DESCRIPTIONS OF ZERO SETS AND PARAMETRIC REPRESENTATIONS OF CERTAIN ANALYTIC AREA NEVANLINNA TYPE CLASSES IN THE UNIT DISK

Transcription

1 Kragujevac Journal of Mathematics Volume 34 (1), Pages DESCRIPTIONS OF ZERO SETS AND PARAMETRIC REPRESENTATIONS OF CERTAIN ANALYTIC AREA NEVANLINNA TYPE CLASSES IN THE UNIT DISK ROMI SHAMOYAN 1 AND HAIYING LI Abstract. A comlete zero set descrition of a scale of analytic area Nevanlinna tye saces in the unit disk and arametric reresentations of these saces are established. These generalize some well-known, classical results. 1. Definitions, Preliminaries and Problem Statement Assuming that D = {z C : z < 1} is the unit disk of the finite comlex lane C, T is the boundary of D and H(D) is the sace of all functions holomorhic in D, introduce the classes of functions: { } Ñα = f H(D) : T (τ, f) f (1 τ) α, τ 1, α, where T (τ, f) = (1/π) T log+ f(rξ) dξ is Nevanlinna s characteristic (see eg. [1]). It is obvious that if α =, then Ñ = N, where N is Nevanlinna s class. The following statement holds by Nevanlinna s classical result on the arametric reresentation of N (see eg. [11]). The analytic subset of N coincides with the set of functions reresentable in the form f(z) = C λ z λ B(z, {z k }) ex { π π } dµ(θ), z D, 1 ze iθ Key words and hrases. Djrbashian roducts, Area Nevanlinna saces; Nevanlinna characteristic, Parametric reresentations, Zero sets. 1 Mathematics Subject Classification. Primary: 3D45. Corresonding author Received: May, 1. Revised: August 4, 1. 73

2 74 ROMI SHAMOYAN 1 AND HAIYING LI where C λ is any comlex number, λ is any nonnegative integer, B(z, {z k }) is the classical Blaschke roduct with zeros {z k } k D enumerated according their multilicities and satisfying the condition k(1 z k ) < + and µ(θ) is any function of bounded variation in [ π, π]. In [3], the following roosition is established (see also []) for the sequences {z k } D satisfying the greater density condition (1.1) (1 z k ) t+ < +, t > 1. Proosition A. If (1.1) is true for some t > 1, then the infinite roduct + (1.) Π t (z, {z k }) = (1 z ) ex zk t (1 ξ ) t ln 1 ξ z k dm π D (1 ξz) t+ (ξ), where m (ξ) is Lebesgue s area measure, converges absolutely and uniformly inside D, where it resents an analytic function with zeros {z k }. We shall be based also on some other known statements which we give below. The next lemma can be found in []. Lemma A. If (1.1) is true for some t > 1, then the following estimate holds for Djrbashian s roduct: ln + + ( 1 zk ) t+ Πt (z, z k ) t, 1 zz k k= where C t > is a constant deending solely on t. To state a theorem recently established in [11], by B, q γ (T), ( < <, < q, γ > ) we denote the standard Besov class on the unit circle T = {z : z = 1} (see, eg. [1,, 1, 11]). We will need Besov saces on the unit circle which we will define with the hel of Besov saces on the real line. The Besov sace Bs, q (R), < <, < q is a comlete quasinormed sace which is a Banach sace when 1, q. Let h f(x) = f(x h) f(x) and the modulus of continuity is defined by ω(f, t) = su hf. h t

3 DESCRIPTIONS OF ZERO SETS AND PARAMETRIC REPRESENTATIONS 75 Let n =, 1,,..., s = n + α with < α 1, the Besov sace Bs, q (R) contains all functions f such that f W n (R), where W n (R), < <, n N is a classical Sobolev sace and The Besov sace B, q s ω (f (n), t) t α q dt t <. (R), 1, q is equied with the norm ( ω f W n (R) + (f (n), t) q ) 1 q dt t α. t To define the Besov sace on unit circle we have to use the revious definition of Besov sace on the real line and the standard ma t e it, t R. Theorem A. If α and β > α 1, then the class Ñ α coincides with the set of functions reresentable in the form { π f(z) = C λ z λ ψ(e iθ } )dθ Π β (z, z k ) ex, z D, π (1 e iθ z) β+ where C λ is a comlex number, λ is a nonnegative integer, Π β (z, {z k }) is Djrbashian s roduct (1.), {z k } D is a sequence satisfying the condition n(τ) = card {z k : z k < τ} and ψ(e iθ ) is a real function of B 1, β α+1. C (1 τ) α+1 We also give the below theorem which is established in [1] and in a sense is similar to Theorem A. Theorem B. If < <, α > 1 and β > (α + 1)/, then (1 τ) α T (τ, f)dτ < + if and only if f(z) = C λ z λ Π β (z, z k ) ex { 1 π ψ(e iθ } )dθ, z D, π π (1 e iθ z) β+1 where C λ is any comlex number, λ is any integer, {z k } D is a sequence such that (1 τ) α+ n(τ) dτ <, and ψ Bs 1, (T), where s = β (α + 1)/.

4 76 ROMI SHAMOYAN 1 AND HAIYING LI One can see that Theorem A gives the arametric reresentations of the saces Ñ α, while Theorem B gives the arametric reresentations of some other analytic area Nevanlinna tye saces in the unit disk. One of the goals of this aer are the arametric reresentations of the larger saces [ 1 } N α, β {f = H(D) : ln + f(z) (1 z ) α dm (z)] (1 R) β dr < +, N α,β 1 = { f H(D) : su R<1 z R [ z R ln + f(z) (1 z ) α dm (z) ] (1 R) β 1 < + where it is assumed that β 1, α > 1, β > 1 and < <. Note that various roerties of N α, are studied in []. In articular, the works [, 1, 11] give comlete descrition of zero sets and arametric reresentations of N α,. Thus, it is natural to consider the roblem on extension of these imortant results to all Nα,β 1 classes. The zero set descrition roblem can be stated in the following simle form: assuming that X is a fixed subsace of H(D), recisely find a class Y of sequences such that the zero set of any function f X is of Y and for any {z k } Y there is a function f X satisfying f(z k ) = (k = 1,..., n) (see [, 7]). Note that for many classical analytic classes, such as the sace A α, this roblem is still oen (see []). On the other hand, the comlete descritions of the zero sets of Nα, and Ñ α are known (see [, 9, 1]). One of the intentions of this aer is to solve this roblem for some new Nevanlinna tye analytic classes in the unit disk and to establish the arametric reresentations of these classes, where the found descrition is used. We mention that several new results of this tye for some classical Nevanlinna-Djrbashian analytic classes in the unit disk are resented in [, 1, 11]. So, it is natural to consider the roblem for N α,β and N α,β 1. Everywhere below, by n(t) = n f (t) we denote the quantity of zeros of an analytic function f in the disk z t < 1 and by Z(X) the zero set of an analytic class X, X H(D). Besides, we assume that (NA),γ,v = { f H(D) : [ 1 su T (f, τ)(1 τ) γ (1 R) v dr < + <τ<r where γ, v > 1 and < <, and [ R } N, α, β {f = H(D) : su ln f( z ξ) dξ] + (1 z ) α d z (1 R) β < +, R<1 T }, },

5 DESCRIPTIONS OF ZERO SETS AND PARAMETRIC REPRESENTATIONS 77 where < <, α > 1 and β. Note that the zero sets of N, α, β in [1] for β =. are described It is not difficult to verify that all the above mentioned analytic classes are toological vector saces with comlete invariant metrics. We note that the mentioned roblems of zero set descrition and arametric reresentation have various alications and are imortant in function theory (see, eg. [4, 5, 6]). Solution of many roblems, for instance on existence of radial limits, is based on descritions of zero sets and arametric reresentations. Zero set descritions and arametric reresentations are used also in sectral theory of linear oerators (see, eg. [8]). All results of this aer are given without roofs in our recent note [1]. The authors intend to ublish searate aers with various alications of these results to the descrition roblem of closed ideals in area Nevanlinna saces over the unit disk and to some other roblems in these saces (see, eg. [13]). Throughout the aer, we write C (sometimes with indexes) to denote a ositive constant which might be different at each occurrence (even in a chain of inequalities) but is indeendent of the functions or variables being discussed.. Theorems on Zero Sets of N α,β, (NA),γ,v and N, α, β Classes This section gives the descritions of the zero sets of the defined above area Nevanlinna tye classes. Theorem.1. For any numbers < <, α > 1 and β > 1, the following conditions are equivalent. (.1) and n k < +, k(+1) kα kβ {z k } Z ( N α,β), where n k = n(1 k ) and n(τ) = card {z k : z k < τ}. If (.1) is true, then Π t (z, z k ) N α,β for t > max [(α + β/) + max {1, 1/}, (α + 1)]. Theorem.. For any numbers < <, v and γ, the following conditions are equivalent {z k } Z ( (NA),γ,v ),

6 78 ROMI SHAMOYAN 1 AND HAIYING LI (.) n k k[(γ+1)+v+1] < +, where n k = n(1 k ). If (.) is true, then Π t (z, z k ) (NA),γ,v for 1, t > v γ 1 and for > 1, t > v + γ. Theorem.3. For any numbers < <, α and β >, the following conditions are equivalent {z k } Z ( N, α, β ) ; (.3) n(τ) (1 τ) (α+β++1)/, τ (, 1). If (.3) is true, then Π t (z, z k ) N, α, β for any t > α + β Theorem.1.3 immediately give as corollaries the following arametric reresentations of mentioned above area Nevanlinna tye classes. Theorem.4. If < <, α > 1 and β > 1, then the class N α,β coincides with the set of functions reresentable for z D as f(z) = C λ z λ (1 z ) t π (1 ρ ) ln 1 ρeiϕ z ex k ρdρdϕ zk π π (1 ρe iϕ z) t+ ex{h(z)}, where t > max {( α + β/ ) + max{1, 1/}, (α + 1) }, C λ is a comlex number, λ, n k < +, k(β++1+α) and h H(D) is a function satisfying the condition [ 1 ( R ) ] π h(τe iϕ ) dϕ (1 τ) α dτ (1 R) β dr < +. π Theorem.5. If < <, v and γ, then the class (NA),γ,v coincides with the set of functions reresentable for z D as f(z) = C λ z λ (1 z ) t π (1 ρ ) ln 1 ρeiϕ z ex k ρdρdϕ zk π π (1 ρe iϕ z) t+ ex{h(z)},

7 DESCRIPTIONS OF ZERO SETS AND PARAMETRIC REPRESENTATIONS 79 where for 1, t > v+1 λ, + γ 1 and for > 1, t > v + γ, C λ is a comlex number, n k k[(γ+1)+v+1] < +, and h H(D) is a function satisfying the condition [ 1 h(τξ) dξ(1 τ) γ (1 R) v dr < +. su <τ<r T Theorem.6. If < <, α and β >, then the class N, α, β the set of functions reresentable for z D as f(z) = C λ z λ (1 z ) ex zk t + 1 π where n(τ) (1 τ) (α+β++1)/, π h H(D) is a function satisfying the condition [ su R<1 π coincides with (1 ρ ) ln 1 ρeiϕ z k ρdρdϕ (1 ρe iϕ z) t+ ex{h(z)}, τ (, 1), C λ is a comlex number, λ, and h(τξ) dξ] (1 τ) α dτ(1 R) β < +. T We omit the roof of Theorem.4.6, since it immediately follows by the descrition of zero sets of the corresonding classes and some standard argument alied in [1] to some other analytic area Nevanlinna classes. The given below roofs of Theorems.1,.,.3 follow mainly by some standard arguments (see, eg. [, 1, 11] and their references) based on Lemma A and the classical Jensen formula, but with more careful examination of estimates. Proofs follow by the same scheme. First we use the classical Jensen inequality to show that the conditions (.1), (.) and (.3) are necessary. Then we rove the converse statements by alication of Lemma A and Proosition A for great enough numbers t deending on α, β or v and γ. Proof of Theorem.1. Let f N α, β. Then, without loss of generality it can be assumed that f() = 1, f(z k ) = (k = 1,,...). Hence, by Jensen s inequality [ 1 ] τ I = (1 τ) α n(u) dτ u du (1 R) β dr [ 1 log + f(z) (1 z ) α dm (z)] (1 R) β dr. z <R

8 8 ROMI SHAMOYAN 1 AND HAIYING LI Further, it is obvious that and τ f N α, β n(u) τ u du τ R τ C 1 ( n(u) 3τ R u du C n ) R τ, [ 1 ( ) 3τ R C (R τ) α+1 n dτ (1 R) β dr C for any numbers R < 3τ, C 1 > C, C < R, C 1 < R < 1, C, C 1 >, α >. Besides, one can see that the following imlications are true: Hence, C 3τ R = ρ τ = ρ + R 3 ( ) 3τ R (R τ) α+1 n dτ C R τ = (3C R)/ (R ρ). 3 n(ρ)(r ρ) α+1 dρ. Suose C = (4R 1)/3. Then (3C R)/ = R (1 R)/ and [ 1 f N C n(ρ)(r ρ) dρ] α+1 (1 R) β dr α, β C 1 R 1 R [ 1 ( ) 3R 1 C n (1 R) (α+1)+β+ dr C 1 C [n(ρ) (1 ρ) (α+1)+β+ n k dρ k(+1), k(α+1)+kβ since C 1 f(ρ)dρ = τk+1 τ k f(τ)dτ for any f L 1 (, 1) and τ k = 1 1 k+1 (k =, 1,,...) and n(s 1 ) n(s ) when s 1 s < 1. For α ( 1, ], a similar argument leads to the estimate [ 1 ( ) R α 3 ρ R f N C n(ρ)(r ρ) dρ (1 R) β dr α, β C 1 R 1 R 3 [ 1 ( ) 3R 1 C n (1 R) (α+1)+β+ dr. C 1 Then, we continue as in the above case α > and come to the desired statement. For roving the converse statement, fix a number t so that Lemma A and Proosition A are alicable. Further, observe that log f and log + f both belong to N α, β if

9 DESCRIPTIONS OF ZERO SETS AND PARAMETRIC REPRESENTATIONS 81 just one of them is of N α, β (D). Hence, for z = τeiϕ, τ = t + we get π π log Π t (z, {z k }) dϕ (1 z k ) t+ dϕ π π 1 z k e iϕ. τ Hence, for great enough values of t T (Π t, ρ)(1 ρ) α dρ (1 z k ) t+ (1 z k ρ) t+1 (1 ρ)α dρ (1 ρ) α (1 s) t+ dn(s)dρ = J(R, f). (1 sρ) t+1 Therefore, and hence Consequently, for 1 J (1 s) t+ (1 sτ) dn(s) (1 s) t+1 n(s)ds, t+1 (1 sτ) t+1 (1 s) t+1 n(s) (1 sρ) t+1 ds (1 ρ) α dρ (1 τ k ρ) t+1 n k k(t+) n k 1 k(t+) (1 τ k ρ). t+1 n k 1 k(t+) (1 τ k R), (t+1) (α+1) and by the inequality [ a k a k ( 1) we get J (f, R)(1 R) β dr n k k(+1+α+β). If > 1, then the following estimates are true: J (f, R)(1 R) β dr [ 1 [ 1 (1 ρ) α dρ n k (1 R) β dr (1 τ k ρ) t+1 k(t+) n k 1 k(t+) (1 τ k R) (t+1 (α+1)) (1 R) β dr.

10 8 ROMI SHAMOYAN 1 AND HAIYING LI Or, which is the same, Hence, M = = J (f, R)(1 R) β dr [ (1 R) β (1 ρ) α (1 s) t+ dn(s)dρ dr (1 sρ) t+1 [ (1 R) β (1 ρ) α (1 s) t+1 n(s)dsdρ dr (1 sρ) t+1 (k+1) [ (1 R) β (1 ρ) α (1 s) t+1 1 k (1 sρ) [ (1 R) β k(t+) (1 ρ) α dρ n k dr (1 ρτ k ) t+1 [ ] (1 R) β n k k(t+) 1 dr, t > α. (1 Rτ k ) t α M = [ (1 R) β n(ρ)(1 ρ) t+1 dρ dr (1 ρr) t α ( ) 1 (1 R) β/ + ψ(r)dr = I 1 + I R n(s)dsdρ dr t+1 for any function ψ such that ψ L q Hölder inequalities, one can be convinced that and hence I 1 ( I 1 = ψ(τ) 1 τ τ ψ q (τ)dτ n(ρ)(1 ρ) t+1 ρ n(ρ)(1 ρ) t+1+ β +α t+1 ρ n(ρ)(1 ρ) β +α+ dρdτ ) 1 I 1 q ( For β < above we used (1 R) β = 1 (1/ + 1/q = 1). Using the Hardy and ( 1 1 τ (1 R) β/ ψ(r) (1 ρr) t α drdρ τ n(ρ) (1 ρ) +β+α+ dρ. ψ(r) 1 R drdρ, n(1 t)t β +α+ dt) dτ ) 1 < (1 ρ) β, R ρ < 1 for β, (1 R) β < (1 ρr) β, ρ, R (, 1), for t > max { (α + β/) + max{1, 1/}, (α + 1) }. Besides for

11 DESCRIPTIONS OF ZERO SETS AND PARAMETRIC REPRESENTATIONS 83 β again by Hölder and Hardy inequalities we will have where I = [ n(ρ)(1 ρ) t+1 ρ (1 R) β/ ψ(r)dr (1 ρr) t α n(ρ)(1 ρ) 1+ β 1 ρ +α ψ(1 u)dudρ n(ρ) (1 ρ) +β+α+ dρ B C ψ L q, q > 1. ] 1/ [ ( 1 ) 1 ρ q ] 1/q ψ(1 u)du 1 ρ [ 1/ [ B = n(ρ) (1 ρ) dρ+β+α n k k(+1) k(α+1)+kβ for t > max { (α + β/) + max{1, 1/}, (α + 1) }. The estimate of I in case of β < needs small modification of mentioned arguments and we omit details. Now we shall show that for great enough numbers t Lemma A and Proosition A are alicable. To this end, we rove that if t > max { (α + β/) + max{1, 1/}, (α + 1) }, then (1 z k ) t+ <. Hence, the condition n k k(β+α++1) < will imly the convergence of the roduct Π t (z, {z k }). Indeed, the obvious inequality imlies that Hence, for β + α + > 1 n (τ)(1 τ) β+α+ dτ < + τ 1 n (τ)(1 τ) β+α+ dτ as τ 1 1. ] 1/ n (τ)(1 τ) β+α++1 as τ 1,

12 84 ROMI SHAMOYAN 1 AND HAIYING LI and therefore n(τ) (1 τ) (β+α++1)/ ( < τ < 1). Consequently, (1 z k ) t+ (1 z k ) t+ n k z k B k (1 z k ) t+ (β+α++1)/ z k B k 1 < + k[t (β+1)/ α] where B k = {z k : z k (τ k, τ k+1 )} and t > max { (α + β/) + max{1, 1/}, (α + 1) }. Thus, the latter requirement on t rovides the convergence of the roduct Π t (z, {z k }), and the roof is comlete. Proof of Theorem.. We start as in Proof of Theorem.1. By the classical Jensen inequality (see, eg. []) [ ( 1 τ su <τ<r ) n(u) u du (1 τ) γ (1 R) v dr f (NA),γ,v. Therefore, the following inequalities are true for any R, R (1/3, 1) such that R = (3R 1)/ < R: τ su <τ<r n(u) τ u du(1 τ)γ C su 1/3<τ<R C su n 1/3<τ<R n(u) du(1 τ)γ u ( 3τ 1 τ 1 τ C su ρ (, R) n(ρ)(1 ρ) γ+1 C Besides, one can see that for R = R (1 R)/ su n(ρ)(1 ρ) γ+1 ρ (C, R) ) (1 τ)(1 τ) γ and f (NA),γ,v f (NA),γ,v C τ (1 R) v su n(ρ) (1 ρ) (γ+1) dr ρ ( R, R) C (1 R) (γ+1)+v n( R) dr C n( R) (1 R) (γ+1)+v d R τ τ n k C, n k[(γ+1)+v+1] k = n ( 1 k), k =, 1,, 3,....

13 DESCRIPTIONS OF ZERO SETS AND PARAMETRIC REPRESENTATIONS 85 For roving the converse statement, we fix a number t such that Lemma A and Proosition A can be used. Then, we observe that for 1 [ and π π f (NA),γ,v = log su T (τ, f)(1 τ) γ (1 R) v dr <τ<r π Π t (z, {z k }) dϕ (1 z k ) t+ π dϕ 1 ττ k e iϕ t+, where it is denoted z k = τ k and z k = τ k ξ k, τ k = 1 1. Hence k+1 [ 1 Π t (1 z k ) t+ (NA),γ,v (1 R) v dr (1 R z k ) t+1 γ [ 1 (1 R) v (1 s) t+1 n(s)ds dr (1 Rs) t+1 γ (1 R) v n k [ ( k[(t+1)+] ) ] (t+1 γ) dr R k+1 n k(t+1) k n k k k[(t+1 γ) v 1] k[(γ+1)+v+1] for t > (v + 1)/ + γ 1, since one can easily verify that that (1 z k ) t+ (1 R z k ) = (1 s) t+ dn(s) t+1 γ (1 Rs) t+1 γ = (1 s)t+ n(s) 1 ( 1 (1 s) t+ (1 Rs) t+1 γ n(s) (1 Rs) t+1 γ [ 1 (t + )(1 s) t+1 = n(s) [ = ) ds s + (1 ] s)t+ ( (t + 1 γ)) ( R) ds (1 Rs) t+1 γ (1 Rs) t+ γ n(s)(t + )(1 s) t+1 ds (1 Rs) t+1 γ n(s)(1 s) t+1 (t + )ds, (1 Rs) t+1 γ n(s)(1 s) t+ (t + 1 γ)rds (1 Rs) t+ γ n(s)(1 s) t+1 [ n ( 1 k 1) k(t+1) k ds (1 Rs) t+1 γ (1 ρ k R) t+1 γ n k k(t+1) k (1 ρ k R) (t+1 γ)

14 86 ROMI SHAMOYAN 1 AND HAIYING LI and that for any τ = (t + 1 γ) (v + 1) > and v > 1 (1 R) v ( ) τ 1 dr, ρ (1 ρ k R) (t+1 γ) k k = 1 1 (k =, 1,,...). k Now, let > 1. Then for the conjugate index q > 1 deduced by 1/ + 1/q = 1 and any t > γ 1 Π t (NA),γ,v [ 1 n(s)(1 s) t+1 ds (1 R) v dr (1 Rs) t+1 γ ] [I 1 + I = C = C (1 R) v ψ(r) ( n(s)(1 s) t+1 ) ds dr, (1 Rs) t+1 γ where ψ is a nonnegative function such that ψ L q = 1, I 1 = I = n(s)(1 s) t+1 ( s (1 Rs) t+1 γ ( n(s)(1 s) t+1 (1 Rs) t+1 γ s ψ(r)(1 R) v/ dr ψ(r)(1 R) v/ dr ) ) ds ds and I 1 n(s)(1 s) t+1 s Further, by Hardy and Hölder inequalities ψ(r)(1 R) v/ (1 R) t+1 γ drds. I 1 n(s)(1 s) t+1 s (1 s) t γ v/ [n(s) (1 s) γ++v ds n k k((γ+1)+v+1) ψ(r) 1 R drds

15 DESCRIPTIONS OF ZERO SETS AND PARAMETRIC REPRESENTATIONS 87 for t > γ + v/. Besides, for t > γ 1 I = n(s)(1 s) t+1 (1 R) v/ ψ(r) ds dr s (1 sr) t+1 γ ( 1 (1 s)t+1 1 (1 R) v/ ) ψ(r) n(s) dr ds (1 s) t+1 s (1 sr) γ ( 1 (1 R) v/ ) ψ(r) = n(s) dr ds s (1 s) γ s n(s)(1 s) v +γ ψ(1 u)duds [ ] 1 1/ [ ( 1 1 s q ] 1/q (n(s)) (1 s) v+γ+ ds ψ(1 u)du) ds 1 s [ ] 1 1/ ψ L q (n(s)) (1 s) v+γ+ ds n k C ψ L q. k[(γ+1)+v+1] As at the end of Proof of Theorem.1, it remains to show that the infinite roduct Π t converges for the considered values of t. This is done in a similar way, and we omit the roof. Proof of Theorem.3. Without loss of generality, we assume that f() = 1, f(z k ) = (k = 1,,...). Then by Jensen s inequality ( [ R τ n(u) J = C u du su C 1 <R<1 su C 1 <R<1 R/3 [ R (1 τ) α dτ ) (1 R) β log f(τξ) dξ] + (1 τ) α dτ(1 R) β, T where C 1 > is some constant and C = τ (R τ)/. Estimating the left-hand side of the above inequality from below, we get ( [ ( )] 3τ R ( ) R τ +α ) J su n dτ (1 R) β C 1 <R<1 R/3 ( ) R su [n(ρ) (R ρ) α+ dρ (1 R) β C 1 <R<1 R 1 R 3 ( [ ( )] 3R 1 ) n (1 R) 1++α+β su C 1 <R<1 C[n(ρ) (1 ρ) 1++α+β,

16 88 ROMI SHAMOYAN 1 AND HAIYING LI where ρ = (3R 1)/ and for any ρ (, 1), α and β >. n(ρ) (1 ρ) (α+β+1+)/ For roving the converse statement, we use Proosition A, Lemma A and the latter inequality for n(ρ). We start by fixing some t for which Lemma A and Proosition A are alicable. Then, similar to Proof of Theorem.1, Π t (z, z k ) su N, α, β C 1 <R<1 su C 1 <R<1 [ R (1 R) β (1 ρ) α [ R (1 R) β (1 ρ) α (1 R) β su C 1 <R<1 (1 R). β (1 t ) t+1 n(t)dt (1 ρt) t+1 dρ (1 s ) t+1 (α++β+1)/ (1 ρs) t+1 ds dρ Now, integrating by arts for t > (α + β + 1)/ 1 and β = t (α + β + 1)/ > 1 we get z k <R (1 z k ) t+ = (1 s) t+ dn(s) (1 s) β ds < +. Thus, these values of t rovide the alicability of Lemma A and Proosition A, and the roof is comlete. Remark.1. It is not difficult to extend the statements and the forthcoming roofs of Theorems.1,.,.3 to more general, slowly varying weights w(1 τ) from the class S (see the works [, 9, 1, 11] and their references). Remark.. The analogs of Theorems.1,.,.3 on zero sets and arametric reresentations are true for the area Nevanlinna tye classes in the uer half-lane C +, which are the analogs of the analytic classes considered above (see [14]). Acknowledgment: The authors thank the referee for his various valuable suggestions concerning the exosition of this note. References [1] R. Adams, Sobolev Saces, Academic Press, [] A. Djrbashian, F. Shamoyan, Toics in the Theory of A α Saces, Leizig, Teubner-Texte zur Mathematik, 1988.

17 DESCRIPTIONS OF ZERO SETS AND PARAMETRIC REPRESENTATIONS 89 [3] M. M. Djrbashian, On Canonical Reresentation of Functions Meromorhic in the Unit Disc, Dokl. Akad. Nauk. Arm. SSR, 3 (1), 3-9 (1945). [4] M. Djrbashian, Integral transforms and reresentations of functions in the comlex lane, Moscow, Nauka, (Russian) [5] M. Djrbashian, V. Zakharyan, Classes and boundary roerties of functions that are meromorhic in the disk, (Russian) Nauka, Moscow, [6] H. Hedenmalm, B. Korenblum and K. Zhu, Theory of Bergman saces, Graduate Graduate Texts in Mathematics, 199, Sringer-Verlag, New York,. [7] C. Horowitz, Zeros of functions in Bergman saces, Duke Math Journal, 41 (1974), [8] A. M. Jerbashian, Functions of α-bounded Tye in the Half-Plane, Advances in Comlex Analysis and Alications (Sringer, 5). [9] E. Seneta, Regularly varying functions, Nauka, Moscow, (Russian) [1] F. Shamoyan, Parametric reresentation and descrition of the root sets of weighted classes of functions holomorhic in the disk, Siberian Math Journal, 4 (6) (1999), [11] F. Shamoyan, E. Shubabko, Parametric reresentations of some classes of holomorhic functions in the unit disk, Oerator theory advances and alications, Basel, Verlag, Birkhauser, 113 (), [1] R. Shamoyan, H. Li, Descritions of zero sets and arametric reresentations of certain analytic Area Nevanlinna tye classes in the unit disk, Proceedings of A. Razmadze Mathematical Institute, 151 (9), [13] R. Shamoyan, H. Li, Characterizations of closed ideals and main arts of some analytic and meromorhic classes of area Nevanlinna tye, International Journal of Statistics and Mathematics, 1, to aear. [14] R. Shamoyan, O. Mihic, On zeros of some analytic saces of area Nevanlinna tye in a halflane, Trudy Petrozavodskogo Universiteta, 1, acceted. 1 Deartment of Mathematics, Bryansk State University, Bryansk 415, Russia. address: rsham@mail.ru College of Mathematics and Information Science, Henan Normal University, Xinxiang 4537, P. R. China address: tslhy1@yahoo.com.cn