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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 [5], Grahl and Nevo obtained a significant improvement for the well-known normality criterion of Montel. They proved that for a family of meromorphic functions F in a domain DC, and for a positive constant ε, if for each fF there exist meromorphic functions af,bf,cf such that f omits af,bf,cf in D and

min{ρ(af(z),bf(z)),ρ(bf(z),cf(z)),ρ(cf(z),af(z))}ε

for all zD, then F is normal in D. Here, ρ is the spherical metric in Cˆ. 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 :={zC:|z|<1} is normal if there are five distinct values a1,,a5 such that

sup{(1|z|2)|f(z)|1+|f(z)|2:zf1{a1,,a5}}<.

Citation

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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

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Rights: Copyright © 2021 The University of Michigan

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