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Value Distribution and Picard-type Theorems for Total Differential Polynomials in $\mathbb{C}^n$

Molla Basir Ahamed, Vasudevarao Allu

TL;DR

The paper extends Nevanlinna value distribution theory to linear total differential polynomials ${L}_k[D]f$ of meromorphic functions in $\mathbb{C}^n$, deriving growth estimates for $T(r, {L}_k[D]f)$ and a Milloux-type inequality. Under sharing conditions for two meromorphic functions $f$ and $g$ and a non-transcendence assumption for $f$, it shows that $\frac{{L}_k[D]f-1}{{L}_k[D]g-1}$ is a non-zero constant. A Picard-type rigidity result is established: if an entire function omits a value $a$ while its linear total differential polynomial omits a non-zero value $b$, then the function must be constant, and the work is extended to generalized operators involving a polynomial $Q_m$ via $f^kQ_m(f)Df$. Together, these results broaden classical one-variable uniqueness and Picard-type theorems to the higher-dimensional setting, highlighting new interactions between value distribution, growth, and differential polynomials in $\mathbb{C}^n$.

Abstract

This paper investigates the value distribution and growth properties of linear total differential polynomials $\mathcal{L}_k[D]f$ for meromorphic functions in several complex variables $\mathbb{C}^n$. By extending the classical Milloux inequality to the framework of total derivatives, we derive a series of fundamental growth estimates for the Nevanlinna characteristic function $T(r, \mathcal{L}_k[D]f)$. We address the value-sharing problem for meromorphic functions $f$ and $g$ sharing values with their differential polynomials. Under the condition $2δ(0,f)+(k+4)Θ(\infty,f)>k+5$, we establish that $\frac{\mathcal{L}_k[D]f-1}{\mathcal{L}_k[D]g-1}$ is a non-zero constant for non-transcendental meromorphic functions. Furthermore, we provide an affirmative answer to several Picard-type inquiries, proving that if an entire function $f$ in $\mathbb{C}^n$ omits a value $a$ while its linear total differential polynomial $\mathcal{L}_k[D]f$ omits a non-zero value $b$, then $f$ must be constant. Our results generalize and extend several existing uniqueness and Picard-type theorems from the classical one-variable setting to the higher-dimensional complex space $\mathbb{C}^n$.

Value Distribution and Picard-type Theorems for Total Differential Polynomials in $\mathbb{C}^n$

TL;DR

The paper extends Nevanlinna value distribution theory to linear total differential polynomials of meromorphic functions in , deriving growth estimates for and a Milloux-type inequality. Under sharing conditions for two meromorphic functions and and a non-transcendence assumption for , it shows that is a non-zero constant. A Picard-type rigidity result is established: if an entire function omits a value while its linear total differential polynomial omits a non-zero value , then the function must be constant, and the work is extended to generalized operators involving a polynomial via . Together, these results broaden classical one-variable uniqueness and Picard-type theorems to the higher-dimensional setting, highlighting new interactions between value distribution, growth, and differential polynomials in .

Abstract

This paper investigates the value distribution and growth properties of linear total differential polynomials for meromorphic functions in several complex variables . By extending the classical Milloux inequality to the framework of total derivatives, we derive a series of fundamental growth estimates for the Nevanlinna characteristic function . We address the value-sharing problem for meromorphic functions and sharing values with their differential polynomials. Under the condition , we establish that is a non-zero constant for non-transcendental meromorphic functions. Furthermore, we provide an affirmative answer to several Picard-type inquiries, proving that if an entire function in omits a value while its linear total differential polynomial omits a non-zero value , then must be constant. Our results generalize and extend several existing uniqueness and Picard-type theorems from the classical one-variable setting to the higher-dimensional complex space .
Paper Structure (4 sections, 15 theorems, 95 equations)

This paper contains 4 sections, 15 theorems, 95 equations.

Table of Contents

  1. Introduction
  2. Basic definitions and some preliminary results:
  3. Main results
  4. Proof of the main results

Key Result

Lemma 1.8

(Second Main Theorem) Let $f$ be a non-constant meromorphic function on $\mathbb{C}^n$, and let $a_1, a_2, \ldots, a_q$ be $q(\geq 3)$ distinct complex number in the whole complex plane $\mathbb{P}$. Then holds for all $z\not\in E.$

Theorems & Definitions (31)

  • Definition 1.1
  • Definition 1.2
  • Definition 1.4
  • Remark 1.1
  • Definition 1.5
  • Lemma 1.8
  • Lemma 1.9
  • Lemma 1.10
  • Lemma 1.11
  • Lemma 1.12
  • ...and 21 more