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Zeros of holomorphic functions in the unit disk and \(\rho \)-trigonometrically convex functions

  • Bulat N. KhabibullinEmail author
  • Farkhat B. Khabibullin
Article
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Abstract

Let M be a subharmonic function with Riesz measure \(\mu _M\) on the unit disk \({\mathbb {D}}\) in the complex plane \({\mathbb {C}}\). Let f be a nonzero holomorphic function on \({\mathbb {D}}\) such that f vanishes on \({\textsf {Z}}\subset {\mathbb {D}}\), and satisfies \(|f| \le \exp M\) on \({\mathbb {D}}\). Then restrictions on the growth of \(\mu _M\) near the boundary of D imply certain restrictions on the distribution of \(\mathsf Z\). We give a quantitative study of this phenomenon in terms of special non-radial test functions constructed using \(\rho \)-trigonometrically convex functions.

Keywords

Holomorphic function Zero set Subharmonic function Riesz measure Uniqueness theorem \(\rho \)-Trigonometrically convex function 

Mathematics Subject Classification

Primary 30C15 31A05 Secondary 31A15 

Notes

Acknowledgements

The authors thank the organizers of International Conferences “Complex Analysis and Related Topics 2018” (April 23–27, 2018, Euler International Mathematical Institute, St. Petersburg, Russia) and “XXVII St.Petersburg Summer Meeting in Mathematical Analysis” (August 6–11, 2018, St. Petersburg, Russia) for the invitation and for the opportunity to report the results related to the content of this article. The authors are also very grateful to the reviewer for useful comments and corrections.

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.

References

  1. 1.
    Djrbashian, M.M.: Integral Transforms and Representation of Functions in the Complex Domain. Nauka, Moscow (1966). (Russian)Google Scholar
  2. 2.
    Grishin, A.F., Malyutin, K.G.: Trigonometrically Convex Functions. South-West State Univ., Kursk (2015). (Russian)Google Scholar
  3. 3.
    Grishin, A.F., Sodin, M.L.: Growth on a ray, distribution of zeros in terms of their arguments for entire functions of finite order, and a uniqueness theorem. Teor. Funktsii, Funktsional. Anal. i Prilozhen. 50, 47–61 (1988) (Russian); English transl. in J. Soviet Math. 49(6), 1269–1279 (1990)Google Scholar
  4. 4.
    Hayman, W., Kennedy, P.: Subharmonic Functions. Academic Press, London (1976)zbMATHGoogle Scholar
  5. 5.
    Khabibullin, B.N.: A uniqueness theorem for subharmonic functions of finite order. Matem. Sb. 182(6), 811–827 (1991); English transl. in Math. USSR-Sb. 73(1) 195–210 (1992)Google Scholar
  6. 6.
    Khabibullin, B.N.: Completeness of systems of entire functions in spaces of holomorphic functions. Mat. Zametki, 66(4), 603–616 (1999); English transl. in Math. Notes, 66(4), 495–506 (1999)Google Scholar
  7. 7.
    Khabibullin, B.N.: Completeness of systems of exponentials and sets of uniqueness, 4th edn. (revised and enlarged). Bashkir State Univ., Ufa (Russian) (2012). http://www.researchgate.net/publication/271841461
  8. 8.
    Khabibullin, B.N., Abdullina, Z.F., Rozit, A.P.: A uniqueness theorem and subharmonic test functions. Algebra i Analiz, 30(2), 318–334 (2018); English transl. in St. Petersb. Math. J.Google Scholar
  9. 9.
    Khabibullin, B.N., Rozit A.P.: On the Distributions of Zero Sets of Holomorphic Functions. Funktsional. Anal. i Prilozhen., 52(1), 26–42 (2018); English. transl. in Funct. Anal. Appl., 52(1), 21–34 (2018)Google Scholar
  10. 10.
    Khabibullin, B.N., Tamindarova, N.R.: Distribution of zeros and masses for holomorphic and subharmonic functions. I. Hadamard- and Blaschke-type conditions (Russian) (2015–2018). arXiv:1512.04610v4
  11. 11.
    Khabibullin, B., Tamindarova, N.: Distribution of zeros for holomorphic functions: Hadamard- and Blaschke-type conditions. In: Abstracts of International Workshop on “Non-harmonic Analysis and Differential Operators” (May 25–27, 2016), Institute of Mathematics and Mechanics of Azerbaijan National Academy of Sciences, Azerbaijan, Baku, 63 (2016)Google Scholar
  12. 12.
    Khabibullin, B.N., Tamindarova, N.R.: Subharmonic test functions and the distribution of zero sets of holomorphic functions. Lobachevskii J. Math. 38(1), 38–43 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Khabibullin, B., Tamindarova, N.: Uniqueness theorems for subharmonic and holomorphic functions of several variables on a domain. Azerb. J. Math. 7(1), 70–79 (2017)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Leont’ev, A.F.: Generalization of Series of Exponentials. Nauka, Moscow (1981). (Russian)zbMATHGoogle Scholar
  15. 15.
    Levin, B. Ya.: Distribution of zeros of entire functions, GITTL, Moscow (Russian) (1956); Am. Math. Soc., Providence, RI (1964); rev. ed. (1980)Google Scholar
  16. 16.
    Levin, B.Y.: Lectures on entire functions. Transl. Math. Monogr. 150, Am. Math. Soc. (1996)Google Scholar
  17. 17.
    Maergoiz, L.S.: Asymptotic Characteristics of Entire Functions and Their Applications in Mathematics and Biophysics. Nauka, SO, Novosibirsk (Russian) (1991), 2nd edn. (revised and enlarged). Kluwer Academic Publishers, Dordrecht (2003)Google Scholar
  18. 18.
    Ransford, Th: Potential Theory in the Complex Plane. Cambridge University Press, Cambridge (1995)CrossRefzbMATHGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of Mathematics and ITBashkir State UniversityUfaRussia

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