Area distortion under meromorphic mappings with nonzero pole having quasiconformal extension

TL;DR

This paper establishes sharp area distortion inequalities for a class of meromorphic functions with quasiconformal extensions, providing new bounds and a sharp estimate for the Hilbert transform.

Contribution

It introduces new sharp area distortion inequalities for meromorphic functions with quasiconformal extensions and derives bounds for the Hilbert transform.

Findings

Sharp area distortion inequalities derived

Weighted area distortion inequalities established

Bounds for the Hilbert transform provided

Abstract

Let $\Sigma_k(p)$ be the class of univalent meromorphic functions defined on $\mathbb{D}$ with $k$-quasiconformal extension to the extended complex plane $\widehat{\mathbb{C}}$, where $0\leq k < 1$. Let $\Sigma_k^0(p)$ be the class of functions $f \in \Sigma_k(p)$ having expansion of the form $f(z)= 1/(z-p) + \sum_{n=1}^{\infty}b_n z^{n}$ on $\mathbb{D}$. In this article, we obtain sharp area distortion and weighted area distortion inequalities for functions in $\Sigma_k^0(p)$. As a consequence of the obtained results, we present a sharp estimate for the bounds of the Hilbert transform.