An asymptotic equality of Cartan's Second Main Theorem and some generalizations

Abstract

Motivated by [19] and [10], we define the modified proximity function \(\overline{m}_{q}(f,r)\) for entire curves in complex projective space \(\mathbf{P}^n\mathbf{C}\), and establish an asymptotic equality of Cartan's Second Main Theorem. This is a generalization of [19, Theorem 1.6] for transcendental meromorphic functions. Moreover, we strengthen the result to entire curves of finite order and holomorphic mappings over multiple variables.