Normality Criteria for Differential Monomials and the Sharpness of Lappan-type Theorems

TL;DR

This paper refines classical normality criteria for meromorphic functions by reducing the size of the set needed to verify boundedness and extends key theorems to differential monomials, broadening the understanding of normality conditions.

Contribution

It introduces new criteria that lower the number of points needed for normality checks and generalizes the Pang-Zalcman theorem to differential monomials with sharp thresholds.

Findings

Normality criteria with fewer points (from five to three)

Boundedness of derivatives on pre-images ensures normality

Extension of Pang-Zalcman theorem to differential monomials

Abstract

A fundamental result of Lappan [Comment. Math. Helv. \textbf{49} (1974), 492-495.] states that a meromorphic function $f$ in the unit disk $\mathbb{D}$ is normal if and only if its spherical derivative is bounded on a five-point subset $E \subset \mathbb{C}$. In this paper, we establish new normality criteria that bridge this classical result with contemporary trends in value distribution theory. We demonstrate that the cardinality of the set $E$ can be reduced from five to as few as three, provided that the spherical derivatives of the function and its successive derivatives $f, f', \dots, f^{(k-1)}$ are bounded on the pre-image of $E$. This shift reveals that analytic data from higher-order derivatives can effectively compensate for a reduction in geometric information from the target set. Furthermore, we extend the Pang-Zalcman theorem to a general class of differential monomials $M[f]$. We prove that if $(M[f])^{\#}$ is bounded on the set of $a$-points ($a \neq 0$), the family $\mathcal{F}$ is normal, provided the degree $d_M$ satisfies a specific sharp threshold relative to the weight $D_M$ and order $k$. These results offer a refined perspective on the natural boundaries of normality and generalize several established findings in the field.