Normal criterion on the polydisc in $ \mathbb{C}^n $

Qing Li; Guobin Lin; Qing Li; Guobin Lin

doi:10.3934/math.2026421

AIMS Mathematics

2026, Volume 11, Issue 4: 10191-10204. doi: 10.3934/math.2026421

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Normal criterion on the polydisc in $ \mathbb{C}^n $

Qing Li ,
Guobin Lin ^,

School of Information Engineering, Fujian Key Lab of Agriculture IOT Application, Sanming University, Sanming 365004, China

Received: 02 February 2026 Revised: 19 March 2026 Accepted: 07 April 2026 Published: 15 April 2026
MSC : 30D45, 32A10, 32A19, 32H02

This paper established a slice characterization for normal families of holomorphic functions on the unit polydisc $ \mathbb{D}^n\subset\mathbb{C}^n $. The main result (Theorem 2.2) showed that a family $ \mathcal{F}\subset\mathcal{O}(\mathbb{D}^n) $ was normal if, and only if, for every point $ a\in\mathbb{D}^n $ and every coordinate direction $ 1\le j\le n $, the corresponding one-dimensional slice family $ \mathcal{F}_{a, j} $ was normal on the unit disc. Building on this characterization, we introduced the notion of normal functions on $ \mathbb{D}^n $ and proved a metric criterion (Proposition 2.7): A function $ f\in\mathcal{O}(\mathbb{D}^n) $ was normal exactly when its spherical derivative grew at most as fast as the Poincaré metric density, i.e., $ f^\sharp(z)\le C\max_j(1-|z_j|^2)^{-1} $ for some constant $ C > 0 $.
- normal criterion,
- polydisc,
- spherical derivative,
- slice characterization,
- Marty's criterion
Citation: Qing Li, Guobin Lin. Normal criterion on the polydisc in $ \mathbb{C}^n $[J]. AIMS Mathematics, 2026, 11(4): 10191-10204. doi: 10.3934/math.2026421

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Abstract

This paper established a slice characterization for normal families of holomorphic functions on the unit polydisc $ \mathbb{D}^n\subset\mathbb{C}^n $. The main result (Theorem 2.2) showed that a family $ \mathcal{F}\subset\mathcal{O}(\mathbb{D}^n) $ was normal if, and only if, for every point $ a\in\mathbb{D}^n $ and every coordinate direction $ 1\le j\le n $, the corresponding one-dimensional slice family $ \mathcal{F}_{a, j} $ was normal on the unit disc. Building on this characterization, we introduced the notion of normal functions on $ \mathbb{D}^n $ and proved a metric criterion (Proposition 2.7): A function $ f\in\mathcal{O}(\mathbb{D}^n) $ was normal exactly when its spherical derivative grew at most as fast as the Poincaré metric density, i.e., $ f^\sharp(z)\le C\max_j(1-|z_j|^2)^{-1} $ for some constant $ C > 0 $.

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