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


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), 101-129
MSC (2000): Primary 30C15.
Published electronically: November 14, 2008
MathSciNet review: 2411972
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Abstract | References | Similar Articles | Additional Information

Abstract: For a domain $ \Omega$ in the complex plane $ \mathbb{C}$, let $ H(\Omega)$ denote the space of functions holomorphic in $ \Omega$, and let $ \mathscr{P}$ be a family of functions subharmonic in $ \Omega$. Denote by $ H_{\mathscr{P}}(\Omega )$ the class of $ f\in H(\Omega)$ satisfying $ \vert f(z)\vert\leq C_f\exp p_f(z)$, $ z\in \Omega$, where $ p_f \in \mathscr{P}$ and $ C_f$ is a constant. The paper is aimed at conditions for a set $ \Lambda \subset \Omega$ to be included in the zero set of some nonzero function in $ H_{\mathscr{P}}(\Omega )$. In the first part, certain preparatory theorems are established concerning ``quenching'' the growth of a subharmonic function by adding to it a function of the form $ \log \vert h\vert$, where $ h$ is a nonzero function in $ H(\Omega)$. The method is based on the balayage of measures and subharmonic functions.


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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/S1061-0022-08-01040-6
Keywords: Holomorphic function, algebra of functions, weighted spaces, nonuniqueness sequence.
Received by editor(s): November 6, 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