13Известия Кабардино-Балкарского научного центра РАН № 2 (106) 2022
ФИЗИКО-МАТЕМАТИЧЕСКИЕ НАУКИ
UDC 517.55+517.33
DOI: 10.35330/1991-6639-2022-2-106-17-21 MSC: 32A10; 32A37 EDN: DCMIAY
Original article
SOME REMARKS ON BLASCHKE TYPE PRODUCTS IN AREA NEVANLINNA TYPE SPACES IN THE UNIT DISK
1 ' 1 * R. SHAMOYAN 1, O. MIHIC 2
1 Bryansk State Academician I.G. Petrovski University 241036, Russia, Bryansk, 14 Bezhitskaya street 2 University of Belgrade 11000, Serbia, Belgrade, Student Square 1
Abstract. The intention of this paper is to introduce and study certain new analytic spaces in disk and to show that certain Blaschke type products belong to such so called new large Nevanlinna type classes in the unit disk. These results extend and complement some previously known assertions of this type obtained earlier by other authors. Our result may be used to get parametric representations of these large spaces of area Nevanlinna type via infinite Ba products.
Keywords: Blaschke type infinite products, area Nevanlinna type spaces, Nevanlinna characteristic, parametric representations, analytic function
The article was submitted 10.03.2022 Acceptedforpublication 15.04.2022
For citation. Shamoyan R., MihiC O. Some remarks on Blaschke type products in area Nevanlinna type spaces in the unit disk. Izvestiya Kabardino-Balkarskogo nauchnogo centra RAN [News of the Kabardino-Balkarian Scientific Center of RAS]. 2022. No. 2 (106). Pp. 17-21. DOI: 10.35330/1991-6639-2022-2-106-17-21
1. INTRODUCTION, BASIC DEFINITIONS AND HISTORY OF PROBLEMS
Assuming that D = { z e (C : | z | < 1 } is the unit disk of the finite complex plane (C , T is the boundary of D , T = { z e (C : | z | = 1 } and H( D ) is the space of all functions holomorphic in D, we introduce the following classes of functions
where , (see [1]). It is obvious that if then where
N ( D ) is the well known classical Nevanlinna class (see [1], [2], [3]). Let , then we define
where by we denote the normalized Lebesgue measure on . Also, by we denote
standard normalized Lebesques area measure.
© Shamoyan R., Mihic O., 2022
* Supported by MNTR Serbia, Project 174017.
1
P
I / (О I Pdm (O ) , Г£( 0 ,1 ), pE ( 0 ,00 ),
Everywhere below, by ry ( t) = n ( t) we denote the quantity of zeros of an analytic function f in the unit disk | z | < t < 1 and by Z (X) the zero set of an analytic class !,!c H( D ) . By let {z/JfcW be a sequence of numbers from D, below we mean that {z^}^ is an arbitrary sequence from unit disk enumerated by it is growth ( | zfc | < | zfc+1 | < ...) according to it is multiplicity.
Let further
) = {f e H): I 1 (/ R On + I f (z)| )( 1 - | z| )adm2(z)) ( 1 - R)4R < go }
(D ) = {f e H (D ) : 0sup i (/* (/ in + | f (z) | d?) ( 1 - |z| ) ad | z | ) ( 1 - R) ^ < oo}, where it is assumed that and , and let
^ (D) = if e H (D) : / ( 1 - | z | ) P ( sup T (r,f) ( 1 - r)^ d | z | < oo},
I J 0 \0<T<|z| J J
where
We refer for basic properties of these new large are Nevanlinna spaces to [4] - [8]. We note that in these papers various results on zero sets and parametric representations can also be seen. Note similar, but less general results can be seen in various papers of various authors, we refer, for example, to [2], [3], [9], [10], [12].
Note that various properties of iV^p( D ) = are studied in [9]. In particular, the works [2], [9] give complete descriptions of zero sets and parametric representations of VV^P( D ) (in [2] for p = 1 ). Thus it is natural to consider the problem of extension of these important results to all VCX ( D ) analytic classes.
It is not difficult to verify that all the above mentioned area Nevanlinna analytic classes are topological vector spaces with complete invariant metric. We note that the mentioned problem of parametric representation has various applications and are important in function theory (see [2], [3], [11]).
Solution of many problems for example the existence of radial limits is based also on parametric representations. Parametric representations are used also in spectral theory of linear operators (see [3], [11]).
Main theorem of this note lead to parametric representations of our new area Nevanlinna type spaces.
We introduce now another infinite product which is the main object of this note. It is known that (see [3]) the following assertion is true. The infinite Blaschke type product Ba(z,{zk}),a > -1
00
(z, {zfc}) = n( 1 - exp( - ^ (z,Zfc) ), a n d k=1
oo
, . r (a + k + 2) Wa (z.O= > ______ ^ X
Г(а + 2)Г(Л + 1)
k=1
" 1 dx ),z,<f £D,
is converges uniformly within D if and only if 1 ( 1 — | zfc | ) a+1 < oo. Moreover it represents an analytic function in .
2. MAIN RESULTS
We formulate now main results of this paper.
Theorem. Let a > — 1, p 6 ( 0, oo) and {zfc} £= 1 be a sequence of complex numbers in the unit disk such that
1 ) Let ( 1 — r) a+p np(r) dr < oo .
Then we have
• r ____ . a+1 „ a+1
if 0 < p < 1, -< t < 2 H--,
p p
if 1 < p < oo , 1+a<t<2+ —
p p
and the infinite product converges absolutely and uniformly within D.
2) Let / .
Then we have and the infinite
product converges absolutely and uniformly within D.
3) Let / ( 1 — r) v n(r) dr < oo .
Then we have and the infinite product
converges absolutely and uniformly within D.
Proofs of our assertions are based on the following propositions.
Proposition 1. Let f 6 H (Z) ) , p > — 1,7 > 0, 0 < q < oo. Then
( 1 —t) p+(v+^ |log +| f (rf) | dm dr) <
< C1 ( f ( 1 — t) P ([ l o g + | f (z) | ( 1 — | z | ) v dm2 (z) ) dr
->0 W|z|<T /
Proposition 2 Let f 6 H (D ) , p > 1, q 6 ( 0, oo) , a > — 1, p > — 1, t = p + ^ ( a + 1) . Then / \ q/p
I ( | (l o g + | f (z) | ) p ( 1 — | z | ) a dm2 (z) ) ( 1 — R) P dR < o
Jo W|z|<R I
if and only if
l / \ 9/p | (l o g + | f (rf) | ) p dm (f)) ( 1—| z | ) Tdr<oo .
We define an extention of T (f, r) to Tq (f, r) in a usual way (see [10-11]). Proposition 3. Let f 6 H (D ) , q > 1 and p < s . Then
( 1-1 z I ) "d I Z I ) <
< i ( 1-r)T (sup r?(p ,/)( 1-I pI ) y) dr,
J 0 \0 <p<r /
for the following values of indexes:
a >—1, p, q, s e (0,oo),y > 0,r = (a + l)(p/s) — yp — 1.
Our assertions in particular extend certain results obtained earlier in [12] and they lead directly to parametric representations of large area Nevanlinna spaces which we introduced in this note. We will discuss this in our next papers.
Related interesting problems in the half plane may also be considered (see for example [13] and various references there). Another complete version of this note with other results can be seen in [14] and with complete proofs.
REFERENCES
1. Hayman W. Meromorphic functions. Oxford University Press, 1964.
2. Djrbashian M.M., Shamoian F.A. Topics in the theory of 4p spaces. Leipzig. Teubner, 1988.
3. Djrbashian M.M. Integral transforms and representation of functions in complex plane. Moscow: Nauka, 1966. 624 p. (In Russian)
4. Shamoyan R., Arsenovic M. On zero sets and parametric representations of some new analytic and meromorphic function spaces in the unit disk. Filomat. 2011. 25(3). Pp. 1-14.
5. Shamoyan R., Li H. Descriptions of zero sets and parametric representations of certain analytic area Nevanlinna type classes in disk. Proceedings of Razmadze Mathematical Institute. 2009. Vol. 151. Pp. 103-108.
6. Shamoyan R., Li H. Descriptions of zero sets and parametric representations of certain analytic area Nevanlinna type classes in the unit disk. Kragujevac Journal of Mathematics. 2010. Vol. 34. Pp. 73-89.
7. Shamoyan R., Mihic O. On zero sets and embeddings of some new analytic function spaces in the unit disc. Kragujevac Journal of Mathematics. 2014. 38(2). Pp. 229-244.
8. Shamoyan R., Mihic O. On zeros of some analytic spaces of area Nevanlinna type in a halfplane. Trudy of Petrozavodsk State University. Ser. Mat. 2010. Vol. 17. Pp. 67-72.
9. Shamoyan F.A. Parametric representation and description of the root sets of weighted classes of functions holomorphic in the disk. Siberian Math. Journal. 1999. Vol. 40. No. 6. Pp. 1452-1470. (In Russian)
10. Shamoyan F.A., Shubabko E.N. Parametric representations of some classes of holomorphic functions in the disk. Complex analysis, operators and related topics. Operator Theory: Advances and Applications. 2000. Vol. 113. Pp. 331-338.
11. Djrbashian M.M., Zakharyan V. Classes and boundary properties of functions that are meromorphic in the disk. Moscow: Nauka, 1993. (In Russian)
12. Shubabko E.N. Phd Dissertation. Bryansk, 2001. (In Russian)
13. Mikayelyan G. Some properties of Blaschke type products in the half - plane. Proceedings of the YSU, Physical and Mathematical Sciences, 54:2 (2020), 101-107.
14. Shamoyan R., Mihic O. On Some properties of Blaschke type products in large area Nevanlinna type spaces: preprint, 2021 (submitted).
Information about the author
Shamoyan Romi F., Candidate of Physical and Mathematical Sciences, Bryansk State University named after Academician I.G. Petrovski;
241036, Russia, Bryansk, 14 Bezhitskaya street;
rsham@mail.ru, ORCID: https://orcid.org/ 0000-0002-8415-9822
Mihic Olivera R., Professor, University of Belgrade;
11000, Serbia, Belgrade, Student Square 1;
oliveradj@fon.rs, ORCID: https://orcid.org/ 0000-0002-6809-5881
УДК 517.55+517.33 Научная статья
НЕКОТ ОРЫЕ ЗАМЕЧАНИЯ К ПРОИЗВЕДЕНИЯМ ТИПА БЛЯШКЕ В ПРОСТРАНСТВАХ ТИПА НЕВАНЛИННА В ЕДИНИЧНОМ КРУГЕ*
Р.Ф. ШАМОЯН 1, О.Р. МИХИЧ 2
1 Брянский государственный университет имени академика И.Г. Петровского 241036, Россия, Брянск, ул. Бежицкая, 14 2 Белградский университет 11000, Сербия, Белград, Студенческая площадь, 1
Аннотация. В работе вводятся новые пространства типа Неванлинна в единичном круге и приводятся новые утверждения о принадлежности бесконечных произведений типа Бляшке этим пространствам. Ранее подобные теоремы для этих же бесконечных произведений были доказаны различными авторами в менее общих классах типа Неваннлина в единичном круге. Наши теоремы могут быть в дальнейшем применены для получения новых параметрических представлений указанных новых широких классов типа Неванлинна в единичном круге через упомянутые нами бесконечные произведения типа Бляшке.
Ключевые слова: аналитическая функция, произведение типа Бляшке, пространства типа Неванлинна, характеристика типа Неванлинна, параметрические представления
Статья поступила в редакцию 10.03.2022 Принята к публикации 15.04.2022
Для цитирования. Шамоян Р.Ф., Михич О.Р. Некоторые замечания к произведениям типа Бляшке в пространствах типа Неванлинна в единичном круге // Известия Кабардино-Балкарского научного центра РАН. 2022. № 2 (106). С. 17-21. DOI: 10.35330/1991-6639-2022-2-106-17-21
Информация об авторах
Шамоян Роми Файзоевич, канд. физ.-мат. наук, Брянский государственный университет имени академика И.Г. Петровского;
241036, Россия, Брянск, ул. Бежицкая, 14; rsham@mail.ru, ОЯСГО: https://orcid.org/ 0000-0002-8415-9822 Оливера Михич, профессор, Белградский университет; 11000, Сербия, Белград, Студенческая площадь 1; oliveradj@fon.rs, ОЯСГО: https://orcid.org/ 0000-0002-6809-5881
* При поддержке МНТР Сербии, проект № 174017
