Quasiconformal extension of meromorphic functions with nonzero pole

Abstract

In this note, we consider meromorphic univalent functions $f(z)$ in the unit disc with a simple pole at $z=p\in (0,1)$ which have a $k$-quasiconformal extension to the extended complex plane $\hat{\mathbb{C}},$ where $0\le k<1$. We denote the class of such functions by ${\Sigma }_k(p)$. We first prove an area theorem for functions in this class. Next, we derive a sufficient condition for meromorphic functions in the unit disc with a simple pole at $z=p\in (0,1)$ to belong to the class ${\Sigma }_k(p)$. Finally, we give a convolution property for functions in the class ${\Sigma }_k(p)$.

8 pages