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Subsequences of zeros for classes of holomorphic functions, their stability, and the entropy of arcwise connectedness. II


Authors: B. N. Khabibullin, F. B. Khabibullin and L. Yu. Cherednikova
Translated by: S. Kislyakov
Original publication: Algebra i Analiz, tom 20 (2008), nomer 1.
Journal: St. Petersburg Math. J. 20 (2009), 131-162
MSC (2000): Primary 30C15
DOI: https://doi.org/10.1090/S1061-0022-08-01041-8
Published electronically: November 14, 2008
MathSciNet review: 2411973
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Abstract | References | Similar Articles | Additional Information

Abstract: Let $ \Omega$ be a domain in the complex plane $ \mathbb{C}$, $ H(\Omega)$ the space of functions holomorphic in $ \Omega$, and $ \mathscr{P}$ a family of functions subharmonic in $ \Omega$. Denote by $ H_{\mathscr{P}}(\Omega )$ the class of functions $ f\in H(\Omega)$ satisfying $ \vert f(z)\vert\leq C_f\exp p_f(z)$ for all $ z\in \Omega$, where $ p_f \in \mathscr{P}$ and $ C_f$ is a constant. Conditions are found ensuring that a sequence $ \Lambda =\{\lambda_k\} \subset \Omega$ be a subsequence of zeros for various classes $ H_{\mathscr{P}}(\Omega )$. As a rule, the results and the method are new already when $ \Omega=\mathbb{D}$ is the unit circle and $ \mathscr{P}$ is a system of radial majorants $ p(z)=p(\vert z\vert)$.


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  • 1. B. N. Khabibullin, F. B. Khabibullin, and L. Yu. Cherednikova, Subsequences of zeros for classes of holomorphic functions, their stability, and the entropy of arcwise connectedness. I, Algebra i Analiz 20 (2008), no. 1, 146-189; English transl., St. Petersburg Math. J. 20 (2009), no. 1, 101-129. MR 2411972
  • 2. V. S. Azarin, Asymptotic behavior of subharmonic functions of finite order, Mat. Sb. (N.S.) 108 (150) (1979), no. 2, 147-167; English transl., Math. USSR-Sb. 36 (1979), no. 2, 135-154 (1980). MR 0525835 (81e:31001)
  • 3. K. Leichtweiss, Konvexe Mengen, Hochschultext, Springer-Verlag, Berlin-New York, 1980. MR 0586235 (81j:52001)
  • 4. T. J. Ransford, Potential theory in the complex plane, London Math. Soc. Student Texts, vol. 28, Cambridge Univ. Press, Cambridge, 1995. MR 1334766 (96e:31001)
  • 5. B. Korenblum, An extension of the Nevanlinna theory, Acta Math. 135 (1975), no. 3-4, 187-219. MR 0425124 (54:13081)
  • 6. K. Seip, On a theorem of Korenblum, Ark. Mat. 32 (1994), 237-243. MR 1277927 (95f:30054)
  • 7. -, On Korenblum's density condition for the zero sequences of $ A^{-\alpha}$, J. Anal. Math. 67 (1995), 307-322. MR 1383499 (97c:30044)
  • 8. J. Bruna and X. Massaneda, Zero sets of holomorphic functions in the unit ball with slow growth, J. Anal. Math. 66 (1995), 217-252. MR 1370351 (97f:32006)
  • 9. D. Luecking, Zero sequences for Bergman spaces, Complex Variables Theory Appl. 30 (1996), 345-362. MR 1413164 (97g:30007)
  • 10. H. Hedenmalm, B. Korenblum, and K. Zhu, Theory of Bergman spaces, Grad. Texts in Math., vol. 199, Springer-Verlag, New York, 2000. MR 1758653 (2001c:46043)
  • 11. B. N. Khabibullin, Zero (sub)sets for spaces of holomorphic functions and (sub)harmonic minorants, Electronic Archive at LANL, 18 Dec 2004, 42 pp., http://arxiv.org/abs/math.CV/0412359.
  • 12. F. A. Shamoyan, A factorization theorem of M. M. Dzhrbashyan and the characteristic of zeros of functions analytic in the circle with a majorant of finite growth, Izv. Akad. Nauk Armyan. SSR Ser. Mat. 13 (1978), no. 5-6, 405-422; English transl. in Soviet J. Contemporary Math. Anal. 13 (1978), no. 5-6. MR 0541789 (80i:30054)
  • 13. W. K. Hayman and P. B. Kennedy, Subharmonic functions. Vol. 1, London Math. Soc. Monogr., No. 9, Academic Press, London-New York, 1976. MR 0460672 (57:665)
  • 14. E. Beller, Factorization for non-Nevanlinna classes of analytic functions, Israel J. Math. 27 (1977), no. 3-4, 320-330. MR 0442234 (56:620)
  • 15. L. Yu. Cherednikova, Nonuniqueness sequences for weighted algebras of holomorphic functions in the unit disk, Mat. Zametki 77 (2005), no. 5, 775-787; English transl., Math. Notes 77 (2005), no. 5-6, 715-725. MR 2178847 (2006i:30074)
  • 16. F. A. Shamoyan, Zeros of functions analytic in the disk and growing near the boundary, Izv. Akad. Nauk Armyan. SSR Ser. Mat. 18 (1983), no. 1, 15-27; English transl., Soviet J. Contemporary Math. Anal. 18 (1983), no. 1, 13-25. MR 0705983 (84g:30032)
  • 17. B. N. Khabibullin, Growth of entire functions with prescribed zeros and representation of meromorphic functions, Mat. Zametki 73 (2003), no. 1, 120-134; English transl., Math. Notes 73 (2003), no. 1-2, 110-124. MR 1993545 (2004i:30023)
  • 18. -, Zero subsets, representation of meromorphic functions, and Nevanlinna characteristics in a disk, Mat. Sb. 197 (2006), no. 2, 117-136; English transl., Sb. Math. 197 (2006), no. 1-2, 259-279. MR 2230094 (2007b:30034)

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Additional Information

B. N. Khabibullin
Affiliation: Bashkir State University, Institute of Mathematics with Computer Center, Urals Scientific Center, Russian Academy of Sciences, Ufa, Bashkortostan, Russia
Email: khabib-bulat@mail.ru

F. B. Khabibullin
Affiliation: Bashkir State University, Institute of Mathematics with Computer Center, Urals Scientific Center, Russian Academy of Sciences, Ufa, Bashkortostan, Russia

L. Yu. Cherednikova
Affiliation: Bashkir State University, Institute of Mathematics with Computer Center, Urals Scientific Center, Russian Academy of Sciences, Ufa, Bashkortostan, Russia

DOI: https://doi.org/10.1090/S1061-0022-08-01041-8
Keywords: Holomorphic function, algebra of functions, weighted space, nonuniqueness sequence
Received by editor(s): November 8, 2006
Published electronically: November 14, 2008
Additional Notes: Supported by RFBR, grant no. 06-01-00067, and by the Program of state subventions for leading scientific schools, grant NSh-10052.2006.1
Article copyright: © Copyright 2008 American Mathematical Society

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