Bruck conjecture for solutions of first-order partial differential equations in Cm

In this paper, we study the Brück conjecture \cite{Bruck-1996} by interpreting it through solutions of first-order partial differential equations in several complex variables. Our results show that the Brück conjecture \cite{Bruck-1996} in ${\mathbb{C}}^m$ holds under certain additional conditions. In pursuit of this objective, we also establish a Borel-Caratheodory theorem in ${\mathbb{C}}^m$ and derive several fundamental results on the order and hyper-order of entire functions in higher dimensions.