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Aequationes mathematicae

, Volume 63, Issue 1–2, pp 110–135 | Cite as

Meromorphic solutions of generalized Schröder equations

  • G. G. Gundersen
  • J. Heittokangas
  • I. Laine
  • J. Rieppo
  • D. Yang
Article

Summary.

We consider meromorphic solutions of functional equations of the form¶¶\( f(cz) = R(z,f(z)) = {\sum_{j=0}^pa_j(z)f(z)^j \over \sum_{j=0}^q b_j(z)f(z)^j} \),¶where the coefficients a j (z),b j (z) are meromorphic functions and c is a complex constant. In fact, for \( |c| > 1 \), any local meromorphic solution around the origin has a meromorphic continuation over \( {\Bbb C} \). We prove a number of results on the growth and value distribution of solutions. In the special case of¶¶\( f(cz) = A(z) + \gamma f(z) + \delta f(z)^2 \),¶where \( c, \gamma, \delta \in {\Bbb C} \), \( |c|>1 \), \( \delta \neq 0 \), and A(z) is entire, we offer a detailed analysis on the number of distinct meromorphic solutions.

Keywords. Complex functional equations, meromorphic solutions, Nevanlinna theory, Schröder equation. 

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Copyright information

© Birkhäuser Verlag, Basel, 2002

Authors and Affiliations

  • G. G. Gundersen
    • 1
  • J. Heittokangas
    • 2
  • I. Laine
    • 2
  • J. Rieppo
    • 2
  • D. Yang
    • 2
  1. 1.University of New Orleans, Department of Mathematics, New Orleans, LA 70148, U.S.A.USA
  2. 2.University of Joensuu, Department of Mathematics, P.O. Box 111, FIN—80101 Joensuu, FinlandFinland

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