Skip to main content
SpringerLink
Log in

ON THE TYPE OF SUBHARMONIC FUNCTIONS OF FINITE ORDER

  • Published:
Journal of Mathematical Sciences Aims and scope Submit manuscript

Abstract

Theorems establishing a connection between the type of an entire function and the distribution of its zeros play an important role in the theory of entire functions. The classical Lindelöf theorem asserts that zeros of an entire function of finite integer order and normal type possess a certain symmetry property. We prove counterparts of the Lindelöf theorem in the spaces of subharmonic functions in the complex plane and in the half-plane the growth of which is determined by the proximate order in the sense of Boutroux. The results are formulated in the terms of the upper density and the upper balance of the Riesz measure and the full measure of functions.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Data Availability

It is not applicable in our paper.

References

  1. B. Ya. Levin, Distribution of Zeros of Entire Functions, Am. Math. Soc., Providence, RI (1964).

  2. E. Lindelöf, “Sur les fonctions entières d’ordre entier”, Ann. scientif. de l’É.N.S. 3\(^{e}\)sér. No. 22, 369–395 (1905).

  3. K. G. Malyutin, M. V. Kabanko, and I. V. Kostenko, "Generalization of the Lindelöf Theorem to the Case of Boutroux Proximate Order. II", J. Math. Sci. 264, No. 5, 609–616 (2022).

  4. P. Boutroux, “Sur quelques propriétés des fonctions entières”, Acta Math. 28, 97–224 (1904).

  5. Yu. F. Korobeinik, “On entire analytic solutions of equations of infinite order with polynomial coefficients”, Dokl. AN SSSR, 157, No. 5, 1031–1034 (1964). (in Russian)

  6. G. G. Braichev, Joint estimates for zeros and Taylor coefficients of entire function, Ufa Mathematical Journal, 13, No. 1, 31–45 (2021).

  7. G. G. Braichev, “On the Lower Indicator of an Entire Function with Roots of Zero Lower Density Lying on a Ray”, Math. Notes, 107, No.6, 877–889 (2021).

  8. G. G. Braichev, “Two-sided estimates for the relative growth of functions and their derivatives”, Ufa Mathematical Journal, 9 , No. 1, 18–26 (2021).

  9. G. G. Braichev, “The least type of an entire function whose zeros have prescribed averaged densities and lie on rays or in a sector”, Sb. Math., 207, No.2, 45–80 (2021).

  10. G. G. Braichev, “Sharp Estimates of Types of Entire Functions with Zeros on Rays”, Math. Notes, 97, No.4, pp. 503–515 (2015).

  11. G. G. Braichev, V. B. Sherstyukov, “Estimates of indicators of an entire function with negative roots”, Vladikavkaz Mathematical Journal, 22, No.3, 30–46 (2020).

  12. G. G. Braichev, V. B. Sherstyukov, “Estimates of indicators of an entire function with negative roots”, Fundamental and Applied Mathematics, 22, No. 1, 51–97 (2020).

  13. G. G. Braichev, O. V Sherstyukova, “Estimates of indicators of an entire function with negative roots”, Ufa Mathematical Journal, 14, No.3, 17–22 (2022).

  14. B. N. Khabibullin, E. B. Menshikova, “Preorders on Subharmonic Functions and Measures with Applications to the Distribution of Zeros of Holomorphic Functions”, Lobachevskii Journal of Mathematics, 43, No.3, 587–611 (2022).

  15. A. E. Salimova, B. N. Khabibullin, “Growth of Entire Functions of Exponential Type and Characteristics of Distributions of Points Along Straight Line in Complex Plane”, Ufa Mathematical Journal, 13, No.3, pp. 116–128 (2021).

  16. A. E. Salimova, B. N. Khabibullin, “Distribution of Zeros of Exponential-Type Entire Functions with Constraints on Growth along a Line”, Math. Notes, 108, No.4, 579–589 (2020).

  17. H. Habib, B. Belaidi, “Hyper-order and fixed points of meromorphic solutions of higher order linear differential equations”, Arab Journal of Mathematical Sciences, 22, No. 1, 96–114 (2016).

  18. L. A. Rubel, “A Fourier series method for entire functions”, Duke Math. J., 30, 437–442 (1963).

  19. L. A. Rubel, Entire and meromorphic functions, New York–Berlin–Heidelberg, Springer (1996).

  20. K. G. Malyutin, “Fourier series and \(\delta\)-subharmonic functions of finite \(\gamma\)-type in a half-plane”, Sb. Math. 192, No. 6, 843–861 (2001).

  21. W. K. Hayman, P. B. Kennedy, Subharmonic Functions. Volume I, London – New York – San Francisco, Academic Press (1976).

  22. M. A. Fedorov and A. F. Grishin, “Some Questions of Nevanlinna Theory for the Complex Half–Plane”, Math. Physics, Analysis and Geometry (Kluwer Acad. Publish.) 1, No 3, 223–271 (1998).

  23. A. F. Grishin and T. I. Malyutina, “General properties of subharmonic functions of finite order in a complex half-plane”, Vestnik Kharkov Nats. Univ. Ser. Mat. Prikl. Mat. Mekh., 475, 20–44 (2000).

  24. K. G. Malyutin and N. Sadik, “Representation of subharmonic functions in a half-plane”, Sb. Math. 198, No. 12, 1747–1761 (2007).

  25. K. G. Malyutin, M. V. Kabanko, and I. V. Kostenko, “Generalization of the Lindelöf theorem to the case of Boutroux proximate order”, J. Math. Sci. 262, No. 3, 301–311 (2022).

  26. N. V. Govorov, Riemann’s Boundary Problem with Infinite Index, Basel; Boston; Berlin: Birkäuser (1994).

  27. A. Edrei and W. H. Fuchs, “Meromorphic functions with several deficient values”, Trans. Amer. Math. Soc. 93, 292–328 (1959).

  28. K. G. Malyutin and M. V. Kabanko, “Inverse formulas for the Fourier coefficients of meromorphic functions in a half-plane”, J. Math. Sci. 260, No. 6, 798–806 (2022).

Download references

Acknowledgements

We gratefully thank the referees for careful reading of the paper and for the suggestions that have greatly improved the paper. The work is supported by the Russian Science Foundation (project No. 22-21-00012, https://rscf.ru/project/22-21-00012/).

Author information

Authors and Affiliations

  1. Kursk State University, 33 Radishcheva St, 305000, Kursk, Russia

    K. G. Malyutin & M. V. Kabanko

Authors
  1. K. G. Malyutin
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. M. V. Kabanko
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to K. G. Malyutin.

Ethics declarations

Conflict of interest

The authors declare no competing interests.

Additional information

Dedicated to the 80 anniversary of Professor Nikolai K. Karapetiants and to the memory of Professor M.M. Dragilev.

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Malyutin, K.G., Kabanko, M.V. ON THE TYPE OF SUBHARMONIC FUNCTIONS OF FINITE ORDER. J Math Sci 266, 981–1001 (2022). https://doi.org/10.1007/s10958-022-06246-4

Download citation

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10958-022-06246-4

Keywords

Mathematics Subject Classification

Search

Navigation