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Normality criterion for a family of holomorphic curves that partially share wandering hyperplanes with their derivatives, and holomorphic functions lifted to curves in $P^2(\mathbb{C})$

Sonam Mehta, Kuldeep Singh Charak

Abstract

In this paper we generalize a result of Ye, Pang and Yang[12] on the normality of a family of holomorphic curves in $P^N(\mathbb{C})$. Further we obtain a normality criterion for family of meromorphic functions that partially share wandering holomorphic functions with their derivatives. We also devise a tractable representation of complex valued holomorphic functions from D as functions from D to $P^2(\mathbb{C})$ obtain a normality criterion that leads to a counterexample to the converse of Bloch's principle.

Normality criterion for a family of holomorphic curves that partially share wandering hyperplanes with their derivatives, and holomorphic functions lifted to curves in $P^2(\mathbb{C})$

Abstract

In this paper we generalize a result of Ye, Pang and Yang[12] on the normality of a family of holomorphic curves in . Further we obtain a normality criterion for family of meromorphic functions that partially share wandering holomorphic functions with their derivatives. We also devise a tractable representation of complex valued holomorphic functions from D as functions from D to obtain a normality criterion that leads to a counterexample to the converse of Bloch's principle.

Paper Structure

This paper contains 8 sections, 12 theorems, 63 equations.

Table of Contents

  1. Introduction
  2. Projective spaces and hyperplanes
  3. Reduced representation
  4. Normal family
  5. Extension of derivative
  6. Sharing of functions and hyperplanes
  7. Main results
  8. Proofs

Key Result

Theorem A

Let $\mathcal{F}$ be a family of meromorphic functions on a domain $D\subseteq \mathbb{C}$ which omit three distinct values $a.b,c$ in $\hat{\mathbb{C}}$. Then $\mathcal{F}$ is normal in $D$.

Theorems & Definitions (30)

  • Theorem A: montel1912familles
  • Theorem B: schwick1992sharing
  • Theorem C: dufresnoy1944theorie
  • Theorem D
  • Definition 1.1
  • Definition 1.2
  • Remark 1
  • Remark 2
  • Definition 1.3
  • Definition 1.4
  • ...and 20 more