On Lappan's five-valued theorem for $\varphi$-normal functions in several variables

Abstract

UDC 517.5

Let $\mathbb{U}^m\subset\mathbb{C}^m$ be а unit ball centered at the origin and let $\mathbb{P}^n$ be an $n$-dimensional  complex projective space with the metric $E_{\mathbb{P}^n}.$ Also, let $\varphi\colon [0,1)\rightarrow(0,\infty)$ be a smoothly increasing function.  A holomorphic mapping $f\colon\mathbb{U}^m\rightarrow \mathbb{P}^n$  is called {\it $\varphi$-normal} if $({\varphi(\|z\|))}^{-1}{(E_{\mathbb{P}^n}(f(z), df(z))(\xi))}$ is bounded above for $z\in\mathbb{U}^m$ and $\xi\in\mathbb{C}^m$ such that $\|\xi\|=1,$ where $df(z)$ is the map from $T_z\big(\mathbb{U}^m\big)$ to $T_{f(z)}\big(\mathbb{P}^n\big)$ induced by $f.$ For $n=1,$ $f$ is called a $\varphi$-normal function. We present an extension of Lappan's five-valued theorem to  the class of $\varphi$-normal functions.