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

TL;DR

This paper extends value distribution and Picard-type theorems to linear total differential polynomials of meromorphic functions in several complex variables, providing growth estimates and uniqueness results in higher dimensions.

Contribution

It generalizes classical results to $ $-dimensional complex space, establishing new growth estimates and uniqueness theorems for differential polynomials.

Findings

Derived growth estimates for $T(r, ext{differential polynomial})$

Proved conditions under which differential polynomials share values

Showed that certain value omissions imply the function is constant

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 $2\delta(0,f)+(k+4)\Theta(\infty,f)>k+5$, 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 $\mathbb{C}^n$.