With the aid of weakly weighted sharing and a recently introduced sharing notion in cite{1b} known as relaxed weighted sharing we investigate the uniqueness of meromorphic functions sharing a small function with its differential polynomials. Our results will improve and supplement all the results obtained by Zhang and Yang cite{12} as well as a substantial part of the results recently obtained by the present author cite{1a1} and thus provide a better answer to the questions posed by Yu cite {10} in this regard.

Keywords: meromorphic function, derivative, small function, weakly weighted sharing