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Let $f$ be a non-constant meromorphic function in the open complex plane $\mathbb{C}$. In this paper we prove under certain essential conditions that $R(f)$ and $P[f]$, rational function and differential polynomial of $f$ respectively, share a small function of $f$ and obtain a conclusion related to $Br\ddot{u}ck$ conjecture. We give some examples in support to our result.
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