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Growth of meromorphic solutions of some functional equations I. (English) Zbl 1009.39022

The authors are concerned with meromorphic solutions of the functional equation \[ \sum^n_{j=0} a_j(z)f(c^jz) =Q(z) \] where \(Q\) and the \(a_j\) are polynomials without common zeros, \(a_n(z)a_0 (z)\neq 0\) and \(0<|c |<1\). They show that each transcendental meromorphic solution \(f(z)\) of this equation satisfies \[ m(r,f)= \sigma_f(\log r)^2 \bigl(1+o(1)\bigr) \] for some constant \(\sigma_f\). Here \(m(r,f)\) signifies the proximity function of \(f(z)\) (standard notations in the Nevanlinna theory).

MSC:

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