Let $mathcal{B}$ be a certain Banach space consisting of analytic functions defined on a bounded domain G in the complex plane. Let ${varphi}$ be an analytic polynomial or a rational function and let $M_{varphi}$ denote the operator of multiplication by ${varphi}$. Under certain condition on ${varphi}$ and G, we characterize the commutant of $M_{varphi}$ that is the set of all bounded operators T such that $TM_{varphi}=M_{varphi}T$. We show that $T=M_{Psi}$, for some function ${Psi}$ in $mathcal{B}$.