In this paper, we give a generalization and improvement of the Pavlović result on the characterization of continuously differentiable functions in the Bloch space on the unit ball in $\mathbb {R}^{m}$. Then, we derive a Holland–Walsh type theorem for analytic normal mappings on the unit disk.

In this paper we give some generalizations and improvements of the Pavlović result on the Holland–Walsh type characterization of the Bloch space of continuously differentiable (smooth) functions in the unit ball in ${{\text{R}}^{m}}$.

While there is a large variety of univalently induced closed range composition operators on the Bloch space, we show that the only univalently induced, closed range, composition operators on the Bloch-type spaces ${{B}^{\alpha }}$ with $\alpha \,\ne \,1$ are the ones induced by a disc automorphism.

In this paper, the boundedness from below of multiplication operators between $\alpha$-Bloch spaces ${{B}^{\alpha }},\,\alpha \,>\,0$, on the unit disk $D$ is studied completely. For a bounded multiplication operator ${{M}_{u}}\,:\,{{B}^{\alpha }}\,\to \,{{B}^{\beta }}$, defined by ${{M}_{u}}f\,=\,uf$ for $f\,\in \,{{B}^{\alpha }}$, we prove the following result:

(i) If $0<\beta <\alpha ,\,\text{or}\,\text{0}<\alpha \le \text{1}\,\text{and}\,\alpha <\beta \text{,}\,{{M}_{u}}$ is not bounded below;

(ii) if $0\,<\,\alpha \,=\,\beta \,\le \,1,\,{{M}_{u}}$ is bounded below if and only if lim ${{\inf }_{z\to \partial D}}\,\left| u\left( z \right) \right|\,>\,0;$

(iii) if $1\,<\,\alpha \,\le \,\beta ,\,{{M}_{u}}$ is bounded below if and only if there exist a $\delta \,>\,0$ and a positive $r\,<\,1$ such that for every point $z\,\in \,D$ there is a point ${{z}^{'}}\,\in \,D$ with the property $d\left( {{z}^{'}},\,z \right)\,<\,r$ and ${{\left( 1\,-\,{{\left| {{z}^{'}} \right|}^{2}} \right)}^{\beta -\alpha }}\left| u\left( {{z}^{'}} \right) \right|\,\ge \,\delta$, where $d\left( \cdot ,\,\cdot \right)$ denotes the pseudo-distance on $D$.

Given a positive continuous function $\mu $ on the interval $0\,<\,t\,\le \,1$, we consider the space of so-called $\mu $-Bloch functions on the unit ball. If $\mu \left( t \right)\,=\,t$, these are the classical Bloch functions. For $\mu $, we define a metric $F_{z}^{\mu }\left( u \right)$ in terms of which we give a characterization of $\mu $-Bloch functions. Then, necessary and sufficient conditions are obtained in order that a composition operator be a bounded or compact operator between these generalized Bloch spaces. Our results extend those of Zhang and Xiao.

We give a necessary and sufficient condition for a composition operator on an $\alpha$-Bloch space with $\alpha \,\ge \,1$ to be bounded below. This extends a known result for the Bloch space due to P. Ghatage, J. Yan, D. Zheng, and H. Chen.