On some results for meromorphic univalent functions having quasiconformal extension

Abstract

We consider the class $\Sigma (p)$ of univalent meromorphic functions $f$ on $\ID$ having simple pole at $z=p\in [0,1)$ 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 $\sphere$ %with $q=\frac{1+k}{1-k}$ where $0\le k<1$. 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 in the class ${\Sigma }_k(p).$ Finally we obtain a sharp distortion result for functions in $\Sigma (p)$ and as a consequence, we get a distortion estimate for functions in ${\Sigma }_k(p).$

arXiv comment15 pages, This article is with a journal

Cited by in corpus (1)

• Area distortion under meromorphic mappings with nonzero pole having quasiconformal extension