×

zbMATH — the first resource for mathematics

Meromorphic solutions of some functional equations. (English) Zbl 0924.39017
Methods Appl. Anal. 5, No. 3, 248-258 (1998); correction ibid. 6, No. 4, 617-618 (1999).
Authors’ abstract: It is shown that transcendental meromorphic solutions \(f(z)\) of the functional equation \[ \sum^n_{j=0} a_j(z)f(c^jz)=Q(z), \] where \(0<| c|<1\) is a complex number and \(a_j(z)\), \(j=0,1,\dots,n\) and \(Q(z)\) are rational functions with \(a_0(z)\not\equiv 0\), \(a_n(z)\equiv 1\), satisfy \[ T(r,f)=O\bigl((\log r)^2\bigr) \quad\text{and}\quad (\log r)^2=O\bigl(T(r,f)\bigr), \] where \(T(r,f)\) is the characteristic function of \(f(z)\). Moreover, in the case \(n=2\) and \(Q(z)\equiv 0\), necessary and sufficient conditions for the existence of solutions are given.

MSC:
39B32 Functional equations for complex functions
30D05 Functional equations in the complex plane, iteration and composition of analytic functions of one complex variable
PDF BibTeX XML Cite
Full Text: DOI