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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.

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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