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Rigidity of entire functions sharing a finite set with their partial derivatives in C^n

Sujoy Majumder, Abhijit Banerjee, Shantanu Panja

Abstract

This paper investigates certain classes of entire functions in C^n that, together with their partial derivatives, share a finite set consisting of three elements. By employing normality criteria, we study the behaviour of such functions and derive the necessary conditions governing their existence. Our results extend those of [4], originally established for functions of a single complex variable, to the setting of several complex variables, thereby providing a comprehensive generalization of the earlier result in a direction not previously explored.

Rigidity of entire functions sharing a finite set with their partial derivatives in C^n

Abstract

This paper investigates certain classes of entire functions in C^n that, together with their partial derivatives, share a finite set consisting of three elements. By employing normality criteria, we study the behaviour of such functions and derive the necessary conditions governing their existence. Our results extend those of [4], originally established for functions of a single complex variable, to the setting of several complex variables, thereby providing a comprehensive generalization of the earlier result in a direction not previously explored.

Paper Structure

This paper contains 4 sections, 14 theorems, 91 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction and main result
  2. Basic Notations in Several Complex Variables
  3. Auxiliary Lemmas
  4. Proofs of Theorem \ref{['t1.1']}

Key Result

Theorem 1.1

Let $f$ be a non-constant entire function in $\mathbb{C}^n$ and let $S=\{a,b,c\}$, where $a$, $b$, $c$$\in \mathbb{C}$. Suppose $E(S,f)=E\left(S,\partial_{z_i}(f)\right)$ for $i=1,2,\ldots,n$.

Figures (2)

  • Figure 1: Geometric correspondence between level lines in the $t$-plane and their images in the $f$-plane.
  • Figure 2: Blue circles indicate trajectories for fixed $\Re(\lambda t)$ and the first circle corresponding to $\Re(\lambda t)=0$. The red inward spiral shows decay, while the green outward spiral shows growth. Rotation around $d$ is governed by $\Im(\lambda t)$, producing the spiraling motion.

Theorems & Definitions (22)

  • Theorem 1.1
  • Remark 1.1
  • Remark 1.2
  • Lemma 3.1
  • Lemma 3.2
  • Lemma 3.3
  • Lemma 3.4
  • Lemma 3.5
  • Lemma 3.6
  • Lemma 3.7
  • ...and 12 more