Growth of Meromorphic Solutions to Some Complex Functional Equations

Let f be a meromorphic function in the complex plane, pj\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p_j$$\end{document} polynomials for (j=0,1,2,…,n\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$j=0, 1, 2, \ldots , n$$\end{document}) and R(z, f) an irreducible rational function in f with small meromorphic functions relative to f as coefficients. Let n be a positive integer and I, J two index sets in Zn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {Z}^n$$\end{document}. In this paper, we systematically study the growth order of meromorphic solutions to the functional equations of the form: ∑μ∈Iαμ(z)∏j=1nf(pj(z))μj∑ν∈Jβν(z)∏j=1nf(pj(z))νj=R(z,f(p0)).\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \frac{\sum _{\mu \in I}\alpha _{\mu }(z)\left( \prod _{j=1}^nf( p_{j}(z))^{\mu _j}\right) }{\sum _{\nu \in J}\beta _{\nu }(z)\left( \prod _{j=1}^nf( p_{j}(z))^{\nu _j}\right) }=R(z, f(p_{0})). \end{aligned}$$\end{document}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.