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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
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