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0Математические заметки СВФУ Июль—сентябрь, 2023. Том 30, № 3
UDC 517.55
SOME REMARKS ON BLASCHKE TYPE PRODUCTS IN LARGE AREA NEVANLINNA SPACES IN THE UNIT DISK R. Shamoyan and O. Mihic
Abstract: The intention of this paper is to introduce and study certain new analytic spaces in the disk and to show that certain Blaschke type products belong to new large Nevanlinna type classes in the unit disk. We also provide parametric representation of such classes. These results extend and complement some previously known assertions of this type obtained earlier by other authors. Our arguments are based on certain new embeddings which relate the well-known SS area Nevanlinna spaces in the unit disk with our new large area Nevanlinna type spaces in the unit disk.
DOI: 10.25587/SVFU.2023.13.41.008 Keywords: Blaschke type infinite products, area Nevanlinna type space, Nevanlinna characteristic, parametric representation, analytic function.
1. Introduction, basic definitions and history of problems
Assuming that D = {z £ C| : |z| < 1} is the unit disk of the finite complex plane C, T is the boundary of D, T = {z £ C : |z| = 1} and H(D) is the space of all functions holomorphic in D we introduce the following classes of functions
N0°(D) = {f £ H(D) : T(t, f) < Cf (1 - t)-a, 0 < t < 1, a > 0},
where T(t, f) is the classical Nevanlinna characteristic defined by
T(T'/) = i / T
where a+ = max{0, a}, a £ R, (see [1]). It is obvious that if a = 0 then N^(D) = N(D), where N(D) is the well known classical Nevanlinna class (see [2-4]). Let f £ H(D), then we define
Mpif'r) = h{J^(M)- pe(0,oo),
T
where by m(£) we denote the normalized Lebesgue measure on T. Also, by m2(£) we denote standard normalized Lebesques area measure.
O. Mihic is supported by MNTR Serbia (Project 174017).
© 2023 R. Shamoyan, O. Mihic
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 H(D). By let {zk}^°=1 be a sequence of numbers from D below we mean that {zkis an arbitrary sequence from unit disk enumerated by it is growth (| zk | < |zk+1| < ...) according to it is multiplicity.
By nk we denote n(1 — 2-k), i.e. nk = n(1 — 2-k), k = 1, 2,..., where n is a number of zeros in the appropriate disk (see definitions above) (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 [2-4]). N(D) class coincides with the set of functions representable in the form
f(z) = CxzxB(z,{zk})exp( I 1 1 , z £ D,
where C\ is a complex number, A is a nonnegative integer, B(z, {zk}) is the classical Blaschke product with zeros {zk}^=1 C D enumerated according to their multiplicities and satisfying the condition Y^ (1 — |zk |) < œ, and ^(6) is a function of
k=i
bounded variation in [—n, n].
In [2,3] the following proposition is established (see also [1]) for sequences {zk}fe=1 C D satisfying the greater density condition
k=1
(1 — |zfc|)t+2 < œ, t> —1. (1)
Proposition A (see [2]). Let {zk}n=1 be a sequence in the unit disc satisfying the density condition (1) for some t > —1. Then the Djrbashian infinite product
=n (i - i) -<«)—•
D (2) converges absolutely and uniformly inside D, where it presents an analytic function with zeros {zk}j!=1.
After the appearance of the classical Nevanlinna's parametric representation in Hayman's book (see [1]) 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 [5] 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 Bg'q (T), 0 < p < ro, 0 < q < ro, a> 0, the classical Besov space on the unit circle T (see [6]).
Theorem A (see [5]). Let a > 0 and ft > a - 1. Then the class N0°(D) coincides with the set of functions representable in the form
f(z) = C\zxIifi{z, {zk}) exp ^ J ' * e D, (3)
where C\ is a complex number, A is a nonnegative integer, n(z, {zk}) is the Djr-bashian infinite product (2), {zk}5?=1 C D is a sequence satisfying the condition
n(r) <
a+1 '
(1 - T)
where c > 0 is a positive constant and ^(e10) is a real function of Bp'-g+^T).
We also give below a theorem which is established in [7] and in a sense is similar to Theorem A.
Let Sp(D) be the class of analytic functions defined by
i
sa(D) = {f G H(D) : 11/lISp = |(1 - T)aTp(T, f )dT < rc, 0 < p < «), a > -1}.
o
Theorem B (see [7]). For p > 0, ¡3 > we have f G S£(D) if and onfy if /(z) admits representation
f (z) = CA(z, {zk})exp I J
^(ei0) d(9
(1 - ze-i0)ß+1
z G D,
where C\ is a complex number, A is a nonnegative integer, |zk}J!=1 C D is a sequence for which
J (1 - T)a+p [n(T)]pdT < œ o
and ip G ßi'p(T), where s = ß -
Note that complete analogues of Theorem A and Theorem B were given also for Npa p and N0°/gP. Npp p area Nevanlinna spaces in the disk (see [8-10] and definitions below).
One can easily see that Theorem A gives the parametric representations of the spaces N0°(D) while Theorem B gives the parametric representations of Sg(D) analytic area Nevanlinna type spaces in the unit disk via same Djrbashian nt(z, {zk}) infinite product.
The main goal of this paper is to obtain new parametric representations of the larger spaces via completely other infinite product. Let further
NP,p(D) = jf G H(D): j^J (ln+ |f (z)|)(1—|z|)a dm2(z^"(1-R)P dR <
I 0 |z|<R J
N
OX (D) = if G H (D):
R
sup 1 m J ln+ |f (z)| P(1 — |z|)ad|z| I (1 — R)^1 < ro
0<R<1
\0 T
where it is assumed that a > —1, ft > —1, ft1 > 0 and 0 < p < ro, and let
np,y,p (D) = If G H (D): /(1 — |z|)^ ( sup T (t, f )(1 — t )y )p d|z| < ro
i j 0<t <|z|
where j > 0, ft > —1.
We refer for basic properties of these new large are Nevanlinna spaces to [10,11]. We note in these papers various results on zero sets and parametric representations can also be seen. Note that similar, but less general results can be seen in various papers of various authors, we refer, for example, to [2, 3, 5,7].
Note that various properties of N0*0P(D) are studied in [7]. In particular, the works [2, 7] give complete descriptions of zero sets and parametric representations of N^0P(D) (in [2] for p = 1). Thus it is natural to consider the problem of extension of these important results to all N(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 [9-11]).
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, 3,12]).
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 [3,12]).
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.
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 < 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.
2. 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 D 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 log+ . The problem of finding conditions on {zk}5?=1 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, [2,3,13] 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, 3,7]). We give one such type example.
Theorem C (see [2]). Let {zk}J!=1 be a sequence from unit disk D. Then if
X)(1 -M)a+2 < rc, a > -1, k=1
then nt(z, {zk}) e (D) for t > a that is
J log+ |nt(z, {zk})|(1 - |z|)adm2(z) < rc,
D
and the reverse is also true. Let (1) holds, then na(z, {zk}) e SKD) if a > t, where n(r) sequence was defined above.
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, {zk}) = A (l - —) exp(-Wa(z, zk)),
and
" Г (a + k + 2)
k=1 Г (a + 2)Г(k + 1)
( 1 . . „ lei
k
V lei 0
is converges uniformly within D if and only if
¿(1 -|zkl)a+1 < œ. k=1
Moreover it represents an analytic function in D.
Remark 1. An interesting generalization of this product can be found in [12].
Our intention is to find conditions on {zk}^°=1 of this product, so that Ba(z, {zk}) G
в(D) or (D) or NP,Y,e(D). We mention that the following result were
given before. It puts in particular direct condition on {zk}'j=1 sequences so that Ba(z, {zk}) G S1 (D).
Let О be a set of positive on (0,1) measurable functions w such that mw < < Mw, for all x G (0,1), Л G [qw, 1] and some fixed Mw, mw, qw such that mw,qw G (0,1), Mw > 0, (see [2]).
We define general S£ area Nevanlinna spaces similarly as Sg spaces by replacing (1 — r)a by w(r).
Theorem D (see [5]). Let {Xk be a sequence from unit disk D. Let w G fi, i
aw = itttt^ > Pw = y2-^, {)< Bw < I and one of the following two conditions holds:
m in ——
n -, aw + ew aw + 1 1 ew 0
0<p<l, p>---, -<a<--h 2,
2 p p
or
l<p<oo, p>aw+pw- 1, — + l<a<——+2.
p p
Then if
1
J w(1 — r)np(r)(1 — r)p dr < ro, 0
then Ba(r£, {Ak}) uniformly converges within D and belongs to that is i p J (y log+ |Ba(r£, {Ak})| dm(o) w(1 — r) dr < ro.
0 T
X
Corollary 1. Let 0 < p < w, a > -1, {zk}^ G D, 0 < |zk| < |zfe+i|, k = 1, 2,..., and
J(1 - r)a+pnp(r) dr < w. 0
Then B(z, {zk}) G Sp that is
i P J (I log+ (r£, {zk})| dm(£)) (1 - r)adr < w,
ifO<p< 1, <ft<2 + ^
if 1 < p < oo, 1 + s < ft < 2 + £±i.
We formulate below in Theorems 1 and 2 new results of such type on infinite products in our new large area Nevanlinna type spaces.
Theorem 1. Let a > -1, ft > —1 and p G (0, w). Let {Akbe a sequence of complex numbers in the unit disk such that
1
J(1 - r)a+pnp(r) dr< w. (4)
0
Then there is an interval (to, t1) for io,i1 G R U {w} and for p < 1 and for p > 1 for which the canonical product Bt(z, {Ak}), t G (t0, t1) converges absolutely and uniformly within D and belongs to (D) class.
Theorem 2. Let a > -1, ft > -1 and p G (0, w). Let {Akbe a sequence of complex numbers in the unit disk such that
1
y (1 - r)a+pnp(r) dr< w. (5)
0
Then there is an interval (t0, t1) for t0, t1 G R U {w} and for p < 1 and for p > 1 for which the canonical product Bt(z, {Ak}), t G (t0,t1) converges absolutely and uniformly within D and belongs to N^^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 NPi7jV 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 Np^(D), N0°/gP(D) and NPi7i)g(D) classes in the unit disk via Ba(zk,z) products. The following theorem provides complete parametric representations for Np ^(D) spaces via Djrbashian products from theorems A and B, (see [8,14]).
1
Theorem E (see [8,14]). If 0 < p < ro, a > —1 and ft > -1, then the class Np p coincides with the set of functions representable for z e D as
„A
f(z)=Cxzxll^--)
jt + 1 }
x exP S — J J (1 _ 2 PdPdV > exPiM^)},
I 0 -n
where t > max{(a + ft/p) + max{1,1/p}, (a + 1)}, CA is a complex number, A > 0,
E:
2k(P + 1+2p+ap) k=1
<,
and h e H(D) is a function satisfying the condition 1 / r / n \ \p
I ( / ( / |h(TelVI (1 — T)"dT I (1 — R)PdR < ro.
0 \0 \-n
Similar results hold for N^°pp(D) and NPYp(D) classes (see [11]). To obtain parametric representations of NPp(D), N^°pp(D) and Np°Y°p(D) classes via Ba(z, {zk}) infinite Blaschke type products we can use some embed-dings and known parametric representations for analytic classes of area Nevanlinna type with quazinorms
1 p
j (/ log+ If (|z|£)| ddj (1 — |z|)a dm2(z) < ro,
0 T
for certain 0 <p< ro, 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 < ro, a > 0. Then Sp(D) coincides with the class of functions f such that
f (z) = e* a+mK?zmBp(z, {afe})
x exp ^ / ({l_Xr+1 ~ l) ^ , ^ D,
{ak }fe=1 and 0 < |ak | < |ak+1|, k =1, 2,... , is an arbitrary sequence of points from D, such that
1
J np(r,f )(1 — r)a+pdr < ro,
0
where ft G (^±1, + 2), ip G Bl<p(T), s = ft -
p ' p ' I ' r S \ n r- p
r
J0\
VKO " 1
lun^Jir-tf-'lnlfite^ldt
p n.
k
and
oo -
*=1 + ft)'
Now it is clear that to obtain 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^(D) or X = N0p(D) or X = Npi7i)3(D), then f G Si(D) for some big enough t > 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 H(D). Let ft > -1, 7 > 0, 0 < q < œ. Then |(1 - tT log+ |f (t^)| dm(o) q dT
< ci I j (1 - T f^j log+ |f (z )|(1 -|z |)Y dm2(z^9 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 N0P(D). Let now
L(AYq )(D) = ( f G H (D) : ||f «)
q/p
yi J (ln+ If (reiv)|)P d^J (1 - r)Y dr< œ
0 \-n
where 0 <p< œ, 0 <q< œ, 7 > -1 and L(Fpq )(D) = (f G H (D) : ||f |l(f-')
p/q
y I y(ln+ |f (re^)|)q(1 - r)Ydr J d^ < œ
where 0 < p < œ, 0 <q< œ, y > -1.
Proofs of Proposition 2 and Proposition 3, as follows from [14] and [15], are based on arguments from [14] and [15] and their are valid for subharmonic function (log+ |f(z)|)s for any s > 1.
Note that Proposition 2 extends Proposition 1 and Proposition 3.
Proposition 2. Let p > 1, q G (0, oo), a > -1, ¡3 > -1, and r = /3 + + 1). Then i
/(/ (log+ If (z)|)p(1 -|z|)adm2(z- R)^ dR< œ
0 |z|<R
if and only if
J (/(log+ If K)|)p dm(0y\l - |z|)T dr < œ.
0 T
Proposition 3. Let 1 < min(p, q) < s and 7 > -1. Then
1/s
J(log+ If (w)|)s(1 - |w|)s(Y+1)/q+s/p-2 dm2(w)^
< ca||f ||l(a-), f G )(D), (6)
1/s
J(log+ |f (w)|)s(1 - |w|)s(Y+1)/q+s/p-2 dm2(w^
D
< C4^f ||L(Fp"), f G L(FT)(D). (7)
Proposition 4. Let q > 1 and p < s. Then
\ p/s 1
|T|(r,/ )(1 - |z|)a d|z|l < CB/ (1 - r)T ^ SUP^ Tq(p,/ )(1 — |p|)Y )P dr,
V0 /0 p r
for the following values of indexes: a > — 1,p,q,s G (0, ro), 7 > 0, t = (a + 1)(p/s) — YP — 1.
The easy proof of Proposition 4 immediately follows from dyadic decomposition of the unit interval and growing of Tq (r, /).
We show only particular case of estimate in Proposition 4, the general case is the same.
Let rn = 1 — 2^7, n G N, p < 1, f(z) = log+ |/(z)|. Then due to basic properties of Nevanlinna characteristics and dyadic decomposition of the unit interval we have
/ /(z)(1 - |z |)adm2(zM 2-kp(a+2) (M^,/))
■I 'i—1
2-kp(a+1) sup (M1(p, /)(1 - p)Y )p2kYP
k=1 0<p<rfc
to 1-2-k 3 ^
E / (1 - |z|)(a+1)p-Yp-1 sup (M1(p,f)(1 - p)Y)p d|z|
>. -, J 0<p<lz
0<p<|z| 1 1 — 2—k —2
< C /"(1 -|z|)(a+i)p-7p-i( sup T(t,/)(1 - T)Y)p d|z J 0<^< I z
0<p<|z| 0
Let us show assertions in Proposition 1:
i
q
|(1 - r)^+(Y+1)q(| /(r£) dmfâ) \t
0 T
2-k(^+(Y+1)q+1)(Mi(Tfc,/))q
k=1
< E( f /(z)(1 -|zl)Y dm2(z^92-k(^+1)
k=1 li-
Tfc<|z|<rfc+1
oo r fc+2
S E f (1 - T)"( / /r(z)(1 -|z|)Y dm2(z)V dT
k=1T \ r, '
Tk+1 |z|<T
1
< J(1 - T f^j /(z)(1 -|z|)Y dm2(z^9 dT.
0 |z|<T
Remark 3. Classes of analytic functions of area Nevanlinna type with quasi-norms that can be seen in the first part of Proposition 4 studied in [10]. 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
1 q
J(^J ln+ |/(z)|(1 -|z|)a dm2(z^ sup(1 - R)^ dR
0 |z|<U
by Proposition 2, 0 < p < oo, r = ft + ^(a + 1) and Theorem D for some values of t parameter.
Note that 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:
u
(A) (1 - J ln+ |/(|z|0| dm(C))P(1 -|z|)a d|z |
0 T
1
< C1 K | ln+ |/(|z|^)| dm^V - |z|)a d|z| = ||/||sp,
0T
0 < p < œ, a > -1, ft > 0;
l
1
(£)/(/ ln+ l/(z)|(1 — |z|)a dm2(z^P(1 — R)^ dR < C2||/||sa,
0 |z|<R
0 <p< w, a > —1, ft > —1;
(C) ||/|na„7,v < C3||/||s1_1 , 0 <p< w, y> 0, v> —1.
Indeed area Nevanlinna type spaces with quasinorms on the right side were studied and assertions of the following type Bt G Sg 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 G (0, w) and {zkbe a sequences of complex
numbers in the unit disk D such that 0 < |zk | < 1, k = 1,....
i
1) /(1 — r)a+pnp(r) dr < w then we have B(z, {zfe}) G Npy, a > —1, ft > 0; 0
if0<p<l, s±i<i<2 + s±i
if 1 < p < oo, 1 + f < t < 2 +
1
2) /(1 — r)a+1n(r) dr < w then we have Bt(z, {zk}) G Np ^, a > —1, ft > —1, 0
t G (a + 1, a + 3); 1
3) /(1 — r)Yn(r) dr < w then we have Bt(z, {zk}) G Npi7jv, v > —1, y > 0,
t G (y, Y + 2).
In our next theorem we provide new interesting parametric representations of our large Na^i7, Np^, Np0^ area Nevanlinna type spaces in the unit disk via Bt(z, zk) infinite products based on Theorem A and embeddings in Propositions 1-4.
Theorem 4. 1. Let / G Npj7jt , 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 / G Np^, y > 0, 0 <p< w, ft> —1. Then assertions of Theorem F are valid for a = ft + (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.
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Submitted November 22, 2022 Revised july 28, 2023 Accepted September 4, 2023
Romi Shamoyan Bryansk State University,
14 Bezhitskaya Street, 241023 Bryansk, Russia rshamoyan@gmail.com
Olivera Mihic
University of Belgrade,
Faculty of Organizational Sciences,
154 Jove Ilica Street, Belgrade, Serbia
oliveradj @fon.rs
