Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Molla Basir Ahamed, Sanju Mandal, Nguyen Van Thin

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.

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

Abstract

A fundamental result of Lappan [Comment. Math. Helv. \textbf{49} (1974), 492-495.] states that a meromorphic function in the unit disk is normal if and only if its spherical derivative is bounded on a five-point subset . 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 can be reduced from five to as few as three, provided that the spherical derivatives of the function and its successive derivatives are bounded on the pre-image of . 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 . We prove that if is bounded on the set of -points (), the family is normal, provided the degree satisfies a specific sharp threshold relative to the weight and order . These results offer a refined perspective on the natural boundaries of normality and generalize several established findings in the field.
Paper Structure (5 sections, 5 theorems, 46 equations)

This paper contains 5 sections, 5 theorems, 46 equations.

Table of Contents

  1. Introduction
  2. Further perspectives on normality and differential operators
  3. Main Results
  4. Proof of the main results
  5. Concluding remarks

Key Result

Theorem 2.1

Suppose that $k$ be a positive integer such that $k>2$. Let $\mathcal{F}$ be a family of meromorphic functions in a domain $D\subset\mathbb{C}$. Assume that for each compact $K\subset D$, there exist a set $E=E(K)\subset\mathbb{C}$ consisting of three distinct points and a positive constant $M=M(K)$ for all $f\in\mathcal{F}$ and $z\in K\cap f^{-1}(E)$. Then $\mathcal{F}$ is normal.

Theorems & Definitions (19)

  • Definition 1.1
  • Definition 1.2
  • Definition 1.3
  • Definition 1.4
  • Theorem 2.1
  • Remark 2.1
  • Example 2.1
  • Example 2.2
  • Theorem 2.2
  • Remark 2.2
  • ...and 9 more