Uniqueness of entire functions and fixed points

Xiao-Guang Qi¹ ; Lian-Zhong Yang¹

¹ School of Mathematics Shandong University Jinan, Shandong, 250100, P.R. China

Annales Polonici Mathematici, Tome 97 (2010) no. 1, pp. 87-100

Résumé

Let $f$ and $g$ be entire functions, $n,$ $k$ and $m$ be positive integers, and $\lambda $, $\mu $ be complex numbers with $|\lambda |+|\mu | \not =0$. We prove that $(f^{n}(z)(\lambda f^{m}(z)+\mu ))^{(k)}$ must have infinitely many fixed points if $n \geq k +2$; furthermore, if $ (f^{n}(z)(\lambda f^{m}(z)+\mu ))^{(k)}$ and $(g^{n}(z)(\lambda g^{m}(z)+\mu ))^{(k)}$ have the same fixed points with the same multiplicities, then either $f\equiv cg$ for a constant $c$, or $f$ and $g$ assume certain forms provided that $n>2k+m^{*}+4,$ where $m^*$ is an integer that depends only on $\lambda .$

Détail

DOI : 10.4064/ap97-1-7

Keywords: entire functions positive integers lambda complex numbers lambda prove lambda have infinitely many fixed points geq furthermore lambda lambda have fixed points multiplicities either equiv constant assume certain forms provided * where * integer depends only lambda