Weighted Yosida Mappings of Several Complex Variables

Abstract

Let $M$ be a complete complex Hermitian manifold with metric $E_{M}$ and let $\varphi: [0,\infty)\rightarrow (0,\infty)$ be positive function such that $$\gamma_r=\sup\limits_{r\leq a<b}\left|(\varphi(a)-\varphi(b))/(a-b)\right|\leq C,~r\in (0,\infty),$$ for some $C\in (0,1],$ and $\lim_{r\rightarrow\infty}\gamma_r=0.$ A holomorphic mapping $f:\mathbb{C}^{m}\rightarrow M$ is said to be a weighted Yosida mapping if for any $z,~\xi\in\mathbb{C}^{m}$ with $\|\xi\|=1,$ the quantity $\varphi(\|z\|)E_{M}(f(z), df(z)(\xi))$ remains bounded above, where $df(z)$ is the map from $T_z(\mathbb{C}^{m})$ to $T_{f(z)}(M)$ induced by $f.$ We present several criteria of holomorphic mappings belonging to the class of all weighted Yosida mappings.