Existence and properties of meromorphic solutions of some q-difference equations

In this paper, we investigate the existence and growth of solutions of the q-difference equation ∏i=1nf(qiz)=R(z,f(z))\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\prod_{i=1}^{n}f(q_{i}z)=R(z,f(z))$\end{document}, where R(z,f(z))\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$R(z,f(z))$\end{document} is an irreducible rational function in f(z)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$f(z)$\end{document}. We also give an estimation of the growth of transcendental meromorphic solutions of the equation ∏i=1nf(qiz)=f(z)m\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\prod_{i=1}^{n}f(q_{i}z)=f(z)^{m}$\end{document}.