AREA DISTORTION UNDER MEROMORPHIC MAPPINGS WITH NONZERO POLE HAVING QUASICONFORMAL EXTENSION

Let ${\Sigma}_k(p)$ be the class of univalent meromorphic functions defined on the unit disc ${\mathbb{D}}$ with k-quasiconformal extension to the extended complex plane ${\hat{\mathbb{C}}}$, where $0{\leq}k<1$. Let ${\Sigma}^0_k(p)$ be the class of functions $f{\in}{\Sigma}_k(p)$ having expansion of the form $f(z)=1/(z-p)+{\sum_{n=1}^{\infty}}\;b_nz^n$ on ${\mathbb{D}}$. In this article, we obtain sharp area distortion and weighted area distortion inequalities for functions in ${\sum_{k}^{0}}(p)$. As a consequence of the obtained results, we present a sharp upper bound for the Hilbert transform of characteristic function of a Lebesgue measurable subset of ${\mathbb{D}}$.