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2021 Higher-Dimensional Generalizations of Some Theorems on Normality of Meromorphic Functions
Tran Van Tan
Michigan Math. J. Advance Publication 1-11 (2021). DOI: 10.1307/mmj/20195842

## Abstract

In , 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

$$\min {\rho ({a_{f}}(z),{b_{f}}(z)),\rho ({b_{f}}(z),{c_{f}}(z)),\rho ({c_{f}}(z),{a_{f}}(z))}\ge \varepsilon$$

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 $\triangle :={z\in \mathbb{C}:|z|\textless 1}$ is normal if there are five distinct values ${a_{1}},\dots ,{a_{5}}$ such that

$$\sup {(1-|z{|^{2}})\frac{|{f^{\prime }}(z)|}{1+|f(z){|^{2}}}:z\in {f^{-1}}{{a_{1}},\dots ,{a_{5}}}}\textless \infty .$$

## Citation

Tran Van Tan. "Higher-Dimensional Generalizations of Some Theorems on Normality of Meromorphic Functions." Michigan Math. J. Advance Publication 1 - 11, 2021. https://doi.org/10.1307/mmj/20195842

## Information

Received: 16 December 2019; Revised: 15 April 2020; Published: 2021
First available in Project Euclid: 9 April 2021

Digital Object Identifier: 10.1307/mmj/20195842

Subjects:  