论文标题
某些定理对Meromorormormormormormormormorthic函数的较高维度的概括
Higher dimensional generalizations of some theorems on normality of meromorphic functions
论文作者
论文摘要
在[以色列J. Math，2014年]中，Grahl和Nevo在Montel的众所周知正态性标准方面取得了重大改进。 They proved that for a family of meromorphic functions $\mathcal F$ in a domain $D\subset \mathbb C,$ and for a positive constant $ε$, if for each $f\in \mathcal F$ there exist meromorphic functions $a_f,b_f,c_f$ such that $f$ omits $a_f,b_f,c_f$ in $D$ and $$ \ min \ {ρ（a_f（z），b_f（z）），ρ（b_f（z），c_f（z）），ρ（c_f（z），a_f（z））\ \} \ geq geq geqε，$ $ $ z \ in d $ in d $，然后$ \ nathcal f $ forne $ \ n $ hscal f $ n ummand n ummand n ummand n um d p $ d n n ummand n $ d d $ d。在这里，$ρ$是$ \ wideHat {\ mathbb c} $中的球形度量。在本文中，我们为上述结果建立了高维版本，以及以下众所周知的Lappan结果：单位光盘$ \ triangle中的meromorphic函数$ f $：= \ {z \ in \ mathbb c：| z | | <1 $ \ sup \ {（1- | z |^2）\ frac {| f'（z）|} {1+ | f（z）|^2}：z \ in f^{ - 1} \ {a_1，\ {a_1，\ dots，\ dots，a_5 \}
In [Israel J. Math, 2014], Grahl and Nevo obtained a significant improvement for the well-known normality criterion of Montel. They proved that for a family of meromorphic functions $\mathcal F$ in a domain $D\subset \mathbb C,$ and for a positive constant $ε$, if for each $f\in \mathcal F$ there exist meromorphic functions $a_f,b_f,c_f$ such that $f$ omits $a_f,b_f,c_f$ in $D$ and $$\min\{ρ(a_f(z),b_f(z)), ρ(b_f(z),c_f(z)), ρ(c_f(z),a_f(z))\}\geq ε,$$ for all $z\in D$, then $\mathcal F$ is normal in $D$. Here, $ρ$ is the spherical metric in $\widehat{\mathbb C}$. In this paper, we establish the high-dimensional versions for the above result and for the following well-known result of Lappan: A meromorphic function $ f$ in the unit disc $\triangle:=\{z\in\mathbb C: |z|<1\}$ is normal if there are five distinct values $a_1,\dots,a_5$ such that $$\sup\{(1-|z|^2)\frac{ |f '(z)|}{1+|f(z)|^2}: z\in f^{-1}\{a_1,\dots,a_5\}\} < \infty.$$
