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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, $a_n(z)a_0 (z)\ne 0$ 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).

39B32Functional equations for complex functions
30D05Functional equations in the complex domain, iteration and composition of analytic functions
30D35Distribution of values (one complex variable); Nevanlinna theory
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