Variation of Subharmonic Function under Transformation of its Riesz Measure

Variation of Subharmonic Function under Transformation of its Riesz Measure

The paper combines two aspects. First, it contains a compressed com- parative review of the well-known results on the change of growth of entire (subharmonic resp.) function under the shifts of its zeros (under T-shift of its Riesz measure resp.). There was B.Ya. Levin who stood at the sources of th...

Ausführliche Beschreibung

Gespeichert in:
Bibliographische Detailangaben
Veröffentlicht in:Журнал математической физики, анализа, геометрии
Datum:2007
Hauptverfasser: Kudasheva, E.G., Khabibullin, B.N.
Format: Artikel
Sprache:English
Veröffentlicht: Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України 2007
Online Zugang:https://nasplib.isofts.kiev.ua/handle/123456789/106438
Tags: Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
Назва журналу:Digital Library of Periodicals of National Academy of Sciences of Ukraine
Zitieren:Variation of Subharmonic Function under Transformation of its Riesz Measure / E.G. Kudasheva, B.N. Khabibullin // Журнал математической физики, анализа, геометрии. — 2007. — Т. 3, № 1. — С. 61-94. — Бібліогр.: 25 назв. — англ.

Institution

Digital Library of Periodicals of National Academy of Sciences of Ukraine
id nasplib_isofts_kiev_ua-123456789-106438
record_format dspace
spelling Kudasheva, E.G.
Khabibullin, B.N.
2016-09-28T17:57:18Z
2016-09-28T17:57:18Z
2007
Variation of Subharmonic Function under Transformation of its Riesz Measure / E.G. Kudasheva, B.N. Khabibullin // Журнал математической физики, анализа, геометрии. — 2007. — Т. 3, № 1. — С. 61-94. — Бібліогр.: 25 назв. — англ.
1812-9471
https://nasplib.isofts.kiev.ua/handle/123456789/106438
The paper combines two aspects. First, it contains a compressed com- parative review of the well-known results on the change of growth of entire (subharmonic resp.) function under the shifts of its zeros (under T-shift of its Riesz measure resp.). There was B.Ya. Levin who stood at the sources of these results. Second, the rst coauthor proves new results obtained in this direction: the estimates of change of subharmonic function under integral restrictions for T-shift of its Riesz measure, and also, in a certain sense, an optimal approximation of an entire function by the entire function with simple zeros.
en
Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
Журнал математической физики, анализа, геометрии
Variation of Subharmonic Function under Transformation of its Riesz Measure
Article
published earlier
institution Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection DSpace DC
title Variation of Subharmonic Function under Transformation of its Riesz Measure
spellingShingle Variation of Subharmonic Function under Transformation of its Riesz Measure
Kudasheva, E.G.
Khabibullin, B.N.
title_short Variation of Subharmonic Function under Transformation of its Riesz Measure
title_full Variation of Subharmonic Function under Transformation of its Riesz Measure
title_fullStr Variation of Subharmonic Function under Transformation of its Riesz Measure
title_full_unstemmed Variation of Subharmonic Function under Transformation of its Riesz Measure
title_sort variation of subharmonic function under transformation of its riesz measure
author Kudasheva, E.G.
Khabibullin, B.N.
author_facet Kudasheva, E.G.
Khabibullin, B.N.
publishDate 2007
language English
container_title Журнал математической физики, анализа, геометрии
publisher Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
format Article
description The paper combines two aspects. First, it contains a compressed com- parative review of the well-known results on the change of growth of entire (subharmonic resp.) function under the shifts of its zeros (under T-shift of its Riesz measure resp.). There was B.Ya. Levin who stood at the sources of these results. Second, the rst coauthor proves new results obtained in this direction: the estimates of change of subharmonic function under integral restrictions for T-shift of its Riesz measure, and also, in a certain sense, an optimal approximation of an entire function by the entire function with simple zeros.
issn 1812-9471
url https://nasplib.isofts.kiev.ua/handle/123456789/106438
citation_txt Variation of Subharmonic Function under Transformation of its Riesz Measure / E.G. Kudasheva, B.N. Khabibullin // Журнал математической физики, анализа, геометрии. — 2007. — Т. 3, № 1. — С. 61-94. — Бібліогр.: 25 назв. — англ.
work_keys_str_mv AT kudashevaeg variationofsubharmonicfunctionundertransformationofitsrieszmeasure
AT khabibullinbn variationofsubharmonicfunctionundertransformationofitsrieszmeasure
first_indexed 2025-11-26T04:45:25Z
last_indexed 2025-11-26T04:45:25Z
_version_ 1850609469500686336
fulltext Journal of Mathematical Physics, Analysis, Geometry 2007, v. 3, No. 1, pp. 61�94 Variation of Subharmonic Function under Transformation of its Riesz Measure E.G. Kudasheva Chair of Mathematics, Bashkir State Agrarian University 34, 50 Let Oktyabrya Str., Ufa, 450001, Bashkortostan, Russia E-mail:Lena Kudasheva@mail.ru B.N. Khabibullin � Department of Mathematics, Bashkir State University 32 Frunze Str., Ufa, 450074, Russia Institute of Mathematics with Computing Centre Ural Branch of the USSR Academy of Sciences 112 Chernyshevskii Str., Ufa, 450077, Russia E-mail:Khabib-Bulat@mail.ru Received June 25, 2006 The paper combines two aspects. First, it contains a compressed com- parative review of the well-known results on the change of growth of entire (subharmonic resp.) function under the shifts of its zeros (under T -shift of its Riesz measure resp.). There was B.Ya. Levin who stood at the sources of these results. Second, the �rst coauthor proves new results obtained in this direction: the estimates of change of subharmonic function under integral restrictions for T -shift of its Riesz measure, and also, in a certain sense, an optimal approximation of an entire function by the entire function with simple zeros.* Key words: entire function, subharmonic function, shift of zeros, Riesz measure, T -shift of measure. Mathematics Subject Classi�cation 2000: 30D15, 31A05. *This research was supported by the Russian Foundation for Basic Research under grant No. 06�01�00067, and by the Russian Foundation �State Support of the Leading Scienti�c Schools� under grant No. 10052.2006.1. c E.G. Kudasheva and B.N. Khabibullin, 2007 E.G. Kudasheva and B.N. Khabibullin Dedicated to the centennial of the birthday of B.Ya. Levin 1. Introduction: Initial Results Denote by D(z; t) an open disk of the radius r centered at z 2 C in the complex plane C , D(r) := D(0; r). For t 6 0, D(z; t) is an empty set ? by de�nition. For S � C , we denote its boundary in C by @S. In particular, @D(z; t) is a circumference of the radius r centered at z. By M+ denote the class of all positive Borel measures � on C , supp� is a support of � 2 M+; �(z; t) := � � D(z; t) � , �rad(r) := �(0; r) = � � D(r) � . By � �� B denote restriction of the measure � to the Borel subset B � C . Let � = f�kg, k = 1; 2; : : : , be a sequence of points in the complex plane without accumulating points in C . With � we associate an integer-valued measure n� on C by the rule n�(D) := X �k2D 1; D � C ; (1.1) i. e., n�(D) is the number of points from � occurring in D. In this connection we set supp� := suppn�. By de�nition, the inclusion � � D means that supp� � D; z 2 � (z =2 � resp.) signi�es z 2 supp� (z =2 supp� resp.); nrad� (r) = n� � D(r) � = X j�kj<r 1; r > 0; is a counting function of sequence �, i. e., nrad� (r) is the number of all points of this sequence from the disk D(r). Our treatment of the sequence di�ers from that commonly used one conside- ring the sequence to be a function of natural or integer argument. Two sequences � � C and � = f k0g � C are equal (we write � = �) if we have the equality n� = n� for the measures n� and n� from (1.1). In other words, we consider each sequence of points as a representative of the equivalence class which consists of the sequences with equal associate integer-valued measures (1.1). A sequence � includes a sequence � � C if n� 6 n�. In this case we write � � �, and � is a subsequence of the sequence �. The union � [ � (the intersection � \ � resp.) is de�ned by the equality n�[� = n� + n� (n�\� = minfn�; n�g resp.). Given � � �, the di�erence �n� is de�ned by the measure n�n� = n��n�. A sequence � consists of single points if n� � fzg � 6 1 for all z 2 C . If a numeration of the points of sequence � is important in principle, then we represent it as � = (�k), i. e., within round brackets. For a nonzero entire function f , by Zerof denote the sequence of zeros of this function counting multiplicities, i. e., nZerof � fzg � is equal to multiplicity of zero of f for every point z 2 C . 62 Journal of Mathematical Physics, Analysis, Geometry, 2007, v. 3, No. 1 Variation of Subharmonic Function under Transformation of its Riesz Measure In Levin's fundamental monograph the results about estimations of change of growth of entire function under shifts of its zeros [1, Ch. II, Lemmas 1, 4] played an important role in creating the theory of the functions of �nite order � of completely regular growth. We represent here only a more simple case of the noninteger order � in the form convenient for parallels with the subsequent results (in the original the formulation is given for the proximate order �(�)). Further images of the point z 2 C and the set D � C under mapping (trans- formation) T : C ! C are frequently written as Tz and TD. Theorem L ([1, Ch. II, Lemma 1]). Let f be an entire function with zero sequence Zerof = f�kg, k = 1; 2; : : : , having �nite density with respect to a non- integer order � > 0, i. e., there exists the �nite limit lim r!+1 nradZerof (r) r� : (1.2) Then, for every " > 0 and � > 0, we can select the number d > 0 such that, for each mapping T : Zerof ! C satisfying the conditions��T�k�� = j�kj; ��arg T�k � arg �k �� 6 d; k = 1; 2; : : : ; (1.3) there is an entire function fT with zero sequence ZerofT = TZerof := fT�kg =: f kg; k = 1; 2; : : : ; (1.4) for which the estimate ��log jfT (z)j � log jf(z)j �� 6 "jzj� holds for all z 2 C n E, where an exceptional set E = 1[ j=1 D(zj ; tj) (1.5) has the upper density 6 �, i. e., lim sup r!+1 1 r P jzj j<r tj 6 �. A.A. Gol'dberg showed in [2, � 6, Lemma] that this result holds if Zerof is only a sequence �nite upper density with respect to order � (or proximate order �(�)), i. e., we can substitute the superior limit for the limit in (1.2). Further more general quantitative results were obtained by I.F. Krasichkov- Ternovski�� [3] who applied them to the problem of spectral synthesis and the Journal of Mathematical Physics, Analysis, Geometry, 2007, v. 3, No. 1 63 E.G. Kudasheva and B.N. Khabibullin problems of completeness of exponential systems [4]. In paper [3] it was supposed, that zeros of entire function moved in arbitrary directions:���1� k �k ��� < d 6 1 2 for all su�ciently large k where T�k = k: (1.6) Then denote various positive constants by const+. Theorem K-T ([3]). Under the condition (1.6), for each entire function f with Zerof = f�kg of �nite upper density with respect to a noninteger order � > 0, there exists an entire function/ fT with ZerofT = TZerof = f kg such that for any �; � 2 (0; 1) and for a const+ independent of �, � 2 (0; 1), the estimate ��log jfT zj � log jf(z)j �� 6 const+ d1�� �� sin�� jzj� (1.7) holds for all z 2 C n E, where the exceptional set E from (1.5) has the upper density 6 �d� 2 . It is easy to see that condition (1.3) follows from (1.6) under the restriction��T�k�� = j�kj with constant 1;033d in place of d. Therefore Th. K-T implies Levin's theorem L by a non-complicated choice of constants �, �, d. A certain development of Th. K-T was obtained later in [5, Th. A]. In [6] V.S. Azarin gave a general subharmonic interpretation for the conception of shifts of zeros of entire function and obtained the result which was used for asymptotic approximation of subharmonic functions by the logarithm of modulus of entire function, for construction of entire function of completely regular growth on arbitrary closed system of rays, and also, as in [4, Cor. 4.3], for a decomposition of entire function into product of entire functions of the prescribed growth at the in�nity [7]. One more Azarin's result [8, � 5, Variational theorem] concerning the subjects studied in the paper is formulated in terms of convergence in the distribution space or, rather, in the language of the theory of limiting sets in the sense of V. S. Azarin. A.F. Grishin [9] used the precise methods in studying asymptotic behavior of the di�erence of subharmonic functions under shift of argument, i. e., the asymp- totic of ju(z + hz)� u(z)j under z !1 depending on h. The results can also be interpreted as an in�uence on change of subharmonic function of special variation of its Riesz measure generated by transformation of the complex plane of the form T : z 7! z + hz, z 2 C , for the �xed h (see the approach in [10], and with generalizations in [11], [12]). The edited version of A.F. Grishin's technique from his recent paper written together with T.I. Malyutina [13, Th. 6 etc.], might be helpful in a number of cases for studying the change of behavior of subharmonic 64 Journal of Mathematical Physics, Analysis, Geometry, 2007, v. 3, No. 1 Variation of Subharmonic Function under Transformation of its Riesz Measure (entire resp.) function under a su�ciently arbitrary variation of its Riesz measure (distribution of zeros resp.). We do not concern here numerous works in which interrelations are between shifts of zeros of entire functions and changes of behavior of these functions for the cases when very special shifts of zero or classes are examined as well as the works on the problems of approximation of subharmonic functions, stability of completeness, minimality, basis properties, etc. for the systems of functions in functional spaces. In part these works are marked in [14, 15]. 2. Subharmonic Interpretation According to V.S. Azarin Let us consider in details subharmonic interpretation of shift of zeros of entire function given by V.S. Azarin. Everywhere below in Sect. 2, for the mapping T : C ! C we assume the following two conditions: � this mapping T is Borel measurable; � the preimage T�1B of every bounded subset B � bC is also bounded in C . For � 2 M+(C ) its T -shift �T 2 M+(C ) or, in other words, the image T� of � under T (see [16, Ch. IV, � 6]) is de�ned by the rule �T (B) := � � T�1B � ; B � C is a Borel subset. (2.1) Thus we have the equalityZ B f d�T = Z T�1B f(Tz) d�(z) (2.2) for every Borel function f on the Borel subset B � C . Without loss of generality, we everywhere assume that supports of the measures do not contain 0. Following L. Schwarz [16, Ch. I, � 4], everywhere we understand the positivity of number, function, measure, etc. as > 0, and > 0 is a strict positivity ; we accept similar agreements also for the negativity and strict negativity. If, for a mapping (a function) f on the set X we have f(x) � a for all x 2 X, then we write �f � a on X�, and if it is not so, then �f 6� a on X�. A function f on a subset X of the real axis R is called increasing (strictly increasing resp.) if the inequality x1 < x2 for x1; x2 2 X implies f(x1) 6 f(x2) ( f(x1) < f(x2) resp.). Similarly, we distinguish decrease and strict decrease. Journal of Mathematical Physics, Analysis, Geometry, 2007, v. 3, No. 1 65 E.G. Kudasheva and B.N. Khabibullin Theorem A ([6, Main Lemma]). Let d : [0;+1) ! (0;+1) be a decreasing function satisfying lim sup t!+1 d(t) d(2t) < +1, and, for T : C ! C , ���1� Tz z ��� 6 d(jzj) for all z 2 C : (2.3) If � 2 M+ is a measure of �nite type with respect to a noninteger order � > 0, i. e., lim sup r!+1 �rad(r) r� < +1; and, by de�nition, p := [�] is an integer part of �, then for each subharmonic function u with the Riesz measure � there exists a subharmonic function uT with the Riesz measure �T such that for any number � 2 (0; 1= p 2), the estimate��uT (z)� u(z) �� 6 const+ jzj� �2 0 @ 1Z 0 d(jzjt) dt t1+p�� + +1Z 1 d(jzjt) dt t2+p�� 1 A (2.4I) + const+jzj�� log 1 � (2.4r) holds for all z 2 C nE, where the exceptional set E from (1.5) has the upper density 6 const+ � �. Here all three arising constants const+ do not depend on �; d. In particular, it follows from Th. A that for an entire function f with sequence of simple * zeros Zerof = f�kg of �nite upper density with respect to an order � > 0, under condition (2.3), there is an entire function fT with zero sequence (1.4), for which the estimate (2.4) with u := log jf j and uT := log jfT j holds outside the exceptional set (1.5) of the upper density 6 const+ � �. Under the conditions of Th. K-T for d(t) � d(0) =: d 2 (0; 1=2], the estimate (2.4) allows to replace the right-hand side of the estimate (1.7) by a somewhat more convenient quantity const+ � d �2 + � log 1 � � jzj� performing this estimate outside some exceptional set (1.5) of the upper density 6 const+�. Moreover, if we choose � = 3 p d and again replace 3 p d by d, it is possible to get rid of the parameters �; � without loss of pithiness: for any number d 2 (0; 1) the left-hand side of (1.7) in Th. K-T can be estimated from above by const+djzj� log(1=d) out- side the exceptional set of a kind of (1.5) of the upper density 6 const+d where the constant const+ does not depend on d. *It is easy to get rid of the restrictions on multiplicity of zeros by the general scheme from Sect. 3. 66 Journal of Mathematical Physics, Analysis, Geometry, 2007, v. 3, No. 1 Variation of Subharmonic Function under Transformation of its Riesz Measure The kind of the last summand in the right part of the estimate (2.4) shows that the order of the estimation juT �uj by Th. A cannot be less than jzj�, even if func- tion d from (2.3) decreases very quickly. It means that the conclusion of Azarin's theorem A "feels" the degree of closeness of measures � and �T insu�ciently. The incompleteness mentioned above is compensated in B.N. Khabibullin's paper [17]. Moreover, the estimations set up in this paper are in a certain degree the best possible ones on terms of behavior of function jTz � zj and cover a multivariate case as well. To formulate one of the versions of these results for the complex plane we will need additional designations. We put Æ(z) := ��Tz � z ��; z 2 C ; ÆT (z) := 8< : sup T�=z jz � �j; z 2 T C ; ÆT (z) := 0; z =2 T C : (2.5) For a measure � 2 M+ with supp� \ f0g = ?, we de�ne the characteristic (cf. (??I)) Kq(r; d�) : = rq r�0Z 0 d�rad(t) tq + r1+q +1Z r�0 d�rad(t) t1+q (2.6d) = qrq rZ 0 �rad(t) dt t1+q + (q + 1)rq+1 +1Z r �rad(t) dt t2+q (2.6p) = q 1Z 0 �rad(rt) dt t1+q + (q + 1) +1Z 1 �rad(rt) dt t2+q ; r > 0; (2.6c) where q should be a positive integer, not smaller than a genus of the measure �. Let us remind, that the genus of the measure � is the least integer q > 0 for which the second integral in (??d) or in (??p), (??c) is �nite. Denote by [�1;+1] an extended real axis equipped by natural order relation. In particular, �1 6 x 6 +1, x 2 [�1;+1]. Given a number " > 0, a function f : C ! [�1;+1], and a subset B � C , de�ne f (")(z) := sup � f(�) : � 2 D(z; "jzj) ; z 2 C ; B" := [ z2B D � z; "jzj � : (2.7) For 0 < " < 1, the ratios � f (") �(") (z) 6 f (3")(z); z 2 C ; � B" �" � B3" (2.8) are valid. Journal of Mathematical Physics, Analysis, Geometry, 2007, v. 3, No. 1 67 E.G. Kudasheva and B.N. Khabibullin Theorem Kh1 ([17, Theorem 2]). Let u be a subharmonic function with the Riesz measure �, 0 =2 supp �, and the measure � 2M+ is de�ned by the equality d� := Æ d� + ÆT d�T (2.9) in designations from (2.5). Let q > 0 be an integer such that q� 6 q where q� is a genus of the measure �. Then there exists a subharmonic function uT with the Riesz measure �T such that, for any Borel function N : C ! (1;+1) and for any " 2 (0; 1), the estimate ��u(z)� uT (z) �� 6 const+N (")(z) Kq � jzj; d� � "2jzj � log 2 + "jzj Kq � jzj; d� � ��(z; "jzj) + �T (z; "jzj) �! ; (2.10) with a constant const+ depending only on q, is ful�lled everywhere outside the exceptional set of a kind of (1.5) satisfying the inequalities X zj2B tj 6 Z B" dm(z) N(z)jzj ; tj 6 " 3 jzj j for all j = 1; 2; : : : , (2.11) and for every Borel subset B of C where m is a Lebesgue measure on C . If the function N depends only on jzj and its restriction on [0;+1) is in- creasing, then, by de�nitions and properties (2.7)�(2.8), it is easy to see that N (")(z) � N � (1 + ")jzj � , z 2 C . In that case, in view of the restrictions* tj 6 " 3 jzj j, j = 1; 2; : : : , the estimate of sum from (2.11) could be replaced by a more convenient and useful estimate X D(zj ;tj) T� D(R)nD(r) � 6=? tj 6 2� (1+3")RZ maxf(1�3")r;0g dt N(t) : (2.12) Let us apply Th. Kh1 to some development of Th. A. Note, that Azarin's theorem A is substantial only for small decreasing func- tion d. Therefore, without loss of generality, the restriction d(0) 6 1=2 can be added to its conditions. Hence the condition (2.3) in notations (2.5) implies *There are no such restrictions in the original formulation [17, Th. 2], but they are obviously present in the proof (see also dissertation [14] or its abstract [15]). 68 Journal of Mathematical Physics, Analysis, Geometry, 2007, v. 3, No. 1 Variation of Subharmonic Function under Transformation of its Riesz Measure Æ(z) 6 d(jzj)jzj and, as a consequence, ÆT (z) 6 const+d(jzj)jzj. The representa- tion and inequality (see de�nition (2.9) of the measure � from Th. Kh1) �rad(t) = tZ 0 Æ(s) d�rad(s) + tZ 0 ÆT (s) d� rad T (s) 6 const+t�+1; (2.13) show that its genus q� is not larger than q = p + 1 = [�] + 1. Hence, for the characteristic (2.6) in version (??d), a series of somewhat tiring integrations by parts and the change of variable allow to obtain the estimate Kq(r; d�) r 6 const+r� 0 @ 1Z 0 d(rt) dt t1+p�� + +1Z 1 d(rt) dt t2+p�� 1 A 6 const+r� 0 @ 1Z 0 d(rt) dt t1+p�� + 1 1 + p� � d(r) 1 A : (2.14) The function x log(2 + a=x) is increasing on [0;+1) when a > 0. Thus, if we choose " = 1=3 and put N(t) � 4�=�, t > 0, then, by Th. Kh1, the estimates (2.10)�(2.12) together with (2.14) entail the following: ��u(z) � uT (z) �� 6 const+ jzj� � 0 @ 1Z 0 d(jzjt) dt t1+p�� + 1 1 + p� � d(jzj) 1 A � log 0 BBB@2 + const+ �(z; jzj=3) + �T (z; jzj=3) jzj� � 1R 0 d(jzjt) dt t1+p�� + 1 1+p�� d(jzj) � 1 CCCA : Some weakening of the last estimation gives Corollary Kh1. Under the conditions of Th. A the estimate (2.4) can be replaced by ��uT (z)� u(z) �� 6 const+ jzj� � 0 @ 1Z 0 d(jzjt) dt t1+p�� + d(jzj) 1 A log � 2 + 1 d(jzj) � (2.15) outside the exceptional set (1.5) of the upper density 6 �. The estimate (2.15) is frequently more re�ned than (2.4) from Th. A. For example, if d(t) � (1 + t)��, t > 0, where 0 < � < � � [�], then the right-hand Journal of Mathematical Physics, Analysis, Geometry, 2007, v. 3, No. 1 69 E.G. Kudasheva and B.N. Khabibullin side of (2.15) can be easily estimated from above by the quantity O � jzj��� log jzj � , z ! 1, whose order of growth at in�nity is strictly less than for jzj� (compare with the comment directly ahead of (2.5)). Using of the genus q� of measure � in Th. Kh1 (below q� 6 [�� ]+ 1) allows to give considerably more general and improved version of Cor. Kh1. Corollary Kh2 ([14, Cor. 3.1]). Let u be a subharmonic function of �nite type with respect to the order � > 0 with the Riesz measure �. If, for a number 2 [0; � + 1] and a function ' : [0;1) ! [0; 1=2] [a;+1) that is decreasing on a ray [a;+1) where a > 0, and ' � 0 on [0; a), the mapping T : C ! C satis�es ���1� Tz z ��� 6 '(jzj)jzj1� ; z 6= 0; (2.16) then there exists a subharmonic function uT of the order 6 � with the Riesz measure �T such that for any increasing function N > 2 on [0;+1) the following estimate ��uT (z)� u(z) �� 6 const+ jzj�� 0 @ 1Z 0 '(jzjt) dt t1+[�� ]�(�� ) +' � jzj=2 � N � 2jzj � log � 2 + jzj ' � jzj=2 �� ! (2.17) is ful�lled outside the exceptional set of a kind of (1.5) satisfying X D(zj ;tj) T� D(R)nD(r) � 6=? tj 6 2RZ r=2 dt N(t) for all su�ciently large r < R: (2.18) Unfortunately, in Cor. Kh2 the function uT can have the in�nite type with respect to the order �. Finiteness of the type of this function can be provided due to any of the following three conditions (see [14, Remark 2, p. 57]): 1) � is a noninteger number; 2) > 0; 3) +1R 1 '(t) dt t < +1. Here is one more addition to Azarin's Theorem A. Corollary Kh3 ([14, Remark 1, p. 56]). Suppose that under conditions of the previous Cor. Kh2 the function ' from (2.16) is increasing on [a;+1), lim t!+1 log'(t) log t = 0, and 0 < 6 � + 1. Then we can draw the conclusions 70 Journal of Mathematical Physics, Analysis, Geometry, 2007, v. 3, No. 1 Variation of Subharmonic Function under Transformation of its Riesz Measure similar to those in Consequence 2, with the same increasing function N > 2, but with the estimate ��uT (z) � u(z) �� 6 const+ jzj�� 0 @ +1Z 1 '(jzjt) dt t2+[�� ]�(�� ) +' � 2jzj � N � 2jzj � log � 2 + jzj ' � 2jzj �� ! (2.19) outside the exceptional set of a kind of (1.5) satisfying (2.18). R e m a r k 1. The right-hand sides of the estimates (2.10) from Th. Kh1, (2.17) from Cor. Kh2, and (2.19) from Cor. Kh3 demonstrate that these results cannot give the order of closeness of the functions u and uT less than O(1=jzj) as z ! 1. Generally speaking, it is impossible to lower this order of closeness without additional conditions. For example, if u(z) � log jz � �j, z 2 C , and T takes � to 6= �, then it is not di�cult to understand that the function uT (z) � log jz � j is asymptotically most close to u. At the same time, for jzj > 2maxfj�j; j jg we have the estimate ��uT (z)� u(z) �� � ��log jz � j � log jz � �j �� > 2j�� j jzj : R e m a r k 2. As well as in the comment following Th. A, all previous sub- harmonic results can be considered as a statement on the change of growth of entire function f under transformation of the sequence of its zeros Zerof = f�kg in the sequence of zeros (1.4) of some entire function fT with the corresponding reformulations for u := log jf j and uT := log jfT j. Thus, for example, conditions (2.3) and (2.16) will be written as���1� k �k ��� 6 d(j�kj); ���1� k �k ��� 6 '(j�kj)j�kj1� ; k = 1; 2; : : : ; T�k := k: As the subharmonic results were formulated for mappings T , application of these results to the entire functions f is possible, generally speaking, only for the case when the sequences Zerof have no multiple (repeating) points. Indeed, if �k = �k0 are two points of � = Zerof , k 6= k0, but k = T�k 6= k0 = T�k0 , then such trans- formation of T is not mapping any more, whereas all the results of V. S. Azarin and B.N. Khabibullin were proved just for mappings T . This di�culty can be overcome in some ways. For example, one of them is to consider the multiple- valued mappings T , i. e., to do all reasonings and calculations once again with the probable complications at least of technical character. An alternative way is o�ered in Sect. 3. Journal of Mathematical Physics, Analysis, Geometry, 2007, v. 3, No. 1 71 E.G. Kudasheva and B.N. Khabibullin 3. Approximation by Entire Functions with Simple Zeros First, we de�ne the joint result of V.V. Napalkov and M.I. Solomeshch [18] which directly relates to our subject. The proof is in the dissertation by M.I. So- lomeshch [19]. Let f be an entire function with Zerof = (�k), 0 =2 Zerof represented by the Weierstrass canonical product f(z) = R(z) 1Y k=1 � 1� z �k � exp pk(z=�k); z 2 C ; (3.1) where R is an entire function without zeros, and pk, k = 1; 2; : : : , are polynomials. As well as in [18], considered is a sequence of points d = (dk) � C such that �k+dk 6= 0 for all k = 1; 2; : : : . Suppose that dk = dk0 in all cases when �k = �k0 . By (3.1) let us construct a formal product fd(z) = R(z) 1Y k=1 � 1� z �k + dk � exp pk(z=�k); z 2 C : (3.2) Let a family of disks D(�k; tk), tk > 0, k = 1; 2; : : : , such that tk = tk0 if �k = �k0 . Theorem N�S ([18, Prop. 1], [19, Props. 7�9]). In the assumed notations and agreements, let jdkj < tk for all k = 1; 2; : : : , and 1X k=1 jdkj tk < +1: (3.3) Then product (3.2) converges if z =2 E = S1 k=1D(�k; tk) and determines an ana- lytic function outside E, and for const+ we have��log jfd(z)j � log jf(z)j �� 6 const+; z 2 C n E: If each connected component of E is bounded, then product (3.2) converges to the entire function fd with Zerofd = (�k + dk), k = 1; 2; : : : . If connected components of the set E from Napalkov�Solomeshch's Theo- rem N�S are unbounded, then product (3.2), generally speaking, can diverge at points z 2 E. Thus, a condition on the connected components of set E in the last paragraph of Th. N�S is essential. Taking into account properties of the sequences (dk) and (tk) in relation to the sequence (�k), we de�ne an auxiliary notion. We say that a sequence (ak), k = 1; 2; : : : , is linked with a sequence (bk), k = 1; 2; : : : , if bk = bk0 implies ak = ak0 . In particular, according to the condition 72 Journal of Mathematical Physics, Analysis, Geometry, 2007, v. 3, No. 1 Variation of Subharmonic Function under Transformation of its Riesz Measure above the sequences (dk), (tk), and (�k + dk) are linked with the sequence (�k). By virtue of the last, Theorem of Napalkov�Solomeshch cannot be used to solve the main problem of Sect. 3 on the approximation of entire function by the entire function with simple zeros. More speci�cally, we cannot "split" multiple zeros of function f to simple zeros of function fd with the help of Th. N�S, because, by the construction of sequence Zerofd = (�k + dk), the coincident points �k = �k0 are transformed to the same points �k + dk = �k0 + dk0 from Zerofd . The main result of this paragraph was announced rather long ago in the paper [20], but its proof is given here for the �rst time. Theorem 1. Let f be an entire function with Zerof = (�k), k = 1; 2; : : : . For every given decreasing function � : [0;+1)! (0;+1) and number " > 0 we can �nd an entire function g with the sequence of simple zeros Zerog = ( k) and (tk) � (0;+1) that is linked with the sequence (�k) such that: 1) for �k 6= �k0, the disks D(�k; tk) and D(�k0 ; tk0) are not intersected; for r > 0, P j�kj>r tk 6 �(r); �nally, j k � �kj < tk for all k = 1; 2; : : : ; 2) the inequality ��log jg(z)j � log jf(z)j �� 6 " jzj2 (3.4) takes place for all z 2 C nS1 k=1D(�k; tk). P r o o f. First, we consider the case when (!) multiplicity of zeros of the function f at any point is an even number . In this case the sequence Zerof = f�kg =: � can be represented as the union � = �0 [ �00, where the sequences �0 = (�0k) and �00 = (�00k), k = 1; 2; : : : , such that �0k = �00k for each k = 1; 2; : : : . Now we choose a sequence of strictly positive numbers (tk) linked with (�0k) such that the disks D(�0k; tk) are mutually disjoint and for all r > 0, P j�0 k j>r tk 6 �(r), i. e., 1) is ful�lled. One can always do it as the imposed conditions are not mutually exclusive in the sense that both restrictions take only a su�cient rapid decrease of the sequence (tk). Given " > 0, we select strictly positive numbers dk 6 tk=2 so small that (cf. (3.3)) 1X k=1 dkj�0kj 6 " 26 ; X j�0kj>r dk t2k j�0kj 6 " 26r2 for all r > 0: (3.5) Journal of Mathematical Physics, Analysis, Geometry, 2007, v. 3, No. 1 73 E.G. Kudasheva and B.N. Khabibullin To each pair of the coincident points �0k = �k" we assign two diametrically opposite points 0k and 00k on the circumference @D(�0k; dk), i. e., 0k + 00k = 2�0k = 2�00k = �0k + �00k; j 0k � �0kj = j 00k � �00kj = dk: (3.6) Since every point of C coincides only with the �nite number of points �0k, it follows that we can construct the distinct diametrically opposite points ( 0k; 00 k ) so that j 0kj 6 j�0kj and (�) the union �0 := �0 [ �00 of the sequences �0 = ( 0k) and �00 = ( 00k ) consists of simple points and at the same time, by construction, � \ �0 = ?. Now we estimate the sum of di�erences �(z) := X k � log ��(z � 0k)(z � 00k) ��� log ��(z � �0k)(z � �00k) ��� for z =2 [ k D(�0k; tk): (3.7) Using (3.6), the identity Lk(z) := log ��(z � 0k)(z � 00k ) ��� log ��(z � �0k)(z � �00k) �� = log ���1� �0k� 00 k � 0k 00 k (z � �0k) 2 ��� implies an upper bound Lk(z) 6 log � 1 + j�0k�00k � 0k 00 k j jz � �0kj2 � 6 j�00kjj�0k � 0kj+ j 0kjj�00k � 00k j jz � �0kj2 6 2dkj�0kj jz � �0kj2 : (3.8) Similarly, it follows from �Lk(z) = log ���1 + �0k� 00 k � 0k 00 k (z � 0k)(z � 00k) ��� 6 log � 1 + j�0k�00k � 0k 00 k j jz � 0kjjz � 00k j � that, in view of (3.6), and for jz � �0kj > dk, �Lk(z) 6 2dkj�0kj� jz � �0kj � j�0k � 0kj �� jz � �00kj � j�00k � 00k j � 6 2dkj�0kj� jz � �0kj � dk �2 : But for jz � �0kj > tk > 2dk we have jz � �0kj � dk > jz � �0kj=2. Hence �Lk(z) 6 23dkj�0kj jz � �0kj2 for jz � �0kj > tk: The last estimate together with (3.8), (3.7) gives ���(z)�� 6X k jLk(z)j 6 X k 23dkj�0kj jz � �0kj2 for z =2 [ k D(�0k; tk) =: E: 74 Journal of Mathematical Physics, Analysis, Geometry, 2007, v. 3, No. 1 Variation of Subharmonic Function under Transformation of its Riesz Measure If we �x the point z =2 E, then ���(z)�� 6 0 @ X j�0kj<jzj=2 + X j�0kj>jzj=2 1 A 23dkj�0kj jz � �0kj2 6 X jz��0 k j>jzj=2 23dkj�0kj jz � �0kj2 + X j�0 k j>jzj=2 23dkj�0kj t2k : Here, using (3.5), we can estimate the �rst sum in the right-hand side as X jz��0kj>jzj=2 23dkj�0kj jz � �0kj2 6 X k 25dkj�0kj jzj2 6 " 2jzj2 ; and the second sum as X j�0 k j>jzj=2 23dkj�0kj t2k 6 " 23 � jzj=2 �2 6 " 2jzj2 : Thus, the last three estimates imply���(z)�� 6 " jzj2 for z =2 E = [ k D(�0k; tk): (3.9) For the case (!), our construction is �nished. Now, if the function f has zeros of odd multiplicity, then we represent the Zerof in the form Zerof = �0 [�, where �0 = f�0kg is a sequence of simple points and � is a sequence of points of even multiplicity , i. e., n� � fzg � is an even number for each point z 2 C . In this case we choose � := �0 [ �0 = ( k) =: Zerog, where the sequence �0 is constructed by the sequence � similarly to that one above. In view of (�), the sequence � consists only of simple points whereas the exceptional set E = S kD(�0k; tk) is identical to �. Besides, considering (3.9), for appropriate renumbering and denotation (if necessary) of the points in Zerof = �0[� := (�k), � := ( k), and (tk), we get���(z; �;�)�� := ���X k � log jz � kj � log jz � �kj ���� 6 " jzj2 (3.10) for z =2 SkD(�k; tk) = E where tk = 0 if the point �k is simple, i.e., n� � f�kg � = 1. To conclude the proof, we use the Weierstrass representation (3.1) of f and de�ne a function g in the form of product (cf. (3.2)) g(z) = R(z) 1Y k=1 k � z �k exp pk(z=�k); z 2 C ; (3.11) Journal of Mathematical Physics, Analysis, Geometry, 2007, v. 3, No. 1 75 E.G. Kudasheva and B.N. Khabibullin for which, according to (3.10), ��log jg(z)j � log jf(z)j �� � ���(z; �;�)�� 6 " jzj2 for z =2 E: Hence, using maximum-modulus principle for increasing sequence of bounded domains with the boundaries disjointed from E, we see that the product (3.11) is uniformly bounded on compacta. Therefore, by the Montel theorem the product (3.11) determines a desired entire function g with simple zeros satisfying (3.4) outside E. This completes the proof of Th. 1. R e m a r k 1. The polynomial f : z 7! z2 shows that the estimate (3.4) is unimprovable. Indeed, for any pair of di�erent points f 1; 2g there exists a constant const+ > 0 such that���log��(z � 1)(z � 2) ��� log jz2j ��� > const+ jzj2 for all jzj > 2max � j 1j; j 2j : R e m a r k 2. Theorem 1 completely solves the problem set in Remark 2 from Sect. 2 and even more, since under Remark 1 from Sect. 2 the highest possible closeness of functions has the order O(1=jzj). To conclude Sect. 3 we note without the proof the result similar to Th. N�S. It is obtained analogously to Th. 1. Theorem 2. Let f be an entire function with Zerof = (�k) =: �, k = 1; 2; : : : , and a sequence of strictly positive numbers (tk) is linked with �. Suppose that all connected components of E := S kD(�k; tk) are bounded. If, for a sequence (dk), 0 6 dk < tk, k = 1; 2; : : : (cf. (3.3)) 1X k=1 dk < +1; X j�kj>r dk tk = O(1=r); r ! +1; (3.12) then, for any sequence of points ( k) � C satisfying the inequalities j�k� kj 6 dk, k = 1; 2; : : : , there exists an entire function g with Zerog = ( k) and a constant const+ such that ��log jg(z)j � log jf(z)j �� 6 const+ jzj for all z 2 C n E: Evidently, if the sequence (tk) is bounded, then the convergence of the �rst sum from (3.12) follows from the convergence of the second sum from (3.12). 76 Journal of Mathematical Physics, Analysis, Geometry, 2007, v. 3, No. 1 Variation of Subharmonic Function under Transformation of its Riesz Measure 4. Integral Condition on T -shift. Main Result As usual, denote by N := f1; 2; : : : g and Z the sets of all natural numbers and all integers resp.; Z+ := f0g [ N. For q < s, by de�nition, Qq m=s � � � := 1,Pq m=s � � � := 0. Given q 2 Z+, the function Eq(z; �) := � 1� z � � qY m=1 exp zm m�m ; z 2 C ; � 2 C n f0g; is called the Weierstrass primary factors of genus q 2 Z+. The following special case of the classical Lindel�of theorem on an intercon- nection between the growth of entire function and the distribution of its zeros [1, Ch. I, � 11, Th. 15] relates to the sources of the main theorem of Sect. 4. This result was announced in [21]. Proposition 1. If, for a sequence � = f�kg � C , k 2 N, the sum P k2N 1 j�kj� is �nite for a number � > 0, then, for q := ( [�] := integer part of � if � is noninteger; �� 1 = [�]� 1 if � is integer; (4.1) the Weierstrass�Hadamard product W�(z) := Q1 k=1Eq(z; �k) of the genus q, z 2 C , is an entire function of zero type with respect to the order � with ZeroW = �. Given q 2 Z+, the function eq(z; �) := log jEq(z; �)j = log ���1� z � ���+ qX m=1 1 m Re zm �m ; z 2 C ; � 2 C nf0g; (4.2) is said to be the subharmonic Weierstrass kernel of genus q. If the function w�(z) := Z C eq(z; �) d�(�); z 2 C ; (4.3) with values in [�1;+1), is locally bounded above, then we may say that the function w� is a Weierstrass�Hadamard potential of genus q of measure �. A subharmonic version of Prop. 1 (particular case of [22, 4.2]) is Proposition 2. If 0 =2 supp � for � 2M+ and R C 1 j�j� d�(�) < +1, then w� is a subharmonic function of zero type with respect to the order � with the Riesz measure �. Journal of Mathematical Physics, Analysis, Geometry, 2007, v. 3, No. 1 77 E.G. Kudasheva and B.N. Khabibullin As the next step for further developing these facts, when � = 1, it is possible to consider the following theorem formulated here with some losses in a substantial and constructive parts in comparison with the original treatment. Theorem Kh2 ([23, Theorem 1]). Let f 6� 0 be an entire function of exponen- tial type with Zerof = (�k), and ( k) be a sequence of points of C , k = 1; 2; : : : . If the series X k��k 6=0 ��� 1 �k � 1 k ��� (4.4) converges, then there exists an entire function of exponential type g 6� 0 with Zerog = �, and with the same indicator function, as f . Natural expansion of the last result on entire functions of the �nite order � was announced in [24]. For the role of condition (4.4), the convergence of seriesX �k 6=0 1 j�kj� ���1� k �k ��� (4.5) was o�ered [24, Cor. 1]. There was also formulated a subharmonic version, but without the proof and in a weaker and less precise form than the main one sub- mitted in our paper. Theorem 3 (partial formulations in [24, Theorem], [21, Theorem]). Let � 2 M+ be a measure of �nite type with respect to the order � > 0, and T : C ! C be a Borel mapping such that the preimage of each bounded set is bounded. If lim inf z!1 jTzj jzj > 0; Z CnD(1) 1 j�j� ���1� T� � ��� d�(�) < +1; (4.6) then, for every subharmonic function u with the Riesz measure �, we can �nd a subharmonic function uT with the Riesz measure �T such that, for any number " > 0, there is an exceptional set E" � C of upper density 6 " for which��uT (z)� u(z) �� 6 "jzj� for all z 2 C n E": (4.7) In particular, for any " > 0,8>< >: uT (z) 6 sup j��zj6"jzj u(�) + "jzj� + const+ ; u(z) 6 sup j��zj6"jzj uT (�) + "jzj� + const+ ; jzj > 1: (4.8) In addition, if u is a function of �nite type with respect to the order � > 0, then the function uT is the same, and the indicator function of u coincides with the indicator function of uT . P r o o f. Fix " > 0. We will prove the theorem in a few steps. 78 Journal of Mathematical Physics, Analysis, Geometry, 2007, v. 3, No. 1 Variation of Subharmonic Function under Transformation of its Riesz Measure 1. Isolating of measure from origin. For any given R > 1, we may suppose that supp �\D(R) = ?. Indeed, if u is a subharmonic function with the Riesz measure �, then we can represent the function u in a form of sum u(z) = Z C log j� � zjd � � �� D(R) � (�) + uR(z) =: pR(z) + uR(z); z 2 C ; where � �� D(R) is a restriction of the measure � to the disk D(R) and uR is a subharmonic function with the Riesz measure � �� CnD(R) = � � � �� D(R) for which D(R) \ supp � � �� CnD(R) � = ?. The logarithmic potential pR satis�es condition [25, Th. 3.1.2] pR(z) = � �� D(R) (C ) log jzj+O(1=jzj) as z !1: (4.9) By (2.1)�(2.2) and by boundness of T�1D(R), under the conditions of theorem, the support of T -shift � � �� D(R) � T of the measure � �� D(R) is a compact set and� � �� D(R) � T (C ) = � �� D(R) (C ). For (pR)T z := Z C log j��zjd � � �� D(R) � T (�) = � � �� D(R) � T (C ) log jzj+O(1=jzj); z !1; in view of (4.9), we have pR(z)� (pR)T z = O(1=jzj) as z !1. The latter means that if functions uR and uRT satisfy (4.7)�(4.8), then the addition of logarith- mic potentials pR and (pR)T to them will give exactly (4.7)�(4.8) under possible increasing of the constant const+, if necessary. It follows from the �rst condition of (4.6) that for the number b > 0 we can choose a number R > 1 so large that bjzj 6 jTzj for all z =2 D(R): (4.10) The second condition from (4.6) impliesZ CnD(R) 1 j�j� ���1� T� � ��� d�(�) =: �(R)! 0 as R!1: (4.11) Now we de�ne more exactly a choice of R depending upon " and other parameters. We consider only that the number R > 1 is chosen so that inequality (4.10) takes place and, besides, supp � \ D(R) = ?: (4.12) Journal of Mathematical Physics, Analysis, Geometry, 2007, v. 3, No. 1 79 E.G. Kudasheva and B.N. Khabibullin Thus, obviously, for some number B > 0 the inequality �rad(t) 6 Bt� (4.13) holds for all t > 0. It is important to note that here the procedure of rejection of restriction of the measure � on the disk D(R) does not increase the constant B under increasing of R. Note also, that inclusion T�1D(t) � D(t=b) follows from (4.10) for all t > R. Hence, according to (2.1) we obtain �radT (t) = �T � D(t) � = � � T�1D(t) � 6 �rad(t=b): In particular, it means that under agreements (4.12)�(4.13) we have supp �T \D(bR) = ?; �radT (t) 6 B b� t� for all t > 0; (4.14) i. e., the measure �T has a �nite type with respect to the order �, and 0 =2 supp �T. 2. The main estimated integral . For q from (4.1), consider the integral* I(z) := Z C eq(z; �) d(�T � �)(�): (4.15) Our goal is to get the estimate jI(z)j 6 "jzj� for all z laying outside some excep- tional set of the upper density 6 ". We set D1=9 := � � 2 C : ���1� T� � ��� < 1 9 � ; �1=9 := � �� D1=9 (4.16) is a restriction of the measure � to the set D1=9, and � 1=9 T := (�1=9)T ; ~�1=9 := � � �1=9; ~� 1=9 T := (~�1=9)T = �T � (�1=9)T : (4.17) Taking into account (2.2), (4.12) and (4.14), we represent I(z) in the form of algebraic sum I(z) = Z C � eq(z; T �)� eq(z; �) � d�1=9(�) + Z C eq(z; �) d~� 1=9 T � � Z C eq(z; �) d~� 1=9(�) =: I1=9(z) + w ~� 1=9 T (z)� w~�1=9(z); (4.18) *Convergence (�niteness) of the integrals arising further for the points z outside some excep- tional set will follow from the estimates. 80 Journal of Mathematical Physics, Analysis, Geometry, 2007, v. 3, No. 1 Variation of Subharmonic Function under Transformation of its Riesz Measure where we use the notation (4.3) for the Weierstrass�Hadamard potentials w ~� 1=9 T and w~�1=9 of the measures ~� 1=9 T and ~�1=9 of genus q. By the second condition from (4.6), for � 2 C nD1=9, i.e., j1� (T�)=�j > 1=9, in view of agreement (4.10), for j�j > R we obtain bj�j 6 jT�j; 1 j�j� 6 9 j�j� ���1� T� � ���; 1 jT�j� 6 1 b�j�j� 6 9 b�j�j� ���1� T� � ���: Hence, for the restriction ~�1=9 of measure � from (4.17), in view of (4.12) and (4.6), we haveZ C 1 j�j� d~� 1=9(�) 6 9 Z CnD(R) 1 j�j� ���1� T� � ��� d�(�) < +1: Similarly, for T -shift ~� 1=9 T from (4.17), using (4.14), (4.12), and (4.6), we getZ C 1 j�j� d~� 1=9 T (�) = Z C 1 jT�j� d~� 1=9(�) 6 1 b� Z C 1 j�j� d~� 1=9(�) < +1: By Proposition 2, the �niteness of these two integrals implies that the Weierstrass� Hadamard potentials w~�1=9 and w ~� 1=9 T are subharmonic functions of zero type with respect to the order �. Proposition 3 (partial case of [12, Th. 2]). Let u be a subharmonic function on C , and N : [0;+1) ! [1;+1) be an increasing function. Then, for some absolute constants a1; a2, the inequality u(z) > �a1 � max j�j=2jzj u(�) � � log � a2N(jzj) � holds for all z 2 C n E0, where E0 is an exceptional set of the form (1.5) such that X jzj j<r tj 6 rZ 0 dt N(t) : If the function N increases to 1 su�ciently slowly, then the application of Prop. 3 to each of functions w~�1=9 and w ~� 1=9 T gives the relationships ��w~�1=9(z) �� + ��w ~� 1=9 T (z) �� = o � jzj� � as z 2 C n E0, z !1; (4.19) where E0 is some set of zero upper density. In other words, the set E0 is a C0-set [1]. Hence, going back to (4.18) for the integral I(z) from (4.15), the problem Journal of Mathematical Physics, Analysis, Geometry, 2007, v. 3, No. 1 81 E.G. Kudasheva and B.N. Khabibullin becomes simpler: we are to prove only the estimate jI1=9(z)j 6 " 2 jzj� outside the set of upper density 6 ". Further, for short, we designate I1=9(z) and � 1=9 as I(z) and �. By de�nitions (4.16) and (4.17), for the proof of estimate jI(z)j 6 " 2 jzj�, we may suppose that the mapping T also satis�es (together with (4.10)�(4.11)) ���1� T� � ��� < 1 9 ; and, hence, also 8 9 j�j 6 jT�j 6 10 9 j�j for all � 2 S� ; (4.20) where S� � C is a supporting set of the measure �. 3. The integral I(z). Let us rewrite the integral I(z) from (4.15) by the rule (2.2), taking into account the de�nition of subharmonic Weierstrass kernel of genus q from (4.2), in the following form: I(z) = Z C � eq(z; T �)� eq(z; �) � d�(�) = Z j�j>4jzj � eq(z; T �)� eq(z; �) � d�(�) + Z j�j<4jzj qX m=1 1 m Re � zm (T�)m � zm �m � d�(�) + Z D(4jzj)nD(z;jzj=2) � log ���1� z T� ���� log ���1� z � ���� d�(�) + Z D(z;jzj=2) � log ���1� z T� ���� log ���1� z � ���� d�(�) =: J1(z) + J0(z) + L(z) + L0(z): (4.21) 3.1. An estimate of the integral J1(z). Under the condition jT�j > 8j�j=9 from (4.20), we use the expansion in series at j�j > 4jzj for eq(z; T �)� eq(z; �) = � 1X m=q+1 1 m Re � zm (T�)m � zm �m � : It implies ��eq(z; T �)� eq(z; �) �� 6 1X m=q+1 jzjm m ��� 1 (T�)m � 1 �m ���: (4.22) 82 Journal of Mathematical Physics, Analysis, Geometry, 2007, v. 3, No. 1 Variation of Subharmonic Function under Transformation of its Riesz Measure For any � 6= 0 and T� 6= 0, we have ��� 1 (T�)m � 1 �m ��� = jT� � �j � ���m�1P k=0 (T�)m�1�k�k ��� jT�jmj�jm ; and also ��� 1 (T�)m � 1 �m ��� 6 jT� � �j � m � maxfjT�j; j�jg �m�1 jT�jmj�jm : Hence, under the conditions (4.20), for each � 6= 0, T� 6= 0, m > 1, we have ��� 1 (T�)m � 1 �m ��� 6 jT� � �j � m � maxfjT�j; j�jg �m jT�jmj�jm+1 6 ���1� T� � ��� � m2m j�jm : (4.23) Using (4.22), for j�j > 4jzj we obtain ��eq(z; T �)� eq(z; �) �� 6 ���1� T� � ��� 1X m=q+1 2mjzjm j�jm 6 ���1� T� � ��� 2q+2jzjq+1j�jq+1 and ��J1(z) �� 6 Z j�j>4jzj ��eq(z; T �)� eq(z; �) �� d�(�) 6 Z j�j>4jzj ���1� T� � ��� 2q+2jzjq+1j�jq+1 d�(�) = jzj� Z j�j>4jzj 2q+2jzjq+1�� j�jq+1�� � 1 j�j� ���1� T� � ���� d�(�) for all z 2 C : By the de�nition from (4.1), for q we have q + 1 � � > 0. Therefore, for all j�j > 4jzj, 2q+2jzjq+1�� j�jq+1�� 6 2q+2 4q+1�� = 22��q 6 4�: So, we obtain the �nal estimate ��J1(z) �� 6 4�jzj� Z j�j>4jzj 1 j�j� ���1� T� � ���d�(�) for all z 2 C : (4.24) Journal of Mathematical Physics, Analysis, Geometry, 2007, v. 3, No. 1 83 E.G. Kudasheva and B.N. Khabibullin 3.2. An estimate of the integral J0(z) from (4.21). Using (4.23) to the integration element of J0(z), for any 0 < j�j < 4jzj and T� 6= 0, we have ����� qX m=1 1 m Re � zm (T�)m � zm �m � d�(�) ����� 6 qX m=1 jzjm m ��� 1 (T�)m � 1 �m ���d�(�) 6 ���1� T� � ��� qX m=1 2mjzjm j�jm = jzj� 1 j�j� ���1� T� � ��� qX m=1 2m � j�j=jzj ���m : Besides, by de�nition of q from (4.1), for all m = 1; 2; : : : ; q, we obtain ��m > 0, and j�j=jzj < 4. Hence����� qX m=1 1 m Re � zm (T�)m � zm �m � d�(�) ����� 6 22� � 1� 1 2q � jzj� 1 j�j� ���1� T� � ���: So, we obtain the �nal estimate jJ0(z)j 6 4�jzj� Z j�j<4jzj 1 j�j� ���1� T� � ���d�(�) for all z 2 C : (4.25) 3.3. An estimate of the integral L(z) from (4.21). For the integration element of the integral L(z), we have the following identity: lT (z; �) := log ���1� z T� ���� log ���1� z � ��� � log ����1 + z� � � z � �1 � � 1 T� ����� ; (4.26) �T � 6= 0. Let us estimate above the right-hand side for � 2 D(4jzj) nD(z; jzj=2), i. e., for 0 < j�j < 4jzj; j� � zj > 1 2 jzj; T � 6= 0: (4.27) Under these conditions, considering the inequality jT�j > 8j�j=9 from (4.20), in view of (4.26), we get lT (z; �) 6 log � 1 + jzjj�j j� � zj � ���1 � � 1 T� ���� 6 jzjj�j j� � zjjT�j � ���1� T� � ��� 6 2j�j jT�j � ���1� T� � ��� 6 3 ���1� T� � ��� 6 3 � 4�jzj� 1 j�j� ���1� T� � ���: (4.28) Under the same conditions, using the identity (4.26), we estimate above �lT (z; �) � log ����1 + z � T� T� � z � � 1 T� � 1 � ����� 6 jzj jT� � zj � ���1� T� � ���: (4.29) 84 Journal of Mathematical Physics, Analysis, Geometry, 2007, v. 3, No. 1 Variation of Subharmonic Function under Transformation of its Riesz Measure In view of (4.20), we obtain j� �T�j < j�j=9. Hence, under the conditions (4.27), jT� � zj > j� � zj � j� � T�j > 1 2 jzj � 1 9 j�j > 1 2 jzj � 4 9 jzj = 1 18 jzj: Thus, we can extend (4.29) just as (4.28): �lT (z; �) 6 18 � ���1� T� � ��� 6 18 � 4�jzj� 1 j�j� ���1� T� � ���: The last one together with (4.28) gives a �nal estimate for the module of the integral jL(z)j 6 Z D(4jzj)nD(z;jzj=2) ��lT (z; �)�� d�(�) 6 18 � 4�jzj� Z j�j<4jzj 1 j�j� ���1� T� � ��� d�(�) for all z 2 C . Hence, using (4.25) and (4.24), under condition (4.11), we get an intermediate estimate jI(z)j 6 19�4�jzj� Z C 1 j�j� ���1�T� � ���d�(�)+jL0(z)j = 19�4�jzj��(R)+jL0(z)j (4.30) for all z 2 bC after simpli�cations of items 1 and 2. The required estimate for the module of L0(z) = Z D(z;jzj=2) lT (z; �) d�(z); (4.31) with lT from (4.26), is possible only outside some exceptional set constructed below. 4. Normal points. Let us give a variant of the de�nition of normal points. De�nition ([17, � 2]). Let f > 0 be a Borel function on C , f > 0 on � 2M+. Let d : C ! (0; 1=2] be a Borel function on C . We shall say that z 2 C is (f; d)- normal with respect to � if �(z; t) 6 f (d)(z) t for all t 6 d(z)jzj where f (d)(z) := sup j��zj6d(z)jzj f(�): (4.32) A partial case of [17, Normal Points Lemma] is Lemma. A set of points z 2 C that are not (f; d)-normal with respect to the measure � 2M+ is contained in a union of the countable set of the disks D(zj ; tj), j = 1; 2; : : : , such that for any �-measurable set D � CX zj2D tj 6 a Z Dd d� f and tj 6 d(zj)jzj j for all j 2 N, (4.33) Journal of Mathematical Physics, Analysis, Geometry, 2007, v. 3, No. 1 85 E.G. Kudasheva and B.N. Khabibullin where a is an absolute constant (we can choose a = 18), and by de�nition Dd :=S z2D D(z; d(z)jzj). In our proof we choose d � 1=2, and f(z) � M jzj��1 for all z 6= 0 where M > 0 is a constant and then we consider the subset E � C of points z that are not (f; d)-normal with respect to � or �T . In other words, z 2 C nE if maxf�(z; t); �T (z; t)g 6 f (1=2)(z) � t = Mc�jzj��1t for all 0 < t 6 jzj=2; (4.34) where c� := maxf3��1;1g 2��1 . By Lemma, the set E can be covered with the disks D(zj ; tj), j 2 N, such that, according to (4.33), we have tj 6 jzj j=2, j 2 N, and X jzj j6r tj 6 a Z � D(r) �1=2 d(� + �T )(z) M jzj��1 = a M 3r=2Z 0 d � �rad(t) + �radT (t) � t��1 : Hence, in view of (4.12)�(4.13), (4.14), using integration by parts, we obtain X jzj j6r tj 6 a M 0 B@�2 3 ���1 �rad(3r=2) r��1 + (�� 1) 3r=2Z 0 �rad(t) dt t� 1 CA + a M 0 B@�2 3 ���1 �radT (3r=2) r��1 + (�� 1) 3r=2Z 0 �radT (t) dt t� 1 CA = 3aB 2M � 1 + 1 b� � � r; r > 0: We choose M > 0 such that the multiplier in front of r is so large that it does not exceed "=6, i. e., X jzj j6r tj < " 6 � r for all r > 1: (4.35) Further, for short, we call the set E constructed here an exceptional set , and the points from C n E normal points. It is signi�cant that the �screening-out of a part� of the measures � and �T under increasing R in 1 does not change conclusions of this item, since the constants B and M , as well as the disks D(zj ; tj) are not changed for all R > 1. Besides, repeating word by word standard reasonings from the �nishing part of [23, item 1)], we can conclude that for any point z0 2 C , jz0j > r0, there exists a number �(z0) 2 (0; ") such that the circumference @D � z0; �(z0)jz0j � contains only normal points. 86 Journal of Mathematical Physics, Analysis, Geometry, 2007, v. 3, No. 1 Variation of Subharmonic Function under Transformation of its Riesz Measure 5. An estimate of the integral L0(z) from (4.31). First, using (4.26) for lT , we estimate the upper bound . For j��zj < jzj=2, under the condition (4.20), we have jT�j > 8j�j=9. Hence lT (z; �) 6 log � 1 + jzjj�j j� � zjjT�j ���1� T� � ���� 6 log � 1 + 9jzj=8 j� � zj ���1� T� � ���� : For M and c� from (4.34), we choose a parameter c > 0 so small that Mc� � c+ c log � 1 + 1 2c �� 6 " 8 and simultaneously c �5 3 �� B b� 6 " 8 : (4.36) Then we write down a previous estimate of lT in a somewhat weakened form lT (z; �) 6 log � 1 + 2jzjc j� � zj � 1 c ���1� T� � ���� 6 log � 1 + 2jzjc j� � zj � + log � 1 + 1 c ���1� T� � ���� 6 log � 1 + 2jzjc j� � zj � + 1 c ���1� T� � ���: (4.37) Integrating this inequality with respect to � over D(z; jzj=2), by (4.31) we obtain L0(z) 6 Z D(z;jzj=2) log � 1 + 2jzjc j� � zj � d�(�) + 1 c Z D(z;jzj=2) ���1� T� � ���d�(�) = jzj=2Z 0 log � 1 + 2jzjc t � d�(z; t) + 1 c Z D(z;jzj=2) j�j� 1 j�j� ���1� T� � ���d�(�) 6 jzj=2Z 0 log � 1 + 2jzjc t � d�(z; t) + 2� c jzj��(R); (4.38) where �(R) is a notation for the integral from (4.11). We estimate the integral in the right-hand side of (4.38) only for normal points z. Integration by parts gives jzj=2Z 0 log � 1 + 2jzjc t � d�(z; t) = log(1 + 4c)�(z; jzj=2) + 2cjzj jzj=2Z 0 �(z; t) dt t(t+ 2cjzj) 6 log(1 + 4c)�(z; jzj=2) + 2cjzj jzj=2Z 0 �(z; t) dt t(t+ 2cjzj) 6 4c �Mc�jzj�=2 Journal of Mathematical Physics, Analysis, Geometry, 2007, v. 3, No. 1 87 E.G. Kudasheva and B.N. Khabibullin +2cjzj jzj=2Z 0 Mc�jzj��1 dt t+ 2cjzj = 2Mc�jzj� � c+ c log � 1 + 1 4c �� : (4.39) Hence, in view of (4.36), (4.38) implies L0(z) 6 " 4 jzj� + 2� c jzj��(R); z 2 C nE; (4.40) where E is an exceptional set from item 4. Now we establish a lower bound . The identity (4.29) with the parameter c, as in (4.37), gives �lT (z; �) 6 log � 1 + jzj jT� � zj � ���1� T� � ���� 6 log � 1 + jzjc jT� � zj � + log � 1 + 1 c ���1� T� � ���� 6 log � 1 + cjzj jT� � zj � + 1 c ���1� T� � ��� Integrating this inequality with respect to � over D(z; jzj=2), similarly to (4.38), we obtain �L0(z) 6 Z D(z;jzj=2) log � 1 + cjzj jT� � zj � d�(�) + 2� c jzj��(R): (4.41) If � 2 D(z; jzj=2), then by (4.20) we have jT� � �j < j�j=9 6 jzj=6 and jT� � zj 6 jT� � �j+ j� � zj 6 1 6 jzj+ 1 2 jzj 6 2 3 jzj: It means that inclusion D(z; jzj=2) � T�1D(z; 2jzj=3) is ful�lled. Therefore, for the integral from the right-hand side of (4.41) we can getZ D(z;jzj=2) log � 1 + cjzj jT� � zj � d�(�) 6 Z T�1D(z;2jzj=3) log � 1 + cjzj jT� � zj � d�(�): By (2.2) and (4.14), for normal points z we obtainZ D(z;jzj=2) log � 1 + cjzj jT� � zj � d�(�) 6 Z D(z;2jzj=3) log � 1 + cjzj j� � zj � d�T (�) = 2jzj=3Z 0 log � 1 + cjzj t � d�T (z; t) = 0 B@ jzj=2Z 0 + 2jzj=3Z jzj=2 1 CA log � 1 + cjzj t � d�T (z; t) 88 Journal of Mathematical Physics, Analysis, Geometry, 2007, v. 3, No. 1 Variation of Subharmonic Function under Transformation of its Riesz Measure 6 jzj=2Z 0 log � 1 + cjzj t � d�T (z; t) + log(1 + 2c)�radT (5jzj=3) 6 log(1 + 2c)�T (z; jzj=2) + cjzj jzj=2Z 0 �T (z; t) dt t(t+ cjzj) + 2c �5 3 �� B b� jzj�: Hence, for normal point z 2 C nE, in view of (4.34), in the same way as in (4.39) we get Z D(z;jzj=2) log � 1 + cjzj jT� � zj � d�(�) 6Mc�jzj� � c+ c log � 1 + 1 2c �� + 2c �5 3 �� B b� jzj�: Then, in view of (4.36), (4.41) implies � L0(z) 6Mc�jzj� � c+ c log � 1 + 1 2c �� + 2c �5 3 �� B b� jzj� + 2� c jzj��(R) 6 " 8 jzj� + " 4 jzj� + 2� c jzj��(R) = 3" 8 jzj� + 2� c jzj��(R): The above and (4.40) give the �nal estimate jL0(z)j 6 3" 8 jzj� + 2� c jzj��(R); z 2 C nE: Thus, by (4.30), we have jI(z)j 6 �(R) � 19 � 4� + 2� c � jzj� + 3" 8 jzj�: As it was said at the end of item 4, we can increase the number R > 1 from 1 without any limits . Considering (4.11), we can choose R to be so large that �(R) � 19 � 4�+2�=c � 6 "=8. Then jI(z)j 6 " 2 jzj� for z 2 C nE. But remembering the arrangements given at the end of item 2, we say that I(z) is I1=9(z) from (4.18). For the initial integral I(z) from (4.15), in view of (4.18) and (4.19), we obtain jI(z)j 6 "jzj� for all normal points z 2 C n (E [E0) where E0 is a C0-set. By (4.35), the set E" := E[E0 has the upper density < "=6. Then the concluding remark of item 4 remains in force in the following form: for any point z0 2 C , jz0j > 1, there exists a number �(z0) 2 (0; ") such that @D � z0; �(z0)jz0j � \E" = ?. Journal of Mathematical Physics, Analysis, Geometry, 2007, v. 3, No. 1 89 E.G. Kudasheva and B.N. Khabibullin 6. From the integral I(z) to the functions u and uT . Let u be a function from Theorem 3. By item 1, we can assume that 0 =2 � and 0 =2 �T . Since the Riesz measure � has a �nite type with respect to the order �, the function u admits the Weierstrass�Hadamard representation of genus p = [�] with a harmonic addition hu (see [22, 4.2]): u(z) = Z C ep(z; �) d�(�) + hu(z); z 2 C : (4.42) Here q from (4.1) is connected with p by the rule q = ( p = [�]; if � is noninteger; �� 1 = p� 1; if � is integer: Now we set uT (z) := Z C ep(z; �) d�T (�) + hu(z) + v�(z); z 2 C ; (4.43) where, by de�nition, v�(z) � 0 if � is noninteger , and v�(z) := Re z� � Z C � 1 �� � 1 (T�)� � d�(�) if � is integer; z 2 C : (4.44) The function uT is well-de�ned. First, the integral from (4.43) is a Weierstrass� Hadamard potential of the measure �T of �nite type with respect to the order � (see (4.14) and [22, 4.2]), and, second, for the integer � the integral from (4.44) is �nite. Indeed, it follows from (4.23) that��� Z j�j>R � 1 �� � 1 (T�)� � d�(�) ��� 6 �2� Z j�j>R ���1� T� � ��� 1 j�j� d�(�); where, by (4.6), the right-hand side tends to 0 as R ! +1. By construction (4.44), the function v� is harmonic. Therefore, by construction (4.43), uT is a subharmonic function with the Riesz measure �T . From the form (4.2) of the subharmonic Weierstrass kernels of genus q and p and the representations (4.42)�(4.44) for u and uT it follows that uT (z)� u(z) � Z C eq(z; �) d(�T � �)(�) = I(z); where I(z) is the integral from (4.15). Therefore, by item 5, for any " > 0 we get��uT (z)� u(z) �� = jI(z)j 6 "jzj� for all z 2 C n E"; (4.45) 90 Journal of Mathematical Physics, Analysis, Geometry, 2007, v. 3, No. 1 Variation of Subharmonic Function under Transformation of its Riesz Measure where the exceptional set E" has the upper density 6 ". Thereby, the main part of Th. 3 is proved. By principle of the maximum for subharmonic functions, the relations (4.8) follow from (4.45) and concluding remark of item 5 on circumferences outside E". Finally, the last assertion on the type and the indicator function of uT is an evident consequence of (4.8). This completes the proof of Th. 3. R e m a r k 3. By representations (4.43)�(4.44), our construction of the function uT by u is completely constructive. R e m a r k 4. If we replace the �rst condition from (4.6) by its nonasymptotic analog inf �2C jT�j=j�j > 0, then there implies a condition �. . . the preimage of any bounded set under T is bounded�. In conclusion, we give a version of Th. 3 for entire functions. Theorem 4 (partial formulations in [24, Corollary 1]). Let f 6� 0 be an entire function with zero set Zerof = (�k) =: � of the �nite upper density with respect to the order � > 0. Let � = ( k) � C , k 2 N, be a sequence. If the series (4.5) converges and lim inf k!1 j kj=j�kj > 0, then there exists an entire function g with Zerog = � such that for any number " > 0 there is a set E" � C of the upper density 6 " that��log jg(z)j � log jf(z)j �� 6 "jzj� for all z 2 C n E": If the function f is of a �nite type with respect to the order �, then the function g is of the same type, and the indicator function (with respect to �) of function f coincides with the indicator function (with respect to �) of function g. P r o o f. By Theorem 1, there exists an entire function ~f with the sequence of simple zeros Zero~f = (~�k) =: ~� such that��log j ~f(z)j � log jf(z)j �� 6 const+ for all z 2 C n ~E; (4.46) where the set ~E is covered with the disks having a �nite sum of radii, and j~�k � �kj 6 1 for all k 2 N (� � 1 chosen su�ciently). Then the condition (4.5) holds if we replace � by ~�, because j~�k=�kj ! 1 as k ! +1 if � is in�nity. Therefore, for su�ciently large k0 2 N, X k>k0 1 j~�kj� ���1� k ~�k ��� 6 max k>k0 ��� ~�k �k �����1 ! X k>k0 1 j�kj� j�k � kj+ j~�k � �kj j�kj 6 const+ 0 @X k>k0 1 j�kj� ���1� k �k ���+ X k>k0 1 j�kj�+1 1 A ; lim inf k!1 j kj j~�kj > 0: (4.47) Journal of Mathematical Physics, Analysis, Geometry, 2007, v. 3, No. 1 91 E.G. Kudasheva and B.N. Khabibullin Here the penultimate sum converges by condition, and the last sum converges since the sequence � has the �nite upper density with respect to �. Now we can consider a mapping T : C ! C that is de�ned by rule T ~�k = k and Tz � z for z =2 ~�. Then, by de�nition (1.1), the integer-valued measure n� is a T -shift of the integer-valued measure n~� , i. e., n� = (n~� )T . The convergence of the �rst sum in (4.47) and the last relation in the same place mean that the measure � := n~� satis�es the �rst and the second conditions from (4.6). Besides, the last relation in (4.47) guarantees that the preimage of each bounded set is bounded. Therefore, by Th. 3, for the subharmonic function u := log j ~f j there exists a subharmonic function uT with the Riesz measure n� such that (4.7) holds, as well as the rest of conclusions of Th. 3. Hence there is an entire function g with Zerog = � such that uT = log jgj, as the Riesz measure n� of function uT is an integer-valued measure. The last together with (4.46) completes the proof of Th. 4. The Authors express their deep gratitude to the reviewer of the paper for important remarks and amendments. References [1] B.Ya. Levin, Distribution of the Zeros of Entire Function. GITTL, Moscow, 1956, p. 632 (Russian) (Engl. transl.: AMS, Providence, RI, 1964.) [2] A.A. Gol'dberg, The Integral with Respect to a Semiadditive Measure, and its Application to the Theory of Entire Functions. IV. � Mat. Sb. 66(108) (1965), 411�457. (Russian) (Engl. transl.: AMS Transl. (2) 88 (1964).) [3] I.F. Krasichkov[-Ternovski��], Comparison of Entire Functions of Finite Order by Means of the Distributions of their Zeros. � Mat. Sb. 70(112) (1966), 198�230; 71(113) (1966), 405�419. (Russian) [4] I.F. Krasichkov-Ternovski��, Invariant Subspaces of Analytic Functions. I, II. � Mat. Sb. 87(129) (1972), No. 4, 459�489; 88(130) (1972), No. 1, 3�30. (Russian) (Engl. transl.: Math. USSR Sb. 16 (1972).) [5] B.N. Khabibullin, Decomposition of Entire Functions of Finite Order into Equivalent Factors. � In: Problems of Approximation for Functions of Real and Complex Variable. Bashkir Branch of AS USSR, Ufa (1983), 161�181 (Russian) (Engl. transl.: Ten Papers in Russian. Ser. AMS, 142 (1989), 61�72.) [6] V.S. Azarin, On Rays of Completely Regular Growth of Entire Function. � Mat. Sb. 79(121) (1969), 463�476. (Russian) 92 Journal of Mathematical Physics, Analysis, Geometry, 2007, v. 3, No. 1 Variation of Subharmonic Function under Transformation of its Riesz Measure [7] V.S. Azarin, On the Decomposition of an Entire Function of Finite Order into Factors Having Given Growth. � Mat. Sb. 90(132) (1973), 225�230. (Russian) (Engl. transl.: Math. USSR Sb. 19 (1973).) [8] V.S. Azarin, On Asymptotic Behavior of Subharmonic Functions of Finite Order. � Mat. Sb. 108(150) (1979), 147�167. (Russian) (Engl. transl.: Math USSR Sb. 36 (1980).) [9] A.F. Grishin, Regularity of Growth of Subharmonic Functions. � Teor. Funkts., Funkts. Anal. i Prilozh. 6 (1968), 3�29; 7 (1969), 59�84; 8 (1969), 32�48. (Russian) [10] I.F. Krasichkov-Ternovski��, On Homogeneity Properties of Entire Functions of Fi- nite Order. � Mat. Sb. 72(114) (1967), 412�419. (Russian) (Engl. transl.: Math. Sb. USSR 1 (1967).) [11] B.N. Khabibullin, Lower Estimates and Properties of Homogeneity of Subharmonic Functions. Dep. VINITI, USSR, (1984), No. 1604-84, 34 p. (Russian) [12] B.N. Khabibullin, Lower Estimates and Properties of Homogeneity of Subharmonic Functions. � In: Investigations on the Theory of Approximation of Functions. Bashkir Branch of AN USSR, Ufa, 1984, 148�159. (Russian) [13] A.F. Grishin and T.I. Malyutina, New Formulas for Indicators of Subharmonic Functions. � Mat. �z., analiz, geom. 12 (2005), No. 1, 25�72. (Russian) [14] B.N. Khabibullin, Distribution of Zeros of Entire Functions and the Balayage. � Dis. . . . Doct. Ph.-Math. Sci., ILTPhE, Kharkov (1993), 322 p. (Russian) [15] B.N. Khabibullin, Distribution of Zeros of Entire Functions and the Balayage. � Dis. . . . Doct. Ph.-Math. Sci., Ufa (1993), 18 p. (Russian) [16] L. Schwartz, Analysis. V. I. Nauka, Moscow (1967). (Russian) [17] B.N. Khabibullin, Comparison of Subharmonic Functions with Respect to their Associated Measures. � Mat. Sb. 125(167) (1984), No. 4(12), 522�538 (Russian) (Engl. transl.: Math. USSR Sb. 53 (1986).) [18] V.V. Napalkov and M.I. Solomeshch, Estimate of Entire Function under Shifts of its Zeros. � Dokl. Akad. Nauk USSR 342 (1995), No. 6, 739�741. (Russian) [19] M.I. Solomeshch, Convolution type operators in some spaces of analytic functions. Dis. . . . Cand. Ph.-Math. Sci., Ufa (1995), p. 110. (Russian) [20] B.N. Khabibullin, Best Approximation of Entire Function by Entire Function with Simple Zeros. � In: Abstr. of Reports on Conference of Young Scientists. Ufa, 1985, 177. (Russian) [21] B.N. Khabibullin and E.G. Kudasheva, Variations of Entire (Subharmonic) Function under Perturbations of its Zero Set (Riesz Measure). � In: Abstr. of the Conference dedicated to the centennial of B.Ya. Levin. Kharkov, August 14�17, 2006, 20�21. Journal of Mathematical Physics, Analysis, Geometry, 2007, v. 3, No. 1 93 E.G. Kudasheva and B.N. Khabibullin [22] W. Hayman and P. Kennedy, Subharmonic Functions. Mir, Mocow (1980). (Rus- sian) [23] B.N. Khabibullin, An Approximation Theorem for Entire Functions of Exponential Type and the Stability of Zero Sequences. � Mat. Sb. 195 (2004), No. 1, 143�156. (Russian) [24] B.N. Khabibullin, Closeness of Subharmonic and Entire Functions, Stability of Completeness of Exponential Systems, Spectral Synthesis. � In: Second Int. Conf. �Mathematical Analysis and Economics� (Book of Abstracts), Sumy, Kharkiv, Kyiv, 2003, 24�25. [25] T.J. Ransford, Potential Theory in the Complex Plane. Cambridge Univ. Press, Cambridge, 1995. 94 Journal of Mathematical Physics, Analysis, Geometry, 2007, v. 3, No. 1

Ähnliche Einträge