Skip to main content
SpringerLink
Log in

The logarithm of the modulus of a holomorphic function as a minorant for a subharmonic function. II. The complex plane

  • Published:
Mathematical Notes Aims and scope Submit manuscript

Abstract

Let u ≢ −∞be a subharmonic function in the complex plane. We establish necessary and/or sufficient conditions for the existence of a nonzero entire function f for which the modulus of the product of each of its kth derivative k = 0, 1,..., by any polynomial p is not greater than the function Ce u in the entire complex plane, where C is a constant depending on k and p. The results obtained significantly strengthen and develop a number of results of Lars Hörmander (1997).

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

References

  1. W. K. Hayman and P. B. Kennedy, Subharmonic Functions (Academic Press, London–New York–San Francisco, 1976; Mir, Moscow, 1980), Vol. 1.

    MATH  Google Scholar 

  2. W. K. Hayman, Subharmonic Functions, Vol. 2, in London Math. Soc.Monogr. (Academic Press, London, 1989), Vol. 20.

    MATH  Google Scholar 

  3. Th. Ransford, Potential Theory in the Complex Plane, in London Math. Soc. Stud. Texts (Cambridge Univ. Press, Cambridge, 1995), Vol. 28.

    Google Scholar 

  4. B. Ya. Levin, Lectures on Entire Functions, in Transl. Math. Monogr. (Amer. Math. Soc., Providence, RI, 1996), Vol.150.

    Google Scholar 

  5. L. Hörmander, “On the Legendre and Laplace transformations,” Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 25 (3-4), 517–568 (1998).

    MathSciNet  MATH  Google Scholar 

  6. B. N. Khabibullin and T. Yu. Baiguskarov, “The logarithm of the modulus of a holomorphic function as a minorant for a subharmonic function,” Mat. Zametki 99 (4), 588–602 (2016) [Math. Notes 99 (3–4), 576–589 (2016)].

    Article  MathSciNet  MATH  Google Scholar 

  7. B. Khabibullin and T. Baiguskarov, “Holomorphicminorants,” in algebra, Analysis and Related Questions of Mathematical Modeling Vladikavkaz, June 26–27, 2015 (SOGU, Vladikavkaz, 2015), pp. 103–105 [in Russian].

    Google Scholar 

  8. B. Khabibullin and T. Baiguskarov, Holomorphic Minorants of (Pluri-)Subharmonic Functions, arXiv: 1506.01746.

  9. I. P. Natanson, Theory of Functions of a Real Variable (Nauka, Moscow, 1974) [in Russian].

    Google Scholar 

  10. A. F. Beardon, “Integral means of subharmonic functions,” Proc. Camb. Philos. Soc. 69, 151–152 (1971).

    Article  MathSciNet  MATH  Google Scholar 

  11. P. Freitas and J. P. Matos, “On the characterization harmonic and subharmonic functions via mean-value properties,” Potential Anal. 32 (2), 189–200 (2010).

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Bashkir State University, Ufa, Russia

    T. Yu. Baiguskarov, B. N. Khabibullin & A. V. Khasanova

Authors
  1. T. Yu. Baiguskarov
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. B. N. Khabibullin
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. A. V. Khasanova
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to T. Yu. Baiguskarov.

Additional information

Original Russian Text © T. Yu. Baiguskarov, B. N. Khabibullin, A. V. Khasanova, 2017, published in Matematicheskie Zametki, 2017, Vol. 101, No. 4, pp. 483–502.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Baiguskarov, T.Y., Khabibullin, B.N. & Khasanova, A.V. The logarithm of the modulus of a holomorphic function as a minorant for a subharmonic function. II. The complex plane. Math Notes 101, 590–607 (2017). https://doi.org/10.1134/S000143461703018X

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S000143461703018X

Keywords

Search

Navigation