Aequationes mathematicae

, Volume 60, Issue 1, pp 148–166

Meromorphic solutions of some linear functional equations

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

DOI: 10.1007/s000100050143

Cite this article as:
Heittokangas, J., Laine, I., Rieppo, J. et al. Aequat. Math. (2000) 60: 148. doi:10.1007/s000100050143

Summary.

We consider meromorphic solutions in the complex plane to functional equations of the form \( \sum\nolimits_{j = 0}^n {a_j } (z)f(c^j z) = Q(z) \), where 0 < |c| < 1, \( n \in {\Bbb N} \) and \( a_0,\ldots ,a_n,Q \) are meromorphic functions. If the coefficients \( a_0,\ldots ,a_n \) are constants and \( \sum\nolimits_{j = 0}^n {a_j } c^{jk} \ne 0 \) for all \( k \in {\Bbb Z} \), then exactly one meromorphic solution exists. In the general case, we give growth estimates for the solutions f as well as for the exponent of convergence \( \lambda (1/f) \) of poles and \( \lambda (f) \) of zeros of f. Similar results hold in the case 1 < |c| < + \( \infty \). Concluding remarks show that the case |c| = 1 is different.

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

Copyright information

© Birkhäuser Verlag, Basel, 2000

Authors and Affiliations

  • J. Heittokangas
    • 1
  • I. Laine
    • 1
  • J. Rieppo
    • 1
  • D. Yang
    • 1
  1. 1. University of Joensuu, Department of Mathematics, P. O. Box 111, FIN-80101 Joensuu, FinlandFinland

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