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Vestnik KRAUNC. Fiz.-Mat. nauki. 2023. vol. 42. no. 1. P. 150-163. ISSN 2079-6641
MATHEMATICS
" https://doi.org/10.26117/2079-6641-2023-42-1-150-163
Research Article
Full text in English
MSC 35K05, 35K15
On Some New Results in Large Area Nevanlinna Spaces in the
Unit Disk
R. Shamoyan*1, O. Mihic*2
1 Bryansk State University, Russia, 241050, Bryansk
2 University of Belgrade, Faculty of Organizational Sciences, Serbia, Belgrade, Jove IliCa 154.
Abstract. The study of various infinite products in various spaces of analytic functions in the unit disk is a well known and well studied problem of complex function theory in the unit disk. The goal of our paper is to study so-called Blashcke type products in new large, general analytic area Nevanlinna spaces in the unit disk. A new approach is suggested in this paper, namely we prove, use and apply various new embedding theorems which relate new general, large analytic area Nevanlinna spaces with less general well-studied and well-known such type analytic spaces in the unit disk. Our theorems can be applied or can be used even in more general situation, when we consider large, general analytic area Nevanlinna spaces not in the unit disk, but in the circular ring. In our paper, using same approach new parametric representations of mentioned large, general analytic area Nevanlinna spaces are presented. These results also can be applied or used in the future to obtain more general theorems on parametric representations of mentioned large, general area Nevanlinna type spaces not in the unit disk, but in more general circular domains.
Key words: Blaschke type infinite products, area Nevanlinna - type spaces, Nevanlinna characteristic, parametric representations, analytic function.
Received: 15.08.2022; Revised: 25.12.2022; Accepted: 24.03.2023; First online: 16.04.2023
For citation. Shamoyan R., MihiC O. On some new results in large area Nevanlinna spaces in the unit disk. Vestnik KRAUNC. Fiz.-mat. nauki. 2023, 42: 1,150-163. EDN: DSMGZK. https://doi.org/10.26117/2079-6641-2023-42-1-150-163. Funding. Not applicable.
Competing interests. There are no conflicts of interest regarding authorship and publication.
Contribution and Responsibility. All authors contributed to this article. Authors are solely responsible for providing the final version of the article in print. The final version of the manuscript was approved by all authors.
* Correspondence: A E-mail: rshamoyan@gmail.com, oliveradj@fon.rs ^
The content is published under the terms of the Creative Commons Attribution 4.0 International License © Shamoyan R., Mihic O., 2023
© Institute of Cosmophysical Research and Radio Wave Propagation, 2023 (original layout, design, compilation)
Вестник КРАУНЦ. Физ.-мат. науки. 2023. Т. 42. №1. C. 150-163. ISSN 2079-6641
МАТЕМАТИКА
" https://doi.org/10.26117/2079-6641-2023-42-1-150-163
Научная статья
Полный текст на английском языке
УДК 517.977.5
О некоторых новых теоремах в классах типа Неванлинны в
единичном круге
Р. Ф. Шамоян*1, О. Михич*2
1 Брянский государственный университет, Россия, 241050, г. Брянск.
2 Университет Белграда, Сербия, ул. Джов Илика 154.
Аннотация. Общая задача о принадлежности тех или иных бесконечных произведений тем или иным аналитическим классам функций хорошо известна в литературе. Цель исследования, в частности, рассмотреть и изучить вопрос о принадлежности бесконечных произведений типа Бляшке к общим новым широким классам типа Неванлинны в единичном круге. Авторы для этого применяют новый метод, а именно доказываются и приводятся в статье различные новые теоремы вложения,связывающие новые общие классы типа Неванлинны с уже хорошо изученными и известными менее общими классами типа Неванлинны в единичном круге. Результаты статьи могут быть обобщены или использованы в более общем случае, когда рассматриваются общие, широкие классы Неванлинны в круговом кольце. В статье тем же методом также получены новые параметрические представления указанных широких классов типа Неванлинны в единичном круге. Вывод: эти результаты также могут быть использованы для получения новых параметрических представлений общих классов типа Неванлинны в круговом кольце.
Ключевые слова: бесконечные произведения типа Бляшке, площадь пространств неванлинновского типа, характеристика Неванлинны, параметрические представления, аналитическая функция.
Получение: 15.08.2022; Исправление: 25.12.2022; Принятие: 24.03.2023; Публикация онлайн: 16.04.2023
Для цитирования. Shamoyan R., MihiC O. On some new results in large area Nevanlinna spaces in the unit disk // Вестник КРАУНЦ. Физ.-мат. науки. 2023. Т. 42. № 1. C. 150-163. EDN: DSMGZK. https://doi.org/10.26117/2079-6641-2023-42-1-150-163.
Финансирование. Исследование выполнялось без финансовой поддержки фондов. Конкурирующие интересы. Конфликтов интересов в отношении авторства и публикации нет. Авторский вклад и ответственность. Авторы участвовали в написании статьи и полностью несут ответственность за предоставление окончательной версии статьи в печать.
* Корреспонденция: А E-mail: rshamoyan@gmail.com, oliveradj@fon.rs ф
Контент публикуется на условиях Creative Commons Attribution 4.0 International License © Shamoyan R., MihiC O., 2023
© ИКИР ДВО РАН, 2023 (оригинал-макет, дизайн, составление)
Introduction, basic definitions and history of problems
Assuming that D = {z G C|: |z| <1} is the unit disk of the finite complex plane C, T is the boundary of D, T = {z G C : |z| = 1} and H(D) is the space of all functions holomorphic in D we introduce the following classes of functions
N~(D) = {f G Hp): T (t, f) < Cf(1 - T)-a, 0 < t<1, a > 0},
where T (T,f) is classical Nevanlinna characteristic defined by T (T,f) = 2nJTlog+ |f(T£Jd£, where a+ = max{0,a}, a G R, (see [5]). It is obvious that if a = 0 then N^(D) = Np), where Np) is the well known classical Nevanlinna class (see [2], [3], [5]).
Let f G Hp), then we define
Mp(f,r) = —
1
2n
1 p
|f(r^)|pdm(^) ) , r G (0,1), p G (0,oo),
T
where by m(£,) we denote the normalized Lebesgue measure on T. Also, by m2(£,) we denote standard normalized Lebesques area measure.
Everywhere below by nf (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 X, X C Hp). By let {zk}k=1 be a sequence of numbers from D below we mean that {zk}k=1 is an arbitrary sequence from unit disk enumerated by it is growth (|zk| < |zk+11 < ...) according to it is multiplicity.
By nk we denote n(1 -2-k), i.e. nk = n(1 -2-k), k = 1,2,..., (see [2]).
In all our assertions below we assume in advance that our functions are not identically zero or infinity.
The following statement holds by Nevanlinna's classical result on the parametric representation of N(D) (see [2], [3], [5]). N(D) class coincides with the set of functions representable in the form
f(z) = CAzAB(z, |zkj)exp
•n
d^(6) 1 - ze-ie
, z G D,
where CÀ is a complex number, À is a nonnegative integer, B(z,{zk}) is the classical Blaschke product with zeros {zk}k=i C D enumerated according to their multiplicities and satisfying the condition Y.k=i(1 — |zk|) < oo, and |x(0) is a function of bounded variation in [—n,n].
In [2], [3] the following proposition is established (see also [5]) for sequences {zk}k= C satisfying the greater density condition
oo
£(1 — |zk|)t+2^, t> —1. (1)
k=1
Proposition A: (see [2]) Let {zk}n=i be a sequence in the unit disc satisfying the density condition (1) for some t > — 1, then the Djrbashian infinite product
nt(z,|zk})=n 1--) exP
^ /-(t +1)
k=^ zk/ I n
(1 1 - £
zk
D
(1 - ^z)t+2
dm2(^) I ,z gD, (2)
n
converges absolutely and uniformly inside D where it presents an analytic function with zeros {zk}£=.
After the appearance of the classical Nevanlinna's parametric representation in Hayman's book (see [5]) which we mentioned above various new results of the same type appeared during past decades where Blaschke products were substituted by more general so called Djrbashian na(z,{zk}) products (see [3]) and we will mention them partially below in Theorem A and Theorem B. In [9] it was shown that these na(z,{zk}) products can be in their turn replaced by other infinite Ba(z, {zk}) products and some aspects of this last development will stand as one of the topics of this paper.
We denote by BO^T), 0<p<oo, 0<q <oo, a > 0, the classical Besov space on the unit circle T, (see [1]).
Theorem A: (see [9]) Let a>0 and |3 > a — 1, then the N^(D) class coincides with the set of functions representable in the form
f(z) = CAzAnß(z, |zkj)exp
-n
4(ei9 )d9 \ (1 _ ze-ie)ß+2J
, z gD, (3)
where Ca is a complex number, À is a nonnegative integer, üß(z,{zk}) is the Djrbashian infinite product (2), {zk}k=i C D is a sequence satisfying the condition
, A C
n(T) <
(1 —T)a+1'
where c > 0 is a positive constant and ^(ei0) is a real function of Bp—(T).
We also give below a theorem which is established in [8] and in a sense is similar to Theorem A.
Let S?(D) be the class of analytic functions defined by
S*(D) = {f G Hp): ||f||Pp =
S a
(1 _T)aïp(T,f)dT< oo, 0<p<oo, a> _1}.
Theorem B: (see [8]) For p > 0, ß > a+1, f G S£(D) if and only if f(z) admits
p
representation f(z) = CAzAnp(z,{zk})exp (j—n (1—) , z gD, where CA is a complex number, A is a nonnegative integer, {zk}k=i C D is a sequence for which jJ(1 — T)a+p[n(T)]pdT< ^ and ^ G B]'p(T), where s = p — a+1.
Note complete analogues of Theorem A and Theorem B were given also for Na p and N^p3. NpYp area Nevanlinna spaces in the disk (see [12], [14], [11] and definitions below).
One can easily see that Theorem A gives the parametric representations of the spaces Na(D) while Theorem B gives the parametric representations of S£(D) analytic area Nevanlinna type spaces in the unit disk via same Djrbashian nt(z,{zk}) infinite product.
One of the goals of this paper are parametric representations of the larger spaces via completely other infinite product.
n
0
Let further
f1
0
= {f e HP):
p
(ln+ |f(z)|)(1 -|z|)adm2(z^ (1 -R)ßdR < oo};
jzj<R )
N«ßP (D) = {f e Hp): sup
" ' 0<R<1
^ |f(z)|d£j (1 - |z|)ad|z|J (1 - R)(l < «>}, where it is assumed that a > —1, (3 > —1, ( > 0 and 0 < p < oo, and let
Np,T,pP)= {f G Hp): [1(1 — |z|)3(sup T (T,f)(1 — t)^ d|z| < oo},
J0 \0<T<|z| /
where y > 0, (3 > —1.
We refer for basic properties of these new large are Nevanlinna spaces to [11] - [11]. We note 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], [8], [9].
Note that various properties of N^P) are studied in [8]. In particular, the works [2], [8] give complete descriptions of zero sets and parametric representations of Na°0P P) (in [2] for p = 1). Thus it is natural to consider the problem of extension of these important results to all (D) analytic classes.
We remark that these analytic classes of area Nevanlinna type in the unit disk was considered by us in our recent paper (see [11], [11], [14]).
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 have various applications and are important in function theory (see [2], [3], [4]).
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], [4]).
The next section will be devoted to study of certain infinite Blashcke type products Ba(z, {zk}) in new analytic area Nevanlinna classes we introduced above, then partially based on these results we will turn to the main topic of paper we mentioned above and we will provide some new parametric representations via these infinite Blaschke type products Ba(z, {zk}).
The main idea to get new results on infinite Ba products and parametric representations via such products in our large new Nevanlinna type spaces to use a group of new embeddings relating them to known Nevanlinna type spaces for which such results were provided by other authors, then apply these known results.
For this we use a simple idea. Namely, we analyze various known embeddings between mixed norm and Bergman spaces and then simply replace in these estimates |f|p by (log+ |f|)p, since both are subharmonic functions for p > 1, and since the already known proof is based only on this fact.
R
Throughout the paper we write C (sometimes with indexes) to denote a positive constant which might be different at each occurrence (even in a chain of inequalities) but is independent of the functions or variables discussed.
The notation A x B means that there is a positive constant C, such that C < A < CB. We will write for two expressions A < B if there is a positive constant C such that A< CB.
We formulated certian assertions below on Nevanlinna spaces after careful analysis of some already known proofs for mixed norm spaces, their proofs will be given below in a sketchy form, since new proofs are almost the same. We leave some arguments in proofs below to readers since they are easy to recover.
On some new theorems on canonical infinite products of Blaschke type in N^p, N^p and Np,y,p(D) classes in the unit disk
First we introduce a new Blaschke type canonical product and list some of it is properties, then based on these properties we will find conditions on sequence
from such that our product belongs to mentioned above new analytic Nevanlinna classes in the unit disk. We remark that the nt(z, {zk}) defined above and the product we are going to consider act as same kind of substitution for Blaschke product in classes with tog+. The problem of finding conditions on {zk}k=i sequences so that the classical Blaschke product belongs to analytic Bergman or Hardy or other spaces is well known and studied by many authors before (see, for example, [2], [3], [16] and the references there).
We would like to note that our results can be considered as analogues of mentioned assertions concerning Blaschke products. Note that similar results for nt(z, {zk}) products are well known (see [2], [3], [8]). We give one such type example.
Theorem C: (see [2]) Let {zk}k=i be a sequence from unit disk D. Then if Y.k=i(1 -|zk|)a+2 < oo, a > -1, then nt(z,{zk}) e S^D) for t > a that is
log+ |nt(z,{zk})|(1 - |z|)adm2(z) <oo,
D
and the reverse is also true. Let (1) holds, then na(z,{zk}) e Sj(D) if a > t.
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
Ba(z,|zkj)=n ( 1 - — )exp(-Wa(z,zk)), and
1 (1 -x)adx (z^k ^ It! xk+1 -
(1 - x)axk-1dx 1 ,z,t eD,
is converges uniformly within D if and only if Y.k=i 0 - |zk|)a+1 < oo. Moreover it represents an analytic function in D.
Remark 1: An interesting generalization of this product can be found in [4].
Our intention is to find conditions on of this product, so that Ba(z,{zk}) G
N^pP) or N^p (D) or Np,T,p(D). We mention that the following result were given before. It puts in particular direct condition on {zk}k= sequences so that Ba(z,{zk}) G
sap).
Let O be a set of positive on (0,1 ) measurable functions w such that mw < w(Àx) <
w(r)
Mw, for all x G (0,1), À G [qw, 1] and some fixed Mw, mw, qw such that
m^ qw G
(0,1), Mw >0, (see [2]).
We define general S^ area Nevanlinna spaces similarly as a S^ spaces by replacing (1 -r)a by w(r).
Theorem D: (see [9])Let {Àk}k= be a sequence from unit disk D. Let w G O, aw =
——w, 3w = lnMW , 0 < 6w < 1 and one of the following two conditions holds: lnqw ln-L
qw
0<p < 1,
aw + ßW aw + 1 „ 1 ßW . T
P >-^-, —"— < a < —---h 2;
2
P
P
or
n -t ^ 1 3w ^
1 <p < oo, p > aw + |3W — 1, — +1 < a< —p--h2.
Then if /¿wO — r)np(r)(1 — r)pdr< oo, then Ba(r^,{Ak}) uniformly converges within D and belongs to S^ that is
log+ |B«(r^,(Akj)|dm(^n w(1 -r)dr<oo.
Corollary 1: Let 0 < p < oo, a > —1, {zk}k= G D, 0 < |zk| < |zk+11, k = 1,2,... and let also Jj(1 — r)a+pnp(r)dr < oo, then Bß(z,{=k}) G Sp that is
Tlog+1BßK,{zk})|dm(£,)J (1 — r)pdr<oo
if 0 < p < 1, < ß < 2 + OH
PP
„ a a +1
if 1 < p < oo, 1 +— < ß < 2 +--
PP
We formulate below in theorems 1 and 2 new such type results on Ba infinite products in our new large area Nevanlinna type spaces.
p
0
Theorem 1: Let a > —1, (3 > —1 and p e (0,oo). Let {Ak|k=i be a sequences of complex numbers in the unit disk such that
•1
(1 — r)a+pnp(r)dr<oo. (4)
0
Then there is an interval (t0,ti) for t0,ti e RU{oo} and for p < 1 and for p > 1 for which the canonical product Bt(z,{Ak}), t e (t0,t1) converges absolutely and uniformly within D and belongs to N^ p(D) class.
Theorem 2: Let a > —1, (3 > —1 and p e (0,oo). Let {Ak}k=1 be a sequences of complex numbers in the unit disk such that
1
(1 — r)a+pnp(r)dr<oo. (5)
0
Then there is an interval (t0 ,tj) for t0 ,tj e R U {oo} and for p < 1 and for p > 1 for which the canonical product Bt(z,{Ak}), t e (t0,tj) converges absolutely and uniformly within D and belongs to N~pp(D) class.
Remark 2: It is not difficult to extend the statements and the proofs of Theorems 1 and 2 to more general, slowly varying weights w(1 — t) from S class (see [2]).
The very similar result is also valid for Np,Y,v area Nevanlinna type spaces, with some restrictions on parameters.
The proof of Theorems 1 and 2 will be given below.
The goal of this section to provide ways to get some new parametric representations for N^ p(D), N^p(D) and Np,y,p(D) classes in the unit disk via Ba(zk,z) products. The following theorem provides complete parametric representations for N^ p(D) spaces via Djrbashian products from theorems A and B, (see [12], [13]).
Theorem E: (see [12], [13]) If 0 < p < oo, a > —1 and (3 > —1, then the class N coincides with the set of functions representable for z e as
a,ß
oo
f(z) = CAzAnh--exP
z \ t +1
k=1 ^ 1 n
•n (1 — p2)ln
1 —
zk
—■ I
(1 — pe—i^z)t+2
pdpd^ > exp{h(z)},
where t > max {(a + |3/p) + max{1,1/pj, (a +1 )}, CÀ is a complex number, À > 0,
oo p
Lnp
2k(p+1+2p+«p) < 00, k=1 2
and h G Hp) is a function satisfying the condition
i7
0 0
" |h(Te^)|d^ (1 — T)ad^ (1 — R)ßdR < oo.
1
0.
Similar results holds for N^pp) and NP)T)pp) classes (see [11]).
To obtain parametric representations of N^pP), N^T(D) and NP)T)p(D) classes via Ba(z,{zk}) infinite Blaschke type products we can use some embeddings and known parametric representations for analytic classes of area Nevanlinna type with quazinorms /J (JTlog+ |f(|z|y|d^)p (1 - |z|)adm2(z) < oo, for certain 0<p < oo, a>-1, that were obtained earlier by other authors.
First we formulate a result that will be used by us via Bt product.
Theorem F: Let 0 < p < oo, a > 0. Then S£(D) coinciding with the class of f functions such that
f(z) = eia+mKßzmBß(z,|ak})exp
-п
2
(1 - e-i^z)ß+1
-1 ,z gD
{ak)k=i and 0 < |ak| < |ak+il, k = 1,2,..., is arbitrary sequences of points from D, so that
n
p(r,f)(1 - r)a+pdr<oo,
where ß g + 2), -ф G Bf^T), s = ß - ^ ^(eie) p^ßy Jr (r -
t)ß-1 In|f(te^)|dt and Kß = ß щщ.
Now it is clear that obtain a parametric representations of classes we study in this paper via Bt(z,{zk}) all we have to do is to show, for example, that if f G X, X = N^ p (D) or X = N^P(D) or X = Np,Y,e(D), then f G ST(D) for some big enough т > 0 then apply Theorem F we just formulated above. To do that we formulate the following propositions.
Note that we collect several such propositions below and they can be interesting also as separate statements and relate various analytic area Nevanlinna type spaces to each other.
Proposition 1: Let f G Hp). Let в > -1, Y > 0, 0 < q < oo. Then
(1 -T)ß+(Y+1)q f
q \ q
log+ |f(t£,)|dm(£) dT
< C1
(1 - T)ß
log+ |f(z)|(1 -|z|)Tdm2(zn dT
|z|<T
Even more general result is valid with the same proof (see Proposition 2 below). We will consider for simplicity only this case. Similar estimate can be proved for N^p(D). Let now
L(Ap,q)p) = {f G HP): МцдГ) =
п
/ . \p \q/p
fln+ |f(rei9)|J dH (1 -r)Ydr < oo},
п
o
o
q
q
o
п
o
where 0 < p < oo, 0 < q < oo, y > —1 and
pn / f 1
\ p/q
L(Fp,q)(D) = {f G Hp): 11%^) = (V |f(re^)|) (1 — r)Ydr d^ < oo},
J — n \ J 0 J
where 0 < p < oo, 0 < q < oo, y > —1.
Proofs of Proposition 2 and Proposition 3 as it follows from [6] and [10] are based on arguments from [6] and [10] and their are valid for subharmonic function (log+ |f(z)|)s for any s > 1.
Note that Proposition 2 extends Proposition 1 and Proposition ??. Proposition 2: Let p > 1, q G (0,oo), a> —1, (3 > —1, t = (3 + q(a +1) then
0
if and only if
\ q/p
(log+|f(z)|)p (1 — |z|)adm2(z) (1 — R)3dR < oo
|z|<R )
1 /r \ q/p
(log+ |f(r£)|)p dm(£u (1 — |z|)Tdr < oo.
Proposition 3: Let 1 < min(p, q) < s and y > —1. Then we have
1/s
d
(log+|f(w)|)s (1 — |w|)s(Y+1)/q+s/p—2dm2(wn < C3|f|L(AY'q), f G L(A?,q)(D),
(log+|f(w)|)s (1 — |w|)s(Y+1)/q+s/p—2dm2(wn < C4|f|L(FÇ.q), f G L(Fp,q)p). (7)
1/s
(6)
d
Proposition 4: Let q > 1 and p < s. Then
Tqs(r,f)(1 -|z|)ad|z^ < C5 f1(1 -rW SUP Tq(p,f)(1 p| )Y] dr,
J Jo \v0<p<r
for the following values of indexes:
a> —1,p, q, s G (0,oo),y > 0, t = (a + 1)(p/s)-YP — 1.
The easy proof of Proposition 4 follows from dyadic decomposition of unit interval and growing properties of Tq (r, f) function immediately.
We show only particular case of estimate in Proposition 4, the general case is the same.
Let Tn = 1 - , n G N, p < 1, f(z) = log+ |f(z)|. Then we have based on basic properties of Nevanlinna characteristics and dyadic decomposition of the unit interval
\p 00 / ~M
f(z)(1 — |z|)adm2(z) < £2—kp(a+2) (M1(Tk,f^ D ' k=1
1
k=1
<y-2—kp(a+1) sup (Mi(p,f)(1 -p)Y 2kyp
0<p<Tk v y
<
L
k=1 "
1 —2
k-3
(1 — |z|)(a+1)p—'Yp—11 sup (Mi(p,f)(1 — p)Y)' d|z|
1—2-k-2 0<p<|z|
p
< C
1
(1 — |z|)(a+1)p—'Yp—11 sup T(t,f)( 1 — t)y d|z|.
0<p<|z|
Let us show assertions in Proposition 1.
(1 — r)ß+(Y+1)q f
' V
f(r^)dm(^M dT
<L2—k(ß+(Y+1)q+1) (M1(Tk,f))' k=1
oo
< L
k=1
f(z)(1 — |z|)Ydm2(zH 2—k(ß+1)
Tk<|z|<Tk+1
<
L
k=1
'Tk+2
(1 — T)ß
Tk+1
|z|<T
f(z)(1 — |z|)Ydm2(z) dT
<
(1 — T)ß
f(z)(1 — |z|)Tdm2(zH dT.
|z|<T
Remark 3: Classes of analytic functions of area Nevanlinna type with quasinorms that can be seen in the first part of Proposition 4 studied in [11]. There complete descriptions od zeros and parametric representations via other nt(z,{zk}) infinite products are given.
Note, for example, obviously Bt(z,zk) belongs to spaces with quasinorms
|z|<R
ln+ |f(z)|(1 — |z|)adm2(z) sup( 1 — R)ßdR
q,
by Proposition 2, 0 < p < oo, t = |3 H—(a +1) and Theorem D for some values of t parameter.
Note to use Theorem F we have to apply reverse embedding in Proposition 2. Estimates of Proposition 1 and Proposition 2 give many new results on parametric representation via Bt(z,zk) product we give such examples below.
To get immediately new assertions of type Bt(z,zk) G X, where X is a certain new large area Nevanlinna type space we simply will use the following elementar embeddings:
(A) ai - R)3 aR (aTln+l f(lz I *)Idm(^)p (1 z|)ad | z|<
< Ci ai (aT|f(|z|^)|dm(^))p (1 - |z|)ad|z| = ||f||sp, 0 < p < oo,a > -1,3 > 0;
(B) aJ (a I z|<Rln+I f(z)I(1 -I z| )adm2(z])p (1 -R)3dR < C2||f||sa, 0<p<oo, a > -1, 3 >-1;
p
0
q
q
q
q
0
q
0
R
(C) llfllNApY.v < Cslfls;.,, 0<p<oo, Y>0, v> —1.
Indeed area Nevanlinna type spaces with quasinorms on the right side were studied and assertions of the following type Bt e So were given by us above. It remains to apply (A) — (C).
We have the following result from Corollary 1 and (A) — (C).
Theorem 3: Let a > —1, p e (0,oo) and {zk}k= be a sequences of complex numbers in the unit disk D such that 0 < |zk| < 1, k = 1,...
1) Jj(1 — r)°+pnp(r)dr<oo
then we have Bt(z,{zk}) e N°°;p, a > —1, B > 0
- , a +1 „ a +1
if 0<p < 1, -< t < 2 +-
pp
• r i a . a +1
if 1<p<oo, 1 + — < t < 2 +--.
p p
2) Jj(1 — r)a+1n(r)dr<oo
then we have Bt(z,{zk}) e N£ß, a>-1, ß >-1, t e (a + 1,a + 3) 3) Xo(1 -r)Tn(r)dr<oo
10»
then we have Bt(z,{zk}) e NP,Y,V, v > —1, y > 0, t e (y,Y + 2).
In our next theorem we provide new interesting parametric representations of our large Na,p,Y, N"p, N^°Bp area Nevanlinna type spaces in the unit disk via Bt(z,zk) infinite products based on Theorem F and embeddings in Propositions 1-4.
Theorem 4: 1) Let f e Np,Y,T, a>0, p<1, a>Y — 1, t = (a + 1 )p — yp — 1. Then assertions of Theorem F (parametric representations) are valid for p = 1.
2) Let f e NYp, y > 0, 0<p<oo, B> —1. Then assertions of Theorem F are valid for a = B + (y + 1)p >0.
Various other assertions similar to those in our last two theorems can be proven also based on estimates above which relate many area Nevanlinna spaces in the unit disk with each other.
Conclusion
Our theorems can be probably used to obtain new results on parametric representations and Blashcke type products in similar type large area Nevanlinna type spaces which we introduced in this paper but in more general situation namely in so-called large area Nevanlinna type spaces in the circular ring.
References
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Information about authors
Shamoyan Romi Fayzovichfa - Ph.D. (Phys. & Math.), Senior Researcher, Department of Mathematical Analysis, Bryansk State University named after Academician I. G. Perovsky, Bryansk, Russia, https: / / orcid.org/0000-0002-8415-9822.
Mihic Oliverafa - Ph.D. (Phys. Associate professor, Department of
University of Belgrade, Belgrade, Republic https://orcid.org/0000-0002-6809-5881.
& Math.), Mathematics, of Serbia,
Информация об авторах
Шамоян Роми Файзович& - кандидат физико-математических наук, старший научный сотрудник кафедры математического анализа, Брянский государственный университет имени академика И. Г. Перовского, Брянск, Россия, Шрв: //orcid.org/0000-0002-8415-9822.
Михич Оливера Ä - Ph.D. (Phys. & Math.), кандидат физико-математических наук, доцент кафедры математики, Белградский университет, г. Белград, Республика Сербия, https://orcid.org/0000-0002-6809-5881.
