Quasiconformal extension of meromorphic functions with nonzero pole

TL;DR

This paper studies meromorphic univalent functions with a simple pole in the unit disk that can be extended to the entire complex plane via a quasiconformal map, providing area theorems, conditions for membership, and convolution properties.

Contribution

It introduces the class al{Sigma}_k(p) of meromorphic functions with a pole at p, establishing an area theorem, sufficient conditions for inclusion, and convolution properties.

Findings

Established an area theorem for al{Sigma}_k(p)

Derived a sufficient condition for functions to belong to al{Sigma}_k(p)

Provided a convolution property for functions in al{Sigma}_k(p)

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 $0\leq k < 1$. 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 $\Sigma_k(p)$.