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Growth of Meromorphic Solutions to Some Complex Functional Equations

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Abstract

Let f be a meromorphic function in the complex plane, \(p_j\) polynomials for (\(j=0, 1, 2, \ldots , n\)) and R(zf) an irreducible rational function in f with small meromorphic functions relative to f as coefficients. Let n be a positive integer and IJ two index sets in \(\mathbb {Z}^n\). In this paper, we systematically study the growth order of meromorphic solutions to the functional equations of the form:

$$\begin{aligned} \frac{\sum _{\mu \in I}\alpha _{\mu }(z)\left( \prod _{j=1}^nf( p_{j}(z))^{\mu _j}\right) }{\sum _{\nu \in J}\beta _{\nu }(z)\left( \prod _{j=1}^nf( p_{j}(z))^{\nu _j}\right) }=R(z, f(p_{0})). \end{aligned}$$

We not only obtain estimates of the growth order of its meromorphic solutions in all possible cases, but also give examples to show these estimates are the best possible.

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Acknowledgments

The authors are grateful to the referee whose suggestions and comments have improved the clarity of the paper. The first two authors also would like to express their thanks for the hospitality of Mathematics Seminar, University of Kiel in Germany, when they visited there.

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Correspondence to Zhuan Ye.

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Communicated by Ilpo Laine.

The first author was supported by the Research Project Supported by Shanxi Scholarship Council of China (No. 2013-045) and the Foundation Research Project of Shanxi Province (No. 2014021009-3). The second author was supported by the NNSF of China (No. 10671109 and No. 11001057).

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Ding, J., Wang, J. & Ye, Z. Growth of Meromorphic Solutions to Some Complex Functional Equations. Comput. Methods Funct. Theory 16, 489–502 (2016). https://doi.org/10.1007/s40315-016-0157-z

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  • DOI: https://doi.org/10.1007/s40315-016-0157-z

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