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
$D$
.