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

G.  Datt

doi:10.37863/umzh.v74i9.6237

G. Datt Babasaheb Bhimrao Ambedkar University, Lucknow, India

DOI: https://doi.org/10.37863/umzh.v74i9.6237

Keywords: Complex projective space, holomorphic mapping, normal functions, $\varphi$-normal function

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.

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