Weighted sharing of a small function by a meromorphic function and its derivative

The paper deals with the problem of meromorphic functions sharing a small function with its derivative and improves the results of Yu [K.W. Yu, On entire and meromorphic functions that share small functions with their derivatives, J. Inequal. Pure Appl. Math. 4 (1) (2003) Art. 21 (Online: http://jipam.vu.edu.au/)], Lahiri–Sarkar [I. Lahiri, A. Sarkar, Uniqueness of meromorphic function and its derivative, J. Inequal. Pure Appl. Math. 5 (1) (2004) Art. 20 (Online: http://jipam.vu.edu.au/)] and a recent result of Zhang [Q.C. Zhang, Meromorphic function that shares one small function with its derivative, J. Inequal. Pure Appl. Math. 6 (4) (2005) Art. 116 (Online: http://jipam.vu.edu.au/)].