Ukrainian Mathematical Journal

, Volume 70, Issue 7, pp 1063–1074 | Cite as

Entire Functions of Order Zero with Zeros on a Logarithmic Spiral

  • M. V. Zabolots’kyi
  • Yu. V. Basyuk
  • S. I. Tarasyuk

We prove the Valiron-type and Valiron–Titchmarsh-type theorems for entire functions of order zero with zeros on a logarithmic spiral.


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  1. 1.
    A. A. Gol’dberg and I. V. Ostrovskii, Distribution of Values of Meromorphic Functions [in Russian], Nauka, Moscow (1970).Google Scholar
  2. 2.
    G. Valiron, “Sur les fonctions entières d’ordre nul et d’ordre fini, et en particulier sur les fonctions à correspondance régulière,” Ann. Fac. Sci. Univ. Toulouse, 5, No. 3, 117–257 (1913).CrossRefGoogle Scholar
  3. 3.
    E. C. Titchmarsh, “On integral functions with real negative zeros,” Proc. London Math. Soc., 26, No. 2, 185–200 (1927).MathSciNetCrossRefGoogle Scholar
  4. 4.
    N. Wiener and R. E. A. C. Paley, Fourier Transforms in the Complex Domain, American Mathematical Society, New York (1934).CrossRefGoogle Scholar
  5. 5.
    N. A. Bowen, “A function theory proof of Tauberian theorem on integral functions,” Quart. J. Math. Oxford Ser., 19, 90–100 (1948).MathSciNetCrossRefGoogle Scholar
  6. 6.
    H. Delange, “Un théorème sur les fonctions entières à zéros réels et négatifs,” J. Math. Pures Appl., 31, No. 1, 55–78 (1952).MathSciNetzbMATHGoogle Scholar
  7. 7.
    M. V. Zabolots’kyi, “Valiron-type and Valiron–Titchmarsh-type theorems for entire function of order zero,” Ukr. Mat. Zh., 48, No. 3, 315–325 (1996); English translation: Ukr. Math. J., 48, No. 3, 354–366 (1996).Google Scholar
  8. 8.
    S. K. Balashov, “On the entire functions of finite order with zeros on the curves of regular rotation,” Izv. Akad. Nauk SSSR, Ser. Mat., 37, No. 3, 603–629 (1973).MathSciNetzbMATHGoogle Scholar
  9. 9.
    A. I. Kheifits, “Analog of the Valiron–Titchmarsh theorem for entire functions with zeros on a logarithmic spiral,” Izv. Vyssh. Uchebn. Zaved., Ser. Mat., No. 12, 74–75 (1980).Google Scholar
  10. 10.
    E. Seneta, Regularly Varying Functions [Russian translation], Nauka, Moscow (1985).Google Scholar
  11. 11.
    N. V. Zabolotskii, “Strongly regular growth of entire functions of order zero,” Math. Notes, 63, No. 2, 172–182 (1998).MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  • M. V. Zabolots’kyi
    • 1
  • Yu. V. Basyuk
    • 1
  • S. I. Tarasyuk
    • 1
  1. 1.Franko Lviv National UniversityLvivUkraine

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