Uniqueness of meromorphic functions concerning fixed points

AIMS Mathematics

article

Jinyu Fan¹  Mingliang Fang¹  Jianbin Xiao¹ 

[1] Department of Mathematics, Hangzhou Dianzi University

关键词: meromorphic function;    entire function;    unicity;    fixed point;   

DOI  :  10.3934/math.20221122

学科分类：地球科学(综合)

来源: AIMS Press

【 摘 要 】

In this paper, we study a uniqueness question of meromorphic functions concerning fixed points and mainly prove the following theorem: Let $ f $ and $ g $ be two nonconstant meromorphic functions, let $ n, k $ be two positive integers with $ n > 3k+10.5-\Theta_{\min}(k+6.5) $, if $ \Theta_{\min}\geq \frac{2.5}{k+6.5} $, otherwise $ n > 3k+8 $, and let $ (f^{n})^{(k)} $ and $ (g^{n})^{(k)} $ share $ z $ CM, $ f $ and $ g $ share $ \infty $ IM, then one of the following two cases holds: If $ k = 1 $, then either $ f(z) = c_{1}e^{cz^{2}} $, $ g(z) = c_{2}e^{-cz^{2}} $, where $ c_{1}, c_{2} $ and $ c $ are three constants satisfying $ 4n^{2}(c_{1}c_{2})^{n}c^{2} = -1 $, or $ f = tg $ for a constant $ t $ such that $ t^{n} = 1 $; if $ k\geq2 $, then $ f = tg $ for a constant $ t $ such that $ t^{n} = 1 $. Our results extend and improve some results due to [ 8 , 9 , 19 , 24 ] .

【 授权许可】

CC BY