Area distortion under meromorphic mappings with nonzero pole having quasiconformal extension

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 $\hat{\mathbb{C}}$, where $0\le k<1$. 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.

16 pages

References in corpus (1)

• On some results for meromorphic univalent functions having quasiconformal extension