On the Nevanlinna's Theory for Vector-Valued Mappings

Abstract

The purpose of this paper is to establish the first and second fundamental theorems for an $E$-valued meromorphic mapping from a generic domain $D\subset ℂ$ to an infinite dimensional complex Banach space $E$ with a Schauder basis. It is a continuation of the work of C. Hu and Q. Hu. For $f\left(z\right)$ defined in the disk, we will prove Chuang's inequality, which is to compare the relationship between $T(r,f)$ and $T(r,f^{\prime })$. Consequently, we obtain that the order and the lower order of $f\left(z\right)$ and its derivative $f^{\prime }\left(z\right)$ are the same.