massiveness of Exceptional Sets of Multi-Term Asymptotic Representations of Subharmonic Functions in the Plane |
| title_full_unstemmed |
Massiveness of Exceptional Sets of Multi-Term Asymptotic Representations of Subharmonic Functions in the Plane |
| title_sort |
massiveness of exceptional sets of multi-term asymptotic representations of subharmonic functions in the plane |
| publisher |
Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
| publishDate |
2006 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/106586 |
| citation_txt |
Massiveness of Exceptional Sets of Multi-Term Asymptotic Representations of Subharmonic Functions in the Plane / P. Agranovich // Журнал математической физики, анализа, геометрии. — 2006. — Т. 2, № 2. — С. 119-129. — Бібліогр.: 9 назв. — англ. |
| series |
Журнал математической физики, анализа, геометрии |
| work_keys_str_mv |
AT agranovichp massivenessofexceptionalsetsofmultitermasymptoticrepresentationsofsubharmonicfunctionsintheplane |
| first_indexed |
2025-07-07T18:43:58Z |
| last_indexed |
2025-07-07T18:43:58Z |
| _version_ |
1837014812651945984 |
| fulltext |
Journal of Mathematical Physics, Analysis, Geometry
2006, vol. 2, No. 2, pp. 119�129
Massiveness of Exceptional Sets of Multi-Term
Asymptotic Representations of Subharmonic Functions
in the Plane
P. Agranovich
Mathematical Division, B. Verkin Institute for Low Temperature Physics and Engineering
National Academy of Sciences of Ukraine
47 Lenin Ave., Kharkov, 61103, Ukraine
E-mail:agranovich@ilt.kharkov.ua
Received February 26, 2004
It was investigated the massiveness of exceptional sets that arise in multi-
term asymptotic representations of subharmonic functions. It was shown
that such exceptional sets be any C0;1+
-sets where
2 [0; 1]. This fact
distinguishes the case of n-term asymptotic from the case of functions of
completely regular growth.
Key words: subharmonic function, asymptotic representation, excep-
tional set.
Mathematics Subject Classi�cation 2000: 30D20, 30D35, 31A05, 31A10.
In the recent decades a number of new e�ects has been found in the studies
of connection between the behavior of a subharmonic function at in�nity and the
growth of the distribution function of its Riesz measure in the terms of multi-term
asymptotic representations. These facts show the essential di�erences between
the multi-term asymptotics and the classical case of the single-term asymptotic
representation (the functions of completely regular growth of Levin�P��uger).
As known [6], the exceptional sets, that appeared in the main theorem of the
function theory of completely regular growth, can be reduced up to C0;0-sets.
Recall that a set E � C is the C0;�-set, � > 0; if it can be covered by the
disks fz : jz � z
j
j < r
j
g such that
lim
R!1
1
R�
X
jzj j�R
r�
j
= 0: (1)
If limit ( 1 ) is zero for every positive �; then E is a C0;0-set.
c
P. Agranovich, 2006
P. Agranovich
It was shown in [3] and [5] that the exceptional sets arising in investigations
of n-term asymptotics cannot in general be less than C0;1 even if the Riesz masses
are concentrated on a �nite system of rays.
If the Riesz masses are distributed on the plane arbitrarily, then the existence
of the asymptotics of a desired form can be guaranteed only outside C0;2-set
([4, 5]).
Then there arises a question about connection between the distribution of the
Riesz masses in the plane and the massiveness of the exceptional sets. The ques-
tion was put by V. Logvinenko after the author's papers ([1, 2]) on the asymptotic
behavior of a subharmonic function with the Riesz measure in a parabolic domain
had been published.
In this article we prove the existence of exceptional C0;�-sets, � 2 [0; 1]; for
the case of multi-term asymptotic representations. This provides one more dif-
ference of the cases of a single- and multi-term asymptotic representations. Thus
a complete description of the massiveness of exceptional sets appeared in multi-
terms asymptotic representations of subharmonic functions in the plane is given.
This result is obtained by using the reasoning from [4]. Note that we omit the
calculations similar to those in [4]. We will assume that the reader is familiar
with the articles [3, 4], and we will point out the aspects of proofs containing the
di�erences.
Before formulation of results let us give the notations and the de�nitions which
will be used below.
We say that a function f(t); t > 0 has multi-term (n-term) asymptotics as
t!1, if f can be represented in the form
f(t) = �1t
�1 +�2t
�2 + : : :+�
n
t�n + '(t);
where �
j
, j = 1; 2; : : : ; n, are real constants; 0 < [�1]
* < �
n
< : : : < �1, and the
function '(t) is small in a certain sense in comparison with the previous term.
Similarly, we understand the expression "polynomial asymptotics of a function
f(z); z ! 1". In the later case the coe�cients �
j
; j = 1; : : : ; n; are functions
only of � = arg z and t = jzj.
Let u(z) be a subharmonic function; � � its measure of Riesz, �(t; �) =
�(fjzj < t; 0 < arg z � �g) .
Put G
!
:= fz : z = rei�; 2k � r < 2k+1; j�j � arctg 2k(!�1); k = 0; 1; : : :g
where ! 2 (0; 1).
We denote various constants by C and the Lebesgue measure in the plane z
by �
z
.
*As usual, [a] is the integral part of a number a.
120 Journal of Mathematical Physics, Analysis, Geometry, 2006, vol. 2, No. 2
Massiveness of Exceptional Sets
The technique used in this paper allows us to generalize all the obtained results
for the multi-term asymptotic representations. Thus without loss of generality
here we consider only the case of two-term asymptotics.
Theorem 1. Let u(z) be a subharmonic function of noninteger order in the
plane C with the support of its Riesz measure concentrated in the domain G
!
.
Assume that the following relation for the measure � is valid
�(t; �) = �1t
�1 +�2t
�2 + '(t; �); � 2 (0; 2�]: (2)
Here p = [�1] < �2 < �1;�1 > 0;�(t; �) = 0 for all � and t � t0, and the function
' satis�es the following estimate for some q � 1
2RZ
R
sup
�2[0;2�]
j'(t; �)jqdt = o(R�2q+1); R!1: (3)
Then the order of the function u is equal to �1 and
u(rei�) = �1
�r�1
sin��1
cos �1(� � �) + �2
�r�2
sin��2
cos �2(� � �) + (rei�);
where (rei�) = o(r�2); r ! 1; uniformly for � 2 [0; 2�]; if the point z = rei�
does not belong to certain C0;1+!-set.
If in (3) the number q > 1; thenZ
G!
T
fR�jzj�2Rg
j (z)jqd�
z
= o(R�2q+1+!); R!1:
P r o o f. Without loss of generality we can assume that measure � has
in�nitely smooth density and its support lies inside G
!
. This follows from the
condition supp� � G
!
and the proof of [4, Th. 4]. Indeed, the proof of Th. 4
in [4] is based on the fragmentation of the plane into the "collars" (parts of the
plane). In these "collars" the density of some measure, "close" in a certain sense
to �, is de�ned. The density of this new measure is equal to zero in the part of
the "collar" adjoining to its boundary.
According to the Riesz�Brelo theorem [7] and conditions (2) and (3), the
order of the function u is equal to �1. Moreover, by this theorem the function u
represented as represented in the form of u = J + P , where
J(z) = Re
Z
C
"
ln
�
1�
z
�
�
+
pX
k=1
1
k
�
z
�
�
k
#
d�(�)
Journal of Mathematical Physics, Analysis, Geometry, 2006, vol. 2, No. 2 121
P. Agranovich
is the canonical potential of the measure �; and P is a harmonic polynomial of
the degree not greater than p: Since u is the function of noninteger order, then,
without loss of generality, we can assume that P (z) � 0; i.e., u � J or
u(rei�) = lim
R!1
Z
f�:j�j<Rg
"
ln
�
1�
rei�
�
�
+
pX
k=1
1
k
�
rei�
�
�
k
#
d�(�)
= Re lim
R!1
�Z
��2�
d�
RZ
0
"
ln
�
1�
rei�
tei�
�
+
pX
k=1
1
k
�
rei�
tei�
�
k
#
@2�
@t@�
dt
= Re lim
R!1
8<
:
�Z
��2�
"
ln
�
1�
rei�
tei�
�
+
pX
k=1
1
k
�
rei�
tei�
�
k
#
@�(t; �)
@�
�����
R
0
d�
�
RZ
0
(rei�)p+1
tp+1
dt
�Z
��2�
e�ip�
tei� � rei�
�
@�(t; �)
@�
d�
9=
;
= Re lim
R!1
(A(R; rei�) +B(R; rei�)):
By integrating by parts the expression A(R; z) and using the condition that
�(t; �) � 0 in some neighborhood of the origin, we obtain A(R; rei�) ! 0 as
R!1 uniformly for each compact set in z-plane, z = rei�:
So
u(rei�) = �Re
8<
:(rei�)p+1
1Z
0
dt
tp+1
�Z
��2�
e�ip�
tei� � rei�
�
@�(t; �)
@�
9=
;
= �Re
8<
:rp+1
1Z
0
�(t; 2�)
tp+1(t� r)
dt
+i(rei�)p+1
�Z
��2�
e�ip�d�
1Z
0
(p+ 1)tei� � prei�
(tei� � rei�)2tp+1
�(t; �)dt
9=
; :
Let us substitute here expression ( 2 ) and take into account that supp� � G
!
.
It is not hard to see that the principal terms of the asymptotics u are
�
j
�r�j
sin��
j
cos �
j
(� � �); j = 1; 2:
122 Journal of Mathematical Physics, Analysis, Geometry, 2006, vol. 2, No. 2
Massiveness of Exceptional Sets
Thus we have only to investigate the behavior of the remainder term :
(rei�) = �Re
8<
:rp+1
1Z
0
'(t; 2�)
tp+1(t� r)
dt
+i(rei�)p+1
�Z
��2�
e�ip�d�
1Z
0
(p+ 1)tei� � prei�
(tei� � rei�)2tp+1
'(t; �)dt
9=
;
= �Ref 1(re
i�) + 2(re
i�)g:
Let us estimate the function 1(re
i�). By the Hardy�Littlewood theorem [8] on
the bound of the Hilbert transform and the reasoning from [3, Th. 1], we conclude
that for q 2 (1;1) 8<
:
2RZ
R
j 1(re
i�)jqdr
9=
;
1
q
=
8<
:
2RZ
R
�����rp+1
1Z
0
'(t; 2�)dt
tp+1(t� r)
�����
q
dr
9=
;
1
q
= o(R
�2+
1
q ); R!1: (4)
If q 2 [1;1) then the set
e1 =
8<
:r 2 [0;1) :
�����rp+1
1Z
0
'(t; 2�)dt
tp+1(t� r)
�����> �(r)r�2
9=
; ;
where the function �(r) tends to zero su�ciently slowly, has the zero relative
Lebesgue measure. Since supp� � G
!
, then the estimate
j 1(z)j � �(jzj)jzj�2
is valid outside the set E1 = fz : jzj 2 e1; z 2 G
!
g. It is easy to see that E1 is
C0;1+!-set.
To estimate the function 2 we split the ray [1;1) into semi-intervals [2k; 2k+1),
k = 0; 1; : : : . For r 2 [2k; 2k+1) we have
2(re
i�) = i
8><
>:(rei�)p+1
0
B@
2k�1Z
0
+
2k+2Z
2k�1
+
1Z
2k+2
1
CA dt
tp+1
Journal of Mathematical Physics, Analysis, Geometry, 2006, vol. 2, No. 2 123
P. Agranovich
�
�Z
��2�
(p+ 1)tei� � prei�
eip�(tei� � rei�)2
'(t; �)d�
9=
;
= i(J
(k)
1 + J
(k)
2 + J
(k)
3 ):
From (3) it is easy to conclude that jzj��2(jJ
(k)
1 j + jJ
(k)
3 j) ! 0 uniformly for
� = arg z 2 [0; 2�] as jzj ! 1.
Let us represent the integral J
(k)
2 as a sum of two integrals estimated similarly.
So we consider only one of them, namely:
~ 2(re
i�) = (rei�)p+1
2k+2Z
2k�1
dt
tp
�Z
��2�
p+ 1
ei(p�1)�(tei� � rei�)2
'(t; �)d�:
Put
�
k
(�) = (p+ 1)e2i arg ���(p+1)'(�)�
k
(�);
where �
k
(�) is a characteristic function of the ring f� : 2k�1 � j�j � 2k+2g. As it
follows from the de�nition of the domain G
!
and estimate (3),
0
B@ Z
G!
T
fz:2k�j�j�2k+1g
j�
k
(�)jqd�
�
1
CA
1
q
= o
�
2
k(�2+
1+!
q
�(p+1))
�
; k !1: (5)
In these notations we have
~ 2(z) = zp+1
Z
j�j2C
�
k
(�)
(z � �)2
d�
�
; z = rei�: (6)
To estimate this function we use the following fact*, which is a special case of [8,
Th. 4, p. 56].
Theorem A. Let f 2 Lp(R2); 1 � p <1. Then a transformation
T
"
(f)(z) =
Z
j�j�"
f(�)
(z � �)2
d�
�
; " > 0;
has the following properties:
a) lim
"!0
T
"
(f)(z) exists for almost all z.
*The reference to this result is missing in [4].
124 Journal of Mathematical Physics, Analysis, Geometry, 2006, vol. 2, No. 2
Massiveness of Exceptional Sets
b) Let T �(f)(z) = sup
">0
jT
"
(f)(z)j. If f 2 L1(R2) then
mesfz : jT �(f)(z)j > �g �
Cjjf jj
L
1
�
; � > 0;
where the constant C does not depend on f and �:
c) If 1 < q <1 then jjT �f jj
q
� C
q
jjf jj
q
, where the constant C
q
depends only
on the number q.
By Theorem A it follows from (5) that for q 2 (1;1)
jjzp+1 ~ 2(z)jjLq(G!
T
fz:2k�jzj�2k+1g) = o
�
2
k(�2+
1+!
q
)
�
; k !1; (7)
and for q 2 [1;1) the measure of the set
E
(2)
k
=
n
z : jzj 2 G
!
\
fz : 2k � jzj � 2k+1
g : jzp+1 ~ 2(z)j > �
k
jzj�2
o
satis�es the estimate
mesE
(2)
k
= �
�q
k
o(2k(1+!)); k !1: (8)
If the sequence of numbers f�
k
g tends to zero su�ciently slowly, then it follows
from (8) that
E2 =
1[
k=1
E
(2)
k
is a C0;1+!-set. Hence E = E1 +E2 is a C0;1+!-set.
If q 2 (1;1) then we obtain by virtue of (7) and (4)Z
G!
T
fz:R�jzj�2Rg
j (z)jqd�
z
= o(R�2q+1+!); R!1:
The theorem is proved.
R e m a r k. In [9, p. 5] the transformation (6), that de�ned the function
~ 2(z) by the function �
k
(�), is called the Berling transformation of the function
�
k
(�):
Now we will show that the massiveness of exceptional sets established in Th. 1
cannot be reduced. Namely, the following fact holds:
Journal of Mathematical Physics, Analysis, Geometry, 2006, vol. 2, No. 2 125
P. Agranovich
Theorem 2. Let ! 2 (0; 1) be some �xed number. Then for any numbers �1
and �2; �1 > �2;
* and a number � 2 (0; !) there exists a subharmonic function u
such that supp� � G
!
;
�(t; �) = �1t
�1 +�2t
�2 + '(t; �);
where �1 > 0 and
2TZ
T
sup
�2[0;2�]
j'(t; �)jqdt = o(T �2q+1); T !1; q � 1;
but the asymptotic representation
u(rei�) =
2X
j=1
�r�j�
j
sin��
j
cos �
j
(� � �) + (rei�);
(rei�) = o(r�2); r !1;
does not take place uniformly for � 2 [0; 2�] on some exceptional set E that is not
a C0;1+�-set.
P r o o f. We follow the same scheme as in [4, Th. 3].
Let us introduce the following notations: a) R = 2
1
�1��2 ; b) Æ
k
= R�1
3�2k
;
k
= 2�
2k
;
c) a sequence f�
k
g, k = 1; 2; : : : ; tends to zero su�ciently slowly.
We proceed in several steps.
1) We begin with the construction of the Riesz measure � of the function to
be determined. Let the distribution function of the measure � be
�(t; �) = �1t
�1 +�2t
�2 + '(t; �);
where �1 = 1 and �2 = 0. We represent the function ' as a sum of two terms
'1 and '2.
Let N be some positive integer which will be chosen later on. Put
'1(t; �) = '2(t; �) = 0 8t < RN ; 8� 2 [0; 2�]:
Let k � N and Rk � t < Rk+1: De�ne the function '1(t; 0) in the following way:
'1(t; 0) = �
k
Rk�2 on the segments [Rk(1 + 3 � 2N jÆ
k
); Rk(1 + (3 � 2Nj + 1)Æ
k
)],
j = 0; 1; : : : ; 2k�N�1; '1(t; 0) = 0 on semi-intervals [Rk(1+(3�2N j+2)Æ
k
); Rk(1+
3 � 2N (j + 1)Æ
k
)), and we de�ne '1(t; 0) as a linear function on the rest of the
ray. Next we require the function '1(t; �) to be independent of � in the angles
*This condition is missing in [4] and is taken for granted.
126 Journal of Mathematical Physics, Analysis, Geometry, 2006, vol. 2, No. 2
Massiveness of Exceptional Sets
j
k�N < � � (j+1)
k�N , j = 0; 1; : : : ;
h
arctg 2k(!�1)
k�N
i
� 1. On the rays � = j
k�N
we de�ne this function in the same way as on the ray � = 0.
We de�ne '2 as
'2(t; �) = �
'1(t; 0)
k�N
�; 0 � � <
"
arctg 2k(!�1)
k�N
#
k�N :
Extend the function ' = '1 + '2 as constant in � on the rest of the interval
[0; 2�):
Show now that the function
�(t; �) = t�1 + '(t; �)
is a distribution function of the Riesz measure for some subharmonic function u
on the plane and this measure is concentrated in the set G
!
.
To prove this fact it is su�cient to investigate the behavior of the expression
S(t1; t2; �1; �2) = �(t2; �2)� �(t2; �1)� �(t1; �2) + �(t1; �1)
for any 0 < t1 < t2; 0 � �1 < �2 < 2�.
Let Rk(1+ 3 � 2N jÆ
k
)) � t1 < t2 < Rk(1+3 � 2N (j+1)Æ
k
)) and j
k�N < �1 <
�2 � (j + 1)
k�N , j = 0; 1; : : : ;
h
arctg 2k(!�1)
k�N
i
� 1, k � N . Then
S(t1; t2; �1; �2) = '(t2; �2)� '(t2; �1)� '(t1; �2) + '(t1; �1)
=
1
k�N
('1(t1; 0)� '1(t2; 0))(�2 � �1) � 0: (9)
If Rk(1 + 3 � 2N jÆ
k
)) � t1 < t2 < Rk(1 + 3 � 2N (j + 1)Æ
k
)) and points �1; �2 >
(j + 1)
k�N , j = 0; 1; : : : ;
h
arctg 2k(!�1)
k�N
i
� 1, k � N , then
S(t1; t2; �1; �2) � 0 (10)
by the construction of '.
If t1; t2 < RN , then it is clear that for any �1; �2
S(t1; t2; �1; �2) � 0: (11)
Obviously, (9), (10) and (11) are su�cient for the con�rmation that the con-
structed function �(t; �) is the distribution function of the measure of a subhar-
monic function on the plane and the support of this measure is concentrated in
the set G
!
:
Journal of Mathematical Physics, Analysis, Geometry, 2006, vol. 2, No. 2 127
P. Agranovich
2) It follows from the construction of ' that
'(t; �) = o(t�2); t!1;
uniformly for � 2 [0; 2�]: Hence in view of Th. 1 the function u has a multi-term
asymptotics of the form:
u(rei�) =
�r�1
sin��1
cos �1(� � �) + (rei�):
Here (rei�) = o(r�2) uniformly for � 2 [0; 2�] as r ! 1; if the point z = rei�
does not belong to some C0;1+!-set.
Consider the set
E0;k =
n
z : jz �Rk(1 + 3 � 2NmÆ
k
)j � �
k
Rk;
m 2
��
2k�N � 1
3
�
+ 1; 2
�
2k�N � 1
3
���
;
where k � N and the number � > 1 will be chosen later on. The estimate of the
remainder term (rei�) on the set E0;k is carried out in the same way as in the
proof [4, Th. 3], where it is shown that on the set E0;k the inequality
� (z) � C(�� 1)�
k
jzj�2 ln jzj � C12
�N�
k
jzj�2 ln jzj
is valid. Here C is a universal constant and C1 does not depend on z, N and k:
It is easy to see that for any � > 1 there exists N 2 N such that
� (z) � ��
k
jzj�2 ln jzj
with � > 0:
It is clear that the analogous estimate is valid also on the sets
E
j;k
=
n
z : jz �Rk(1 + 3 � 2NmÆ
k
)eij
k�N j � �
k
Rk;
m 2
��
2k�N � 1
3
�
+ 1; 2
�
2k�N � 1
3
���
; j = 0; 1; 2; : : : ;
"
arctg 2k(!�1)
k�N
#
� 1:
So for such choice of N and for all z 2 E =
S
j;k
E
j;k
the relation
(z) � ���
k
jzj�2 ln jzj
holds. Therefore, if �
k
decreases su�ciently slowly, then we obtain the inequality
(z) < �jzj�2
for all great by modulus z 2 E.
128 Journal of Mathematical Physics, Analysis, Geometry, 2006, vol. 2, No. 2
Massiveness of Exceptional Sets
3) We estimate now a relative (1 + �)-measure of the set E: We have
rel mes
1+�
(E
\
fz : jzj � Rk+1
g)
�
1
R(k+1)(1+�)
22k2k(!�1)(�
k
Rk)1+�
= C2k(1+!��(1+�)):
Hence the relative (1+�)-measure of the set E is equal to1 if �(1+�) < 1+!:
It is easy to see that for any 0 � � < ! there exists � > 1 such that the relative
(1 + �)-measure of the set E is in�nity.
This completes the proof of the theorem.
References
[1] P. Agranovich, Approximation of Subharmonic Functions and the Questions of Multi-
Term Asymptotics that Connected with its. � Mat. �z., analiz, geom. 7 (2000),
255�265. (Russian)
[2] P. Agranovich, About Precision of Multi-Term Asymptotics of a Subharmonic Func-
tion with Masses in Parabola. �Mat. �z., analiz, geom. 11 (2004), 127�134. (Russian)
[3] P. Agranovich and V. Logvinenko, Analogous of the Theorem of Valiron�Titchmarsh
for Two-Term Asymptotics of a Subharmonic Function with Masses on the Finite
System of Rays. � Sibirsk. Mat. Zh. 24 (1985), No. 5, 3�19. (Russian)
[4] P. Agranovich and V. Logvinenko, Polynomial Asymptotic Representation of a Sub-
harmonic Function in the Plane. � Sibirsk. Mat. Zh. 32 (1991), No. 1, 1�16. (Russian)
[5] P.Z. Agranovich and V.N. Logvinenko, Exceptional Sets for Entire Functions. �
Math. Stud. 13 (2000), 149�156.
[6] V. Azarin, About Asymptotic Behavior of Subharmonic Functions of Finite Order.
� Mat. Sb. 108(150) (1979), No. 2, 147�167. (Russian)
[7] L. Ronkin, Introduction in the Theory of Entire Functions of Several Variables.
Nauka, Moscow, 1971. (Russian)
[8] E. Stein, Singular Integrals and Di�erentiability Properties of Functions. Mir,
Moscow, 1973. (Russian)
[9] A. Zygmund, Integrales Singulieres. Lect. Notes Math. Springer�Verlag, Berlin, 204
(1971).
Journal of Mathematical Physics, Analysis, Geometry, 2006, vol. 2, No. 2 129
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