Power of a meromorphic function sharing value with its k-th order directional derivative in C^m
Abjijit Banerjee, Sujoy Majumder, Debabrata Pramanik
TL;DR
The paper addresses the global uniqueness problem in several complex variables for powers of meromorphic functions that share a finite value with their $k$-th order directional derivative along a fixed direction. Employing Nevanlinna theory in $\mathbb{C}^m$, the authors prove a sequence of rigidity theorems (Theorems 2.1–2.5) under CM or IM sharing, showing that either $f^n$ and $\partial_u^k(f^n)$ force $f \equiv c\,\partial_u(f)$ with $(c/n)^k=1$, or $f^n \equiv \partial_u^k(f^n)$, with coordinate-reduced forms $f(z)=e^{c z_j+\alpha(z')}$ when the direction aligns with a coordinate axis. They provide corollaries under weakened growth assumptions and construct numerous examples to demonstrate the sharpness of the bounds and the necessity of the sharing conditions. The results extend classical one-variable uniqueness theorems to the higher-dimensional setting and illustrate the power of value-distribution methods for directional derivatives and multivariable meromorphic mappings.
Abstract
In the context of several complex variables, we investigate the uniqueness problem for a power of a meromorphic function that shares a value with its $k$-th order directional derivative in $\mathbb{C}^m$. Our results extend previous uniqueness theorems from the one-variable case to higher dimensions. Furthermore, we provide numerous examples to demonstrate that our results are, in certain senses, best possible.