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28Trudy Petrozavodskogo gosudarstvennogo universiteta
Seria “Matematika” Vypusk 18, 2011
UDK 517
ON CHARACTERIZATIONS OF MAIN PARTS OF SOME MEROMORPHIC CLASSES OF AREA NEVANLINNA TYPE IN THE UNIT DISK
Romi Shamoyan, Haiying Li
We characterize main parts of Loran expansions of certain mero-morphic spaces in the unit disk defined with the help of Nevanlinna characteristic.
§ 1. Introduction
Let D be the unit disk in the complex plane C. Let T = {z, |z| = 1} be unit circle. Let H(D) be the space of all functions holomorphic in D, dm2(z) be a normalized Lebesgue measure in D, T(t, f) be Nevanlinna characteristic of f G H(D) (see [4]). We introduce several area Nevanlinna-type spaces that will be mentioned in this paper. Let further
NPi/8 = {f G H(D) :
I Na
,0
ln+ |f (z)|(1 — |z|)“dm2(z)
0 \z\<R
(1 - Rf dR <
where a > —1, ¡3 > —1, 0 < p < <x.
(NA)r
{f G H(D) :
1
l(NA)p
sup T(t, f )(1 — t)y
0<t<R
(1 — R)vdR <
1
p
p
(§ Romi Shamoyan, Haiying Li, 2011
where y > 0, v > -1,0 <p < x.
T^,p
a,3
Na,Pp — {f € H(D) :
N^,p — sup a,& 0<R<1,
0
ln+ |f (|z|e)|de
(1 - |z|)ad|z|(1 - R)3 < +x
where a > -1,0 > 0,0 < p < x.
It is not difficult to prove that all mentioned above analytic classes are topological vector spaces with complete invariant metric.
Let M(D) be the space of all meromorphic functions in D and X C M(D) be its subspace. Let also a sequence {zk}*!=i be from a certain class of sequences (|zk| < 1). The problem that we will consider in this paper is the following one. We would like to find sharp conditions on {ak,n} and {zk } sequences of complex numbers so that under it there always exists a function f € X with main parts
ut \ — ak,n ak,1
H (z-zk-ak) — (Z-IkF +... + (J-51.
For meromorphic functions of bounded type such problems were considered and solved previously by A. Naftalevich in [3]. We also remark that the recently such a problem was considered in [1] for some meromorphic Ma C M(D) classes. Our intention is to develop these ideas from
[1] solving such a problem for mentioned above meromorphic algebras of Np,p and N0°pp, (NA)p,Y,v type.
We define new classes of meromorphic functions in the unit disk. Let further
MNpap — {f € M(D) :
/(/ log+ |f (z)|(1 -|z|)“dm2(z)) P(1 - R)p dR< +xj> ,
a,p
1
log+
0 |z|<R
where a > -1, 0 > -1,0 < p < x.
M(NA)p,Y,v — f € M(D) :
sup T(t, f )(1 - t)Y ) (1 - R)vdR < +x
0<t<R
0
where y > 0, v > -1, 0 < p < œ.
MN^'P = {f e M (D) :
R
I
(/
P
sup
0<R<1
ln+ |f (|z|£M (1 - |z|)“d|z|(l - R)P < +œ
0 T
where a > -1,p > 0, 0 < p < to.
Let further B(z) be a classical Blaschke product (see [2]). For our exposition we will need two types of special sequences in the unit disk D sampling sequences and Carleson sequences.
The sampling sequence in D is a sequence in unit disk D such that for t e (0,1],D = UT D(ak,t), the sets D(ak, 4) are mutually disjoint; each point z e D belongs to at most N of the sets D(ak, 2t), where N is independent from {ak} for fixed t e (0,1]. Such a sequence in D exists (see [5]). Let {zk}fc=o be a sequence of complex numbers in D, {zk}fc=o is a Carleson sequence (see [2]) if
We denote by C in this note all constants which depend only on various parameters like p,q, a.
The goal of this section is to characterize main parts of Loran expansions of certain meromorphic spaces in the unit disk defined with the help of Nevanlinna characteristic.
The following theorem provides a solution of mentioned problem of description of main parts of Loran expansions of some classes of mero-morphic functions on the unit disk D based on Nevanlinna characteristic. First we will consider the general situation.
Theorem 1. Let {zk}fc=1 be a sequence of unit disk D. Let X c M(D) be a class of meromorphic functions in the unit disk. We assume that HX = X p| H(D) is closed under the operation of differentiation (if f e HX, then f1 e HX ). Let f e X,f (z)(B(z))n e HX,n = 1, 2,.... Let
infTT Zj —k = s > o.
fc^1 1 - zj zk
§ 2. Main results
k=0
for some a > 0, for every f G HX. Let also the S map, S is acting from HX to the class of all {wk}fc=i sequences such that
¿(1 - |Zk I2)a ln+ \wk\ < (2)
k=0
be onto map, this is for every sequence of complex numbers with condition
(2), there is a function f,f G HX so that f (zk) = wk. Then for every expression of type
m \ ak,n ak,1 i , 0
H (z,zk, ak) = ---------+ ... +-----------;—, k = 1, 2,...
(z - Zk)n Z - Zk
there is a function G(z) G X with H(z, zk,ak) as the main parts of it is Loran expansion if and only if
I |2\ai + \ak,i\\Bk (zk)|n • -i 0 io^
2J1 -\zk \) ln+ (1-\zfc\2)n < +~, * = 1, 2,...,n, (3)
k=1
where zk G D = {\z\ < 1}, {ak,n} G C, Bk(z) is an ordinary Blaschke product but without k factor
Bk(z) = n -z-j-j.k > 1.
i = 1,z, =.. 1 z’z 'z’ '
In our exposition below as X we will take various concrete spaces of meromorphic functions in the unit disk for which (1) and (2) (but for some fixed zk) can be checked directly.
Theorem 2. Let {zk}'ik=1 is a Carleson sequence in D. Then for every expression of type
TT/ \ ak,n ak,1 1 , 0
H (z, zk ,ak ) = 7----m + ... +---------, k = 1, 2,> . . .
( z - zk ) n z - zk
there is a function G G MN1 ^,a> -1, ft > —1 with H(z, zk, ak) as the
main parts of it is Loran expansion if and only if
¿(1 - ln+ ^ z2)nn < +,», i = 1,2,...,n,
where {ak,n} € C.
Theorem 3. Let {zk}tt=1 is a Carleson sequence in D. Then for every expression of type
tt/ \ ak,n ak,i 1 , 0
H (z,zk,ak) — 7-------rn + ■ ■ ■ +------, k — 1, 2,...
(z — zk ) z — zk
there is a function f € M(NA)1,Y,V> 0, v > — 1 with H(z,zk,ak) as the main parts of it is Loran expansion if and only if
g(1 — |zk | )Y+V« ln+ ^< +”• • — 1-2:- "■
where {akn} € C.
Theorem 4. Let {zk}tt=1 is a Carleson sequence in D. Then for every expression of type
tt/ \ ak,n ak,i 1 , 0
H (z,zk ,ak j — 7-----zn + ■■■ +--------------• k — 1- ■ ■ ■
(z — zk)n z — zk
there is a function G € MN^'p-a > —1,fl > 0 with H(z,zk,ak) as the main parts of it is Loran expansion if and only if
g<1 — | zk | r+»2ln+ ^-Bk:F)F < +”• * = 1-2- " ■• n-
where {ak'n} € C.
The proofs of mentioned assertions will be based on following propositions. These assertions as separate statements are interesting from our point of view as separate propositions.
Lemma 1. Let {zk}tt=1 be a sampling sequence. Then
tt
£(1 — |zk|)T+2 ln+ |f (zk)|< C||f |
k = 1
where p > 1, t — (1 + a) + p,r > 0, a > — 1, ^ > — 1 tt
£(1 — |zk |)T ln+ |f (zk )|< C ||f H(NAW„ , k=1
|N l,P ,
where 0 <p < to, v — Tp — YP — 1, v > —1,7 > 0,t> 0^
Lemma 2. Let p > —1,7 > —1,0 < q < to Let f € H(D),f — log+ |f(w)|. Then
1 q
/(1 — T)^ j f(z)(1 — |z|)Ydm2(z)Sj dT <
0 |z|<r
1
< c|(1 — T f+q(Y+1)^J f(T£)d£) 9dT■
0 T
Lemma 3. 1) Let {zk }tt=1 be a sampling sequence. Then for every
sequence {wk }tt=1,
tt
^(1 — |zk|)T ln+ |wk| < TO,
k=1
there is a function f (z) so that ||f ||(na)1y„ < to and so that f (zk) —
wk,k — 1, 2, ■■■ ,t — y + v + 2, y > 0, v > — 1
2) Let {zk}tt=1 be a sampling sequence. Then for every sequence {wk}tt=1 so that
tt
^(1 — |zk|)T ln+ |wk| < oo, k=1
there is a function f (z) so that ||f ||N^, 1 < to and so that f (zk) — wk, k —
1, 2, ■■■ ,t — a + p + 2, a > — 1, p > 0.
Complete proofs of provided assertions will be given by authors elsewhere.
Acknowledge. We thank Professor F. Shamoyan for various discussions concerning results of this note.
Bibliography
[1] Bednag V., Shamoyan F. Descriptions of main parts of Loran expansion of some classes of meromorphic functions in the unit disk // Vestnik BGU. 2004. V. 4. P. 84-92. (in Russian).
[2] Garnett John B. Bounded Analytic Functions. New York: Springer, 2006.
[3] Naftalevich A. G. On interpolation of functions of bounded type // Uchenie Zapiski Vilnus University. 1956. V. 18. P. 19-27. (in Russian).
[4] Nevanlinna R. Analytic functions. Moscow: GITTL, 1991.
[5] Zhu K. Spaces of holomorphic functions in the unit ball. New York: SpringerVerlag, 2005.
Bryansk State University,
Department of Mathematics,
241050, Bryansk, Russia
E-mail: [email protected], [email protected] Henan Normal University,
College of Mathematics and Information Science, 453007, Xinxiang, P. R. China E-mail: [email protected]
