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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

$\sum _{j=0}^{n}{a}_{j}\left(z\right)f\left({c}^{j}z\right)=Q\left(z\right)$

where $Q$ and the ${a}_{j}$ are polynomials without common zeros, ${a}_{n}\left(z\right){a}_{0}\left(z\right)\ne 0$ and $0<|c|<1$. They show that each transcendental meromorphic solution $f\left(z\right)$ of this equation satisfies

$m\left(r,f\right)={\sigma }_{f}{\left(logr\right)}^{2}\left(1+o\left(1\right)\right)$

for some constant ${\sigma }_{f}$. Here $m\left(r,f\right)$ signifies the proximity function of $f\left(z\right)$ (standard notations in the Nevanlinna theory).

##### MSC:
 39B32 Functional equations for complex functions 30D05 Functional equations in the complex domain, iteration and composition of analytic functions 30D35 Distribution of values (one complex variable); Nevanlinna theory