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Zbl 0963.39028
Heittokangas, Janne; Laine, Ilpo; Rieppo, Jarkko; Yang, Degui
Meromorphic solutions of some linear functional equations.
(English)
[J] Aequationes Math. 60, No.1-2, 148-166 (2000). ISSN 0001-9054; ISSN 1420-8903/e

The authors consider the linear functional equation $$\sum_{j=0}^n a_j(z) f( c^j z)= Q(z) \tag FE$$ where $c \in \bbfC \setminus \{ 0\}$, $n \in \bbfN$, the coefficients $a_0, a_1, \dots ,a_n, Q$ are given complex functions, and $f : \bbfC \to \bbfC$ is the unknown function to be determined. The authors show that if $0< |c|< 1$, the coefficients $a_0, a_1, \dots ,a_n$ are complex constants, $Q(z)$ is a meromorphic function, and $\sum_{j=0}^n a_j c^{jk} \neq 0$ for all $k \in \bbfZ$, then exactly one meromorphic solution of the functional equation (FE) exists. In the general case, the authors give growth estimates for the solution $f$ as well as the exponent of convergence $\lambda (1/f)$ of poles and $\lambda (f)$ of zeros of $F$.
[Prasanna Sahoo (Louisville)]
MSC 2000:
*39B32 Functional equations for complex functions
30D05 Functional equations in the complex domain
30D35 Distribution of values (one complex variable)

Keywords: complex functional equations; meromorphic solutions; Nevanlinna theory; linear functional equation; growth estimates; exponent of convergence

Cited in: Zbl 1052.30027

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