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Let $G$ be a bounded domain in the complex plane $C$. A Banach
space of analytic functions on $G$ is a Banach space $\cal{B }$
consisting of functions that are analytic on G such that $1 \in
\cal{B} $, the functional $e(\lambda ) : \cal{B} \rightarrow C $
of evaluation at $\lambda \in G $ given by $e (\lambda ) (f) = f
(\lambda ) $ is bounded, and if $f \in \cal{B}$ then $zf \in
\cal{B} $. The collection of all multipliers of $\cal{B }$ is
denoted by $\cal{M} (\cal{B} )$. In this article, we give
sufficient conditions so that $\cal{M} (\cal{B} )$ $= H^{\infty }
(G ) \cap \cal{B} $$= H^{\infty } $. Also we show that if the
number of connected components of $\partial G $ is finite and
$H^{\infty } (G)$ is dense in $L^1_a (G) $ then $H^{\infty } (G)
$ is dense in $L^p_a (G) , p > 1$.
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