Value Distribution and Picard-type Theorems for Total Differential Polynomials in $\mathbb{C}^n$
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866914268449865728 |
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| author | Ahamed, Molla Basir Allu, Vasudevarao |
| author_facet | Ahamed, Molla Basir Allu, Vasudevarao |
| contents | 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 $\mathbb{C}^n$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2601_14308 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Value Distribution and Picard-type Theorems for Total Differential Polynomials in $\mathbb{C}^n$ Ahamed, Molla Basir Allu, Vasudevarao Complex Variables Primary 32A22, Secondary 30D35, 32A20, 32W99 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 $\mathbb{C}^n$. |
| title | Value Distribution and Picard-type Theorems for Total Differential Polynomials in $\mathbb{C}^n$ |
| topic | Complex Variables Primary 32A22, Secondary 30D35, 32A20, 32W99 |
| url | https://arxiv.org/abs/2601.14308 |