Ukrainian Mathematical Journal

, Volume 69, Issue 11, pp 1749–1761 | Cite as

On Rationally Loxodromic Holomorphic Functions

  • Dz. V. Lukivs’ka
  • A. Ya. Khrystiyanyn

We consider a functional equation of the form f(qz) = R(z)f(z), where R(z) is a rational function, z 𝜖 ℂ\{0}, q 𝜖 ℂ\{0}, |q| < 1. Holomorphic solutions of this equation are obtained. These solutions can be regarded as generalizations of p-loxodromic functions.


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  1. 1.
    D. G. Crowdy, “Geometric function theory: a modern view of a classical subject,” Nonlinearity, 21, No. 10, T205–T219 (2008).MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    J. Marcotte and M. Salomone “Loxodromic spirals in M. C. Escher’s sphere surface,” J. Humanist. Math., 4, No. 2, 25–46 (2014).MathSciNetCrossRefGoogle Scholar
  3. 3.
    Y. Hellegouarch, Invitation to the Mathematics of Fermat–Wiles, Academic Press, London (2002).zbMATHGoogle Scholar
  4. 4.
    O. Hushchak and A. Kondratyuk, “The Julia exceptionality of loxodromic meromorphic functions,” Visn. Lviv Univ., Ser. Mech., Math., 78, 35–41 (2013).Google Scholar
  5. 5.
    V. S. Khoroshchak, A. Ya. Khrystiyanyn, and D. V. Lukivska, “A class of Julia exceptional functions,” Carpath. Math. Publ., 8, No. 1, 172–180 (2016).MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    V. S. Khoroshchak and A. A. Kondratyuk, “The Riesz measures and a representation of multiplicatively periodic 𝛿-subharmonic functions in a punctured Euclidean space,” Mat. Stud., 43, No. 1, 61–65 (2015).MathSciNetzbMATHGoogle Scholar
  7. 7.
    V. S. Khoroshchak and N. B. Sokulska, “Multiplicatively periodic meromorphic functions in the upper half plane,” Mat. Stud., 42, No. 2, 143–148 (2014).MathSciNetzbMATHGoogle Scholar
  8. 8.
    V. S. Khoroshchak and A. A. Kondratyuk, “Stationary harmonic functions on homogeneous spaces,” Ufimsk. Mat. Zh., 7, No. 4, 155–159 (2015).MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    A. Ya. Khrystiyanyn and A. A. Kondratyuk, “Meromorphic mappings of torus onto the Riemann sphere,” Carpath. Math. Publ., 4, No. 1, 155–159 (2012).zbMATHGoogle Scholar
  10. 10.
    A. Ya. Khrystiyanyn and A. A. Kondratyuk, “Modulo-loxodromic meromorphic function in C\0,Ufimsk. Mat. Zh., 8, No. 4, 156–162 (2016).MathSciNetCrossRefGoogle Scholar
  11. 11.
    F. Klein, “Zur Theorie der Abel’schen Functionen,” Math. Ann., 36, 1–83 (1890).MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    A. A. Kondratyuk and V. S. Zaborovska, “ Multiplicatively periodic subharmonic functions in the punctured Euclidean space,” Mat. Stud., 40, No. 2, 159–164 (2013).MathSciNetzbMATHGoogle Scholar
  13. 13.
    A. A. Kondratyuk, “Loxodromic meromorphic and 𝛿-subharmonic functions,” in: Proc. of the Workshop on Complex Analysis and Its Applications to Differential and Functional Equations, University of Eastern Finland, Reports and Studies in Forestry and Natural Sciences, Joensuu, 14 (2014), pp. 89–99.Google Scholar
  14. 14.
    O. Rausenberger, Lehrbuch der Theorie der Periodischen Functionen einer Variabeln, Teubner, Leipzig (1884).zbMATHGoogle Scholar
  15. 15.
    S. Kos and T. K. Pogány, “On the mathematics of navigational calculations for meridian sailing,” Electron. J. Geogr. Math. (2012).Google Scholar
  16. 16.
    F. Schottky, “Über eine specielle Function welche bei einer bestimmten linearen Transformation ihres Arguments unveraändert bleibt,” J. Reine Angew. Math., 101, 227–272 (1887).MathSciNetzbMATHGoogle Scholar
  17. 17.
    G. Valiron, Cours d’Analyse Mathematique. Theorie des fonctions, Masson et. Cie., Paris (1947).Google Scholar

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

Authors and Affiliations

  • Dz. V. Lukivs’ka
    • 1
  • A. Ya. Khrystiyanyn
    • 1
  1. 1.Franko Lviv National UniversityLvivUkraine

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