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8
УДК 517.5
On New Parametric Representations of Analytic Area Nevanlinna Type Classes in a Circular Ring K on a Complex Plane C
Romi F. Shamoyan*
Bryansk State University, 50 October av., 7, 241035, Bryansk,
Russia
Received 27.07.2012, received in revised form 30.09.2012, accepted 17.11.2012 We define certain new large area Nevanlinna type spaces in circular ring K on a complex plane C and provide complete parametric representations for these new scales of analytic function spaces. Our results complement certain previously known assertions.
Keywords: analytic function, Nevanlinna characteristic, area Nevanlinna type spaces, circular ring.
Introduction
Assuming that D = {z G C : |z| < 1} is the unit disk of the finite complex plane C, T is the boundary of D and H(D) is the space of all functions holomorphic in D, introduce the following classes of functions:
NV0° = { f G H(D) : T(r,f) < Cf (1 — t)-a, 0 < t < lj, a > 0,
where T(t, f) is Nevanlinna's characteristic (see eg. [1,2]). It is obvious that if a = 0, then N0° = N, where N is the well-known Nevanlinna's class. The following statement on parametric representation in spaces in unit disk is known. It holds by Nevanlinna's classical result on the parametric representation of N class (see eg. [1,2]) and it serves (see [1]) as a base of all theory of area Nevanlinna type spaces. The N class coincides with the set of functions representable in the following form
f (z) = C\zxB(z, {zk})exp{J* J—ZeF^} ' z G D,
where C\ is any complex number, A is any nonnegative integer, B(z, {zk}) is the classical Blaschke product with zeros {zk}+=i C D enumerated according their multiplicities and satisfying the condition Y1 +=i(1 — |zk|) < and p(6) is any function of bounded variation in [—n, n].
Later this important result was extended to weighted Nevanlinna spaces we defined above and other similar spaces, for example, to so-called Nevanlinna-Djrbashyan spaces (see eg. [1,3,4]). The goal of this paper is to continue our earlier investigation started in [5-7] of certain new large analytic area Nevanlinna type spaces in the unit disk and to obtain similar parametric representation for these spaces. Our intention here is to extend some results from [8] and [3] on spaces in the circular ring to larger spaces of area Nevanlinna type in the same circular ring. Note
* [email protected] © Siberian Federal University. All rights reserved
in our previous mentioned papers we provided already such an extention procedure in the unit disk and we extended some known classical results about zero sets in certain analytic Nevanlinna spaces from [1] and from [2] to larger spaces, but in case of the unit disk. For formulation of our theorems we need to introduce first these analytic spaces in the unit disk. We define below three different scales of large area Nevanlinna spaces in the unit disk and then based on that their direct analogues in circular ring. These large spaces in unit disk appeared and studied for the first time in [5-7]. Later in [9,10] meromorphic classes of this type were also introduced and studied by authors from mentioned papers. Through out this paper dm2(z) denotes the normalized Lebegues measure in the unit disk on the complex plane. Let further
Np p
a, p
N
na,p i
f e H(D) :
= { f e H(D): sup
0<r<1
ln+ |f (z)|(1 -|z|)a dm2(z)
\<R
(1 - R)pdR < +to i ,
|z|<r
ln+ |f (z)|(1 -|z|)a dm2(z)
(1 - R)p1 < +to
where it is assumed that [1 > 0, a > -1, [ > —1 and 0 < p < to. Note that various properties of N^°0 are studied in [2]. In particular, both books [1] and [2] provide complete descriptions of zero sets and parametric representations of N^0. Later these assertions were extended in [2] and [5] by us to larger analytic spaces as we defined above and these results were used in [7].
Hence it is natural to consider the problem on extension of various known results, for example,
to all N^°Pi classes.
Let
(NA)
pyv = {f e H(D) : j0
sup T(f,T)(1 - T)Y
0 <t<r
(1 - R)vdR < +to
where y > 0, v > -1 and 0 < p < to, and
r
tiJ0 l^t
N
f e H(D): sup f
0<r<u0
in+ |f (|z|e)|de
(1 - |z|)ad|z|(1 - R)p < +to
where 0 < p < to, a > -1 and [ > 0. Note that the zero sets of N^'J are described in [4] for 3 = 0 and later for all positive values of [.
It is not difficult to verify that all these analytic classes are topological vector spaces with complete invariant metrics.
Throughout the paper, we write C for constant which is independent of the functions or variables being discussed.
The following assertion from [1] is crucial for our investigation and in the study of analytic area Nevanlinna spaces in the unit disk. Let (zk) be an arbitrary sequence of complex numbers from the unit disk D. so that the following condition holds
]T((1 -|zfc|)p+2) < +to,
k= 1
3 > -1, then the Djrbashian infinite product (see [1])
np(z, zk)
is an analytic function and converges uniformly in D and have zeros only at sequence (zk).
1
0
p
Complete parametric representations of classes we defined above in case of unit disk were given in [5]and [6]. The intention of this paper to extend further these results to the case of circular rings. The theory of analytic function spaces in circular rings was developed in [8]. Note that similar problems were considered in [3,12] and [13] for other analytic area Nevanlinna type spaces in circular rings. Hovewer, the classes we introduced in circular rings are larger and we can consider our theorems as direct extentions of previously known results formulated in [3].
For 0 < R1 < R2 < we denote by K the usual circular ring on the complex plane K = K(R1,R2) = {z € C, R1 < |z| < R2}, we also denote by H(K) the space of all analytic functions in this circular ring and for f function so that f € H(K) we denote as usual the Nevanlinna characteristic of f in K by
T (T,f).
We will also use the following definition. We denote by Z(f) the set of zeros of f function in K. Let further r0 = (Ri + R2), by Z(R1; r0, f) we denote all those points from circular ring K(r0, R1) which belong to Z(f). Similarly we define Z(R2, r0, f). Let also further Z(R1, r0, f) = (zk), Z(R2,r0,f) = (wk).
We define new spaces in circular ring K. The idea is keeping the same conditions on parameters is to replace in quazinorms the integration intervals, namely the unit interval (0,1) by (R1?R2) and (0, R) by (R1,R). Let
(NA)
P,H,72,vi,V2
f e H(K): /R
suPri<t<rT(f,T)(R - T)Yi (t - Ri)
y2
(R - R1)ßl (R2 - R)ß2 dR <
we will write f € (NA)Pi7ljVl meaning the quazinorm we defined without factors with y2 and v2 and in the same way we will use the notation (NA)Pi72jV2 meaning the quazinorm we just provided, but without factors with y1 and v1. In other spaces in K we assume the same. We
define (Nas space of analytic functions in K, with finite quazinorm
sup
Ri<R<R2
'Ri
T(f, |z|)|(R - |z|)ai (|z| - Ri)a2d|z|
(R - R1)ßl (R2 - R)ß2
and (N)ai a2 as space of functions analytic in circular ring K so that
/■R2
'Ri
CR
/ T(f, |z|)(|z| - Ri)a2(R - |z|)aid|z|
■JRi
(R - R1)ßi (R2 - R)ß2dR <
We will use later the following expression. It provides a parametric representation of the f function analytic in K. We will call it standard parametric representation of f function in K = K(R1; R2). Let further
f (z) = cmzmnti (v1, V2)nt2 (v3, V4) exp ^1(^1) exp h2(V3),
v1 = —, v2 = —, v3 = —, v4 = wr, z € K(R1; R2), where cm is an arbitrary complex number, z zk r2 r2 m in a nonnegative integer, h1, h2 are analytic functions in the unit disk D. In this case we say
that the f function in K allows standard parametric representation in K. If X is a certain class
of analytic functions in K and each function f from X admits such a parametric representation
then we will say that the X class admits standard parametric representation.Various results on
parametric representations are well-known (see, for example, [1] and [2]).
p
R
1. Theorems on Parametric Representations of ^aia20l02, (NA)P,71, 72, Vl,V2 and N<, pa2, 0i, 02 Classes in Circular Rings
In this section we formulate our main results for spaces in circular ring. These are three theorems providing parametric representation for each space in circular ring we defined. We consider these results for analytic spaces in circular ring as direct extentions of our earlier theorems in the unit disk from [5-7].
0- + 1
Theorem 1. Let p e (0 , <), tj > —--+ aj, aj > -1, 0j > -1, j = 1, 2.
Then let
nk = {cardzk : |zfc | > RiTu},
mu = {cardwk : |wfc| < R2Sk} ,
where sk = 1 — 2-i, Tk = 1 + 2-i and where (zk) and (wk) are arbitrary sequences from halfrings of K, and
ntl (z ,zk) and ni2 (z ,wk)
are mentioned Djrbashyan products constructed via (zk)or (wk) sequences. We also assume that the following sum is finite
TO p
nk pu = 2k(Pi+<*iP+2p+1)
and we also assume that the same condition but for (mu) instead of (nu) with a2, 02 instead of a1 and 0i is also true.
Under these conditions a f function from the
N p
ai,a2
space allows standard parametric representation in circular ring K. In such a parametric representation h1(^) belongs to NPi ^ and h2() belongs to NP2 ^2.
And moreover each f function from the mentioned class in circular ring admits such a standard parametric representation.
In the following two assertions we again deal with two sequences (zu) and (wu) in circular halfring of K and use again notations (nu) and (mu) as they were defined in theorem 1. We assume also again that two infinite products
n^ (z,zu) and nt2 (z,wk) are consructed via these given sequences.
Vj + 1
Theorem 2. Let p e (0, <), Vj > 0, Yj > 0, and also tj > —--+ Yj — 1, j = 1, 2. Let further
(zu) and (wu) be two sequences in the unit circular ring K, we also assume that the following two sums are finite
+TO p +TO p
y-ni sr-'mi h lu' ¿i du'
lu = 2k(p+i+vi+pYi), du = 2fc(p+i+v2+pt2). Then there is a f function from
(NA)
p,Yi ,Y2,vi,V2
space which admits standard parametric representation in circular ring K. Moreover in such a parametric representation h1 and h2 are analytic in the unit disk, and h1(—) belongs to (NA)P}11}Vl, and h2( ) belongs to (NA)P}l2}V2. And each f function from these classes in circular ring K admits such a standard parametric representation.
Theorem 3. Let p G (0, aj > 0, fy > — 1, tj > aj + fy + 1 + p, j = 1, 2.
Let further
(n1)p(T) < c(T — R1)-ai-^1-p-1,
where c is a constant.
We assume the same condition holds for (n2) instead of (n1) and with a2 and instead of a1 and and R2 — t instead of t — R1, where
ni(T) = {cardzk : |zfc| > t}, t G (R1, R2), n2(T) = {cardwk : |wfc| < t}, t G (R1, R2) Then there is a f function from (Nspace which admits standard parametric representation in circular ring K.
Moreover in such a representation h2 and h1 are analytic in the unit disk and h1( —) belongs to (N, and h2( —2) belongs (N)™;%.
And moreover each f function from this class in circular ring K admits standard parametric representation.
Proofs of these assertions will be presented elsewhere. Results we formulated in case of the unit disk were already known (as we indicated above). In case of similar type but narrower spaces these results were obtained previously in [3]. Our arguments follow similar strategy as in [3], but with more technically involved estimates in all proofs.
Proofs of our assertions are based on a simple idea to replace functions analytic in circular ring by functions analytic in the unit disk via factorization. Note the same procedure was applied in [3]. We formulate this procedure in the following assertions.
Lemma 1. Each f function analytic in K admits representation
f = f1f2,
where each f is an analytic function in the unit disk and f1 = f 1 (—1 ), f2 = f2\ —
z G K(R1,R2). Moreover
T (f,r)= TU*\ + O(1), r G (R1 ,ro),
z ' V R
2
T(f,r)= T^f2,R~J + O(1), r G (ro,R2),
r0 G (R1,R2). In addition in proofs we constantly use the fact that both T(f,r) and T(fi,r) functions are growing functions.
We formulate one more assertion which is a base of proofs.
Lemma 2. Let X be one of the spaces in our theorems (for example (NA)Pj71j72jV1jV2 space which admits standard parametric representation for some restrictions on parameters- see formulations of theorems).
Then the following are equivalent f G X(K) and
f = f1f2,
where f1 = f1(Rz1), f2 = f2 ^ —^, z G K(R1,R2) and fi belongs to Xi(D) where Xi(D) is a
parallel to X(K) analytic class (without two factors in quazinorm of X(K) as it was indicated above).
For example, if X = N01'02 then X1 is equal to (N'p/3l and X2 is equal to (N)™2Pi32 .
References
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[2] H.Hedenmalm, B.Korenblum, K.Zhu, Theory of Bergman spaces, Graduate Texts in Mathematics, Springer-Verlag, New York, 2000.
[3] I.Kursina, Factorization and parametric representation of weighted spaces of analytic functions in the unit disk, Bryansk State University, 2000, PhD, Dissertation (in Russian).
[4] F.Shamoian, The root sets and parametric representations of certain analytic function spaces in the unit disk, Siberian Mathematical Journal, 23(1999) (in Russian).
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[7] R.Shamoyan, H.Li, Characterizations of closed ideals and main parts of some analytic and meromorphic classes of area Nevanlinna type in the unit disk, International Journal of Mathematics and Statistics, 10(2011), 109-121.
[8] V.Zmorovich, On certain spaces of analytic functions in circular rings, Mat. Sbornik, 32(1953), no. 74, 643-652 (in Russian).
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[10] R.Shamoyan, M.Arsenovic, On zero sets and parametric representations of some new analytic and meromorphic function spaces of area Nevanlinna type in the unit disk, Filomat, 25:3(2011), 1-14.
[11] R.Shamoyan, O.Mihic, On zeros of some analytic spaces of area Nevanlinna type in a halfplane, Trudy Petrozavodskogo Universiteta, 17(2010), 67-72.
[12] G.U.Matevosyan, An analogue of N(w) class in case of circular rings, 21:2(1977), 173-182 (in Russian).
[13] S.Ya.Kasyaniuk, On functions from A and H class in case of circular rings, Mat. Sbornik, 42(84)(1957), 301-326 (in Russian).
О некоторых новых параметрических представлениях аналитических классов типа Неванлинны в круговом кольце K на комплексной плоскости C
Роми Ф. Шамоян
Мы определяем новые пространства типа Неванлинны в круговом кольце и даем их параметрические представления. Эти результаты дополняют ранее известные теоремы о параметрических представлениях аналитических классов типа Неванлинны.
Ключевые слова: аналитическая функция, характеристика Неванлинны, пространства типа Неванлинны, круговое кольцо.
