zbMATH — the first resource for mathematics

Meromorphic solutions of generalized Schröder equations. (English) Zbl 1012.30016
The authors consider meromorphic solutions of functional equations of the form \[ f(cz) = R(z,f(z)) = \frac{\sum\limits_{j=0}^p a_j(z)f(z)^j} {\sum\limits_{k=0}^q b_k(z)f(z)^k} \tag{1} \] where the coefficients \(a_j\), \(b_k\) are meromorphic functions and \(c\) is a complex constant. If \(|c|>1\), then any local meromorphic solution around the origin has a meromorphic continuation over \(\mathbb C\). The authors prove a number of results on the growth and value distribution of solutions \(f\) of (1).
For a meromorphic function \(f\) in the complex plane, let \(T(r,f)\) denote its Nevanlinna characteristic, and \(\rho(f)\) its order of growth. Furthermore, let \(d:=\max{\{p,q\}} \geq 1\), and \(\Psi(r) := \max\limits_{j,k}{\{T(r,a_j),T(r,b_k)\}}\). Then, for example, one of the results reads as follows.
Theorem. Let \(f\) be a transcendental meromorphic solution of (1) with \(|c|>1\), and assume that \(\Psi(r)=S(r,f)\). Then \[ \rho(f) = \frac{\log{d}}{\log{|c|}} . \]
Furthermore, the authors consider the special case \[ f(cz) = A(z) + \gamma f(z) + \delta f(z)^2 , \tag{2} \] where \(c\), \(\gamma\), \(\delta \in \mathbb C\), \(|c|>1\), \(\delta \neq 0\), and \(A\) is an entire function. They show that every meromorphic solution of (2) is entire, and they give a detailed analysis on the number of distinct meromorphic solutions.

30D05 Functional equations in the complex plane, iteration and composition of analytic functions of one complex variable
39B32 Functional equations for complex functions
30D35 Value distribution of meromorphic functions of one complex variable, Nevanlinna theory
Full Text: DOI