Fixed points of meromorphic functions and of their differences and shifts

Zong-Xuan Chen¹

¹ School of Mathematical Sciences South China Normal University 510631, Guangzhou, P.R. China

Annales Polonici Mathematici, Tome 109 (2013) no. 2, pp. 153-163

Résumé

Let $f(z)$ be a finite order transcendental meromorphic function such that $\lambda (1/f(z))\sigma (f(z))$, and let $c\in \mathbb {C}\setminus \{0\}$ be a constant such that $f(z+c)\not \equiv f(z)+c$. We mainly prove that $$\eqalign {\max\{\tau (f(z)), \tau (\Delta _c f(z))\}=\max\{\tau (f(z)), \tau (f(z+c))\} \cr =\max\{\tau (\Delta _c f(z)), \tau (f(z+c))\}=\sigma (f(z)), }$$where $\tau (g(z))$ denotes the exponent of convergence of fixed points of the meromorphic function $g(z)$, and $\sigma (g(z))$ denotes the order of growth of $g(z).$

Détail

DOI : 10.4064/ap109-2-4

Keywords: finite order transcendental meromorphic function lambda sigma mathbb setminus constant equiv mainly prove eqalign max tau tau delta max tau tau max tau delta tau sigma where tau denotes exponent convergence fixed points meromorphic function sigma denotes order growth