Weighted Yosida Mappings of Several Complex Variables

Let $M$ be a complete complex Hermitian manifold with metric $E_M$ and let $\phi :[0,\mathrm{\infty })\to (0,\mathrm{\infty })$ be positive function such that $${\gamma }_r=\underset{r\le a<b}{\sup }\mid (\phi (a)-\phi (b))\mathrm{/}(a-b)\mid \le C,r\in (0,\mathrm{\infty }),$$ for some $C\in (0,1],$ and ${\lim }_{r\to \mathrm{\infty }}{\gamma }_r=0.$ A holomorphic mapping $f:{\mathbb{C}}^m\to M$ is said to be a weighted Yosida mapping if for any $z,\xi \in {\mathbb{C}}^m$ with $\mathrm{\parallel }\xi \mathrm{\parallel }=1,$ the quantity $\phi (\mathrm{\parallel }z\mathrm{\parallel })E_M(f(z),df(z)(\xi ))$ remains bounded above, where $df(z)$ is the map from $T_z({\mathbb{C}}^m)$ to $T_{f(z)}(M)$ induced by $f\mathrm{.}$ We present several criteria of holomorphic mappings belonging to the class of all weighted Yosida mappings.