# Article

Full entry | PDF   (0.4 MB)
Keywords:
Analytic function; area Nevanlinna type spaces; zero sets; factorization representation; meromorphic function; subharmonic function.
Summary:
The survey collects many recent advances on area Nevanlinna type classes and related spaces of analytic functions in the unit disk concerning zero sets and factorization representations of these classes and discusses approaches, used in proofs of these results.
References:
[1] Avetisyan, K.: On representation of some classes of subharmonic functions in the unit disk and in the half plane. Izv. Nats. Acad. Armenii, Matematika., 29, 1, 1994, 1-13, (In Russian). MR 1438866
[2] Bednazh, V.A., Karbanovich, O.V., Shamoyan, F.A.: On some weighted spaces of meromorphic functions with restrictions of the Nevanlinna characteristic. Modern problems of complex and harmonic analysis: Abstracts of conference, Bryansk St. Univ., 2014, p. 56, (In Russian).
[3] Beller, E.: Zeros of $A^p$ functions and related classes of analytic functions. Israel Jour. Math., 22, 1975, 68-80, 1. DOI 10.1007/BF02757275 | MR 0385103
[4] Beller, E.: Factorization for non-Nevanlinna classes of analytic functions. Israel Jour. Math., 27, 3--4, 1977, 320-330, MR 0442234
[5] Chyzhykov, I.E.: Growth of analytic functions in the unit disc and complete measure in the sense of Grishin. Mat. Stud., 29, 1, 2008, 35-44, MR 2424597
[6] Chyzhykov, I.E.: Zero distribution and factorization of analytic functions of slow growth in the unit disc. Proc. Amer. Math. Soc., 141, 2013, 1297-1311, DOI 10.1090/S0002-9939-2012-11463-7 | MR 3008877
[7] Chyzhykov, I.E.: Growth of $p$-th means of analytic and subharmonic function in the unit disk and angular distribution of zeros. arXiv:1509.02141, 2015, 1-19, MR 2385361
[8] Chyzhykov, I.E., Skaski, S.: Growth, zero distribution and factorization of analytic functions of moderate growth in the unit disc, Blashke products and their applications. Fields Inst. Comm., 65, 2013, 159-173, MR 3052292
[9] Djrbashian, M.M.: On a problem of representation of analytic functions. Soobsch. Inst. Mat. i Meh. AN Arm. SSR, 2, 1948, 3-40, (In Russian).
[10] Djrbashian, M.M.: On parametric representation of some classes of meromorphic functions in the unit disk. Dokl. AN SSSR., 157, 1964, 1024-1027, (in Russian).
[11] Djrbashian, M.M.: Integral transforms and representation in complex plane. 1966, p. 671, Nauka, GITTL. Moscow, (In Russian).
[12] Djrbashian, M.M.: The factorization theory of functions, meromorphic in the unit disk. Mat. Sb., 79, 121, 1969, 517-615, (In Russian).
[13] Djrbashian, A.M., Mikaelyan, G.V.: On boundary properties of Blaschke type products. Izv. Nats. AN Armenii, Matematika, 26, 1991, 435-442, (In Russian). MR 1328283
[14] Goldberg, A.A., Ostrovskii, I.V.: On distribution of values of meromorphic functions. 1970, p. 457, Nauka. Moscow, (In Russian). MR 0280720
[15] Hardy, G., Littlwood, J., Polia, G.: Inequalities. 1948, p. 456, GITTL, Moscow, (In Russian). MR 0083530
[16] Hayman, W.: Meromorphic functions. 1966, p. 447, Mir, Moscow, (In Russian). MR 0214768
[17] Heipler, A.: The zeros of functions in Nevanlinna's area class. Israel Jour. Math., 34, 1--2, 1979, 1-11, DOI 10.1007/BF02761820 | MR 0571391
[18] Kasyaniuk, S.Ya.: On functions from $A$ and $H$ class in case of circular rings. Mat. Sb., 42, 84, 1957, 301-326, (In Russian).
[19] Krylov, V.I.: On functions regular in half plane. Mat. Sb., 6, 48, 1939, 95-138, (In Russian).
[20] Kursina, I.S.: Factorization and parametric representation of weighted classes of analytic functions: the thesis abstract. Voronezh State Univ., Voronezh, 4, 2000, 43-54, (In Russian).
[21] Matevosyan, G.U.: On an factorization of functions meromorphic in multiconnected domains and applications. Izv. AN Armenii, Matematika, 9, 1974, 387-408, (In Russian).
[22] Matevosyan, G.U.: An analogue of $N(w)$ class in case of circular rings. Izv. AN Armenii, Matematika., 21, 2, 1977, 173-182, (In Russian).
[23] Nevanlinna, R.: Single-Valued Analytic Functions. 1941, p. 388, Gostehizdat, Moscow, (In Russian).
[24] Ohlupina, O.V.: Description of a class of subharmonic functions in the unit disk, whose Nevanlinna's characteristic is in wheigted $L^p$-spaces. Vestnik Samarskogo Univ., 9/1, 59, 2008, 108-120, (In Russian). MR 2436023
[25] Ohlupina, O.V.: Parametric representation of some classes of subharmonic functions in a half plane with characteristic from weighted $L^p$-spaces. Vestnik Bryanskogo Univ., Bryansk State Univ., 4, 2010, 24-36, (In Russian).
[26] Ohlupina, O.V.: Green type potentials and integral representation of wheigted classes of subharmonic functions: the PhD thesis abstract. 38, 2012, 56-77, Saratov St. Univ., Saratov, (In Russian).
[27] Ortega-Cerda, J., Bruna, J.: On $L^p$-solutions of the Laplace equation and zeros of holomorphic functions. Annali della Scuola Normale de Pisa, 24, 1997, 571-591, MR 1612330
[28] Prihodko, O.V.: On root sets of some area Nevanlinna type spaces in angular domains on a complex plane. Vestnik Bryanskogo Univ. Bryansk State Univ., Bryansk, 4, 2010, 36-40, (In Russian).
[29] Privalov, I.I.: Boundary properties of single-valued analytic functions. 1941, 206 p., Moscow St. Univ., Moscow, (In Russian). MR 0047765
[30] Privalov, I.I.: Boundary properties of analytic functions. 1950, 337 p., Gostekhizdat, Moscow and Leningrad, (In Russian). MR 0047765
[31] Privalov, I.I., Kuznetsov, P.I.: Boundary properties and various classes of harmonic and subharmonic functions, defined in arbitrary domains. Mat. Sbornik, 6, 48, 1939, 345-376, (In Russian).
[32] Rodikova, E.G.: Factorization and description of zero sets of a class of functions, analytic in the unit disk. Sib. Elektron. Mat. Izv., 11, 2014, 52-63, (In Russian). MR 3488405
[33] Rodikova, E.G.: Factorization, root sets and interpolation in weighted classes of analytic functions. 2014, 121 p., Bryansk, Dissertation, Bryansk, (2014) ,121 p. (In Russian).
[34] Rodikova, E.G.: On zeros of analytic classes of Privalov. Abstracts of Saratov winter school, Saratov, 1, 39, 2012, 141-142, (In Russian).
[35] Seneta, E.: Regularly varying functions. 1985, 141 p., Nauka, Moscow, (In Russian). MR 0815924
[36] Shamoyan, F.A.: Description of closed ideals and some issues of factorization in algebras of growing functions, analytic in the disk. Izv. Acad. nauk ArmSSR, Matematika, 5, 1970, 450-472, (In Russian). MR 0291815
[37] Shamoyan, F.A.: Djrbashian's factorization theorem and characterization of zeros of analytic functions in a disk with bounded growth rate. Izv. Acad. nauk ArmSSR, Matematika, 13, 1978, 405-422, (In Russian).
[38] Shamoyan, F.A.: Some remarks to parametric representation of Nevanlinna-Djrbashian. Mat. zametki., 52, 1992, 128-140, (In Russian).
[39] Shamoyan, F.A.: Parametric representation and description of zero sets of weighted classes of holomorphic functions in the disk. Sib. Math. Journ., 40, 1999, 1422-1440, (In Russian). DOI 10.1007/BF02677545 | MR 1741095
[40] Shamoyan, F.A.: Weighted spaces of analytic functions with mixed norm. 2014, 250 p., Bryansk St. Univ., Bryansk, Bryansk State University Publishing House, (In Russian).
[41] Shamoyan, F.A., Bednazh, V.A., Karbanovich, O.V.: On classes of analytic functions in a disk with a characteristic R. Nevanlinny and $\alpha$-characteristic of weighted $L^p$ spaces. Sib. Elektron. Mat. Izv., 12, 2015, 150-167, (In Russian). MR 3493791
[42] Shamoyan, F.A., Bednazh, V.A., Prihodko, O.V.: On zero sets of some weighted classes of analytc functions in the disk. Vesthik Bryanskogo Universiteta, Bryansk, 4, 2008, 85-92, (In Russian).
[43] Shamoyan, F.A., Rodikova, E.G.: On characterization of root sets of a weighted class of analytic functions in a disk. Vladikavkaz Mat. Jour., 16, 3, 2014, 64-75, (In Russian). MR 3540665
[44] Shamoyan, F.A., Shubabko, E.N.: Parametrical representations of some classes of holomorfic functions in the disc. Complex Analysis, Operators, and Related Topics, Oper. Theory Adv. Appl, 113, 2000, 331-338, (In Russian). DOI 10.1007/978-3-0348-8378-8_26 | MR 1771772
[45] Shamoyan, F.A., Shubabko, E.N.: On a class of holomorphic functions in a disk. Zapiski Nauchnykh Seminarov POMI., 282, 2001, 244-255, (In Russian). MR 1874891
[46] Shamoyan, F.A., Shubabko, E.N.: Introduction in the theory of $L^p$-classes of meromorphic functions. 2009, 152 p, Bryansk State University Publishing House, (In Russian).
[47] Shamoyan, R.F.: On new parametric representations of analytic area Nevanlinna type classes in a circular ring $K$ on a complex plane $\mathbb {C}$. Jour. Sib. Fed. Univ. Math. Phys., 6, 1, 2013, 114-119, MR 3168451
[48] Shamoyan, R.F., Li, H.: Descriptions of zero sets and parametric representations of certain new analytic area Nevanlinna type spaces in the unit disk. Kragujevac Journal of Mathematics, 34, 2010, 73-89, MR 2897897
[49] Shamoyan, R.F., Mihic, O.R.: On zeros of some analytic spaces of area Nevanlinna type in halfplane. Trudy Petrozavodsk St. Univ., Matematika., 17, 2010, 67-72, MR 3260491
[50] Shamoyan, R.F., Mihic, O.R.: On some new theorems on certain analytic and meromorphic classes of Nevanlinna type on the complex plane. Kragujevac Math. Journal, 37, 1, 2013, 65-85, MR 3073698
[51] Shamoyan, R.F., Mihic, O.R.: On zeros sets and embeddings of some new analytic function spaces in the unit disk. Kragujevac Math. Journal, 38, 2, 2014, 229-244, DOI 10.5937/KgJMath1402229S | MR 3301453
[52] Shamoyan, R., Shipka, A.: Integral operators, embedding theorems and Taylor coefficients of area Nevanlinna type spaces of several variables and related problems. 2018, Preprint..
[53] Shvedenko, S.V.: Canonical products of Blashke type for spaces of area Nevanlinna type. Mat. Zametki., 37, 2, 1985, 212-219, (In Russian). MR 0784366
[54] Shvedenko, S.V.: The Hardy classes and related spaces of analytic functions in the unit circle, in the ball and in the polydisk. Itogi nauki i tehniki, Mat. analiz., 23, 1985, 3-124, MR 0824267
[55] Stein, E.: Singular integrals and differentiability properties of functions. 1970, 290 p., Princeton Univ. Press, New Jersey, MR 0290095 | Zbl 0207.13501
[56] Tsuji, M.: Canonical product for a meromorphic functions in a unit circle. Journ. Math. Jap., 8, 1, 1956, 7-19, MR 0103273
[57] Tsuji, M.: Potential Theory in modern function theory. 1959, p. 345, Maruzen, Tokyo, MR 0114894
[58] Zmorovich, V.: On certain spaces of analytic functions in circular rings. Mat. Sbornik, 32, 1953, 643-652, (In Russian). MR 0055448

Partner of