Area distortion under meromorphic mappings with nonzero pole having quasiconformal extension

Let ${\mathrm{\Sigma }}_k(p)$ be the class of univalent meromorphic functions defined on $\mathbb{D}$ with $k$-quasiconformal extension to the extended complex plane $\widehat{\mathbb{C}}$, where $0\le k<1$. Let ${\mathrm{\Sigma }}_k^0(p)$ be the class of functions $f\in {\mathrm{\Sigma }}_k(p)$ 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 ${\mathrm{\Sigma }}_k^0(p)$. As a consequence of the obtained results, we present a sharp estimate for the bounds of the Hilbert transform.