On some results for meromorphic univalent functions having quasiconformal extension

TL;DR

This paper studies meromorphic univalent functions with simple poles, providing representation formulas, asymptotic Laurent coefficient estimates, and distortion results, especially for those with quasiconformal extensions.

Contribution

It introduces a representation formula for meromorphic univalent functions with poles and derives new asymptotic and distortion estimates for functions with quasiconformal extensions.

Findings

Representation formula for functions in $oldsymbol{ ext{Σ}(p)}$

Asymptotic Laurent coefficient estimates for $oldsymbol{ ext{Σ}_k(p)}$

Sharp distortion bounds for functions in $oldsymbol{ ext{Σ}(p)}$

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