On Generalized Fermat Type Functional Equations

Let p be a positive integer not less than 2. It is shown that a necessary condition for the generalized Fermat type functional equation \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sum_{j=1}^pa_j(z){f_j}^{k_j}(z)\equiv 1$$\end{document} having non-constant meromorphic solutions f1, f2, …, fp is \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sum_{j=1}^{p} {1\over k_{j}} \geq {1\over (p-1)+A_{p}}$$\end{document}, where A2 = 1,2, Ap = (2p − 3)/3 if p = 3, 4, 5, Ap = (2p + 1 − 2√2p)/2 if p ≥ 6 and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T\left( {r,a_j } \right) = \mathcal{O}\left( {T\left( {r,f_j } \right)} \right)$$\end{document}, 1 ≤ j ≤ p, as r → + ∞, r ∉ E and E is a set of finite linear measure. This improves the result of Yu and Yang [14] in 2002. Next we discuss a question of Hayman [7] and give a partial answer to it.