@article{Li2026, 
author = {Qing Li and Guobin Lin},
title = {Normal criterion on the polydisc in              C        n},
year = {2026},
journal = {AIMS Mathematics},
volume = {11},
number = {4},
pages = {10191-10204},
keywords = {normal criterion, polydisc, spherical derivative, slice characterization, Marty's criterion},
url = {https://www.sciopen.com/article/10.3934/math.2026421},
doi = {10.3934/math.2026421},
abstract = {This paper established a slice characterization for normal families of holomorphic functions on the unit polydisc              D        n    ⊂            C        n  . The main result (Theorem 2.2) showed that a family        F    ⊂      O    (            D        n    ) was normal if, and only if, for every point    a  ∈            D        n   and every coordinate direction    1  ≤  j  ≤  n, the corresponding one-dimensional slice family              F              a      ,      j       was normal on the unit disc. Building on this characterization, we introduced the notion of normal functions on              D        n   and proved a metric criterion (Proposition 2.7): A function    f  ∈      O    (            D        n    ) was normal exactly when its spherical derivative grew at most as fast as the Poincaré metric density, i.e.,        f    ♯    (  z  )  ≤  C      max    j    (  1  −      |        z    j              |        2        )          −      1       for some constant    C  >  0.}
}