On some results for a class of meromorphic functions having quasiconformal extension

We consider the class (Sigma (p)) of univalent meromorphic functions f on ({mathbb D}) having a simple pole at (z=pin [0,1)) with residue 1. Let (Sigma _k(p)) be the class of functions in (Sigma (p)) which have k-quasiconformal extension to the extended complex plane ({hat{mathbb C}}), where (0le k < 1). We first give a representation formula for functions in this class and using this formula, we derive an asymptotic estimate of the Laurent coefficients for the functions in the class (Sigma _k(p)). Thereafter, we give a sufficient condition for functions in (Sigma (p)) to belong to the class (Sigma _k(p).) Finally, we obtain a sharp distortion result for functions in (Sigma (p)) and as a consequence, we obtain a distortion estimate for functions in (Sigma _k(p).)