Area distortion under meromorphic mappings with nonzero pole having quasiconformal extension
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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 $\widehat{\mathbb C}$, 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 upper bound for the Hilbert transform of characteristic function of a Lebesgue measurable subset of $\mathbb D$.
Abstract
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 $\widehat{\mathbb C}$, 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 upper bound for the Hilbert transform of characteristic function of a Lebesgue measurable subset of $\mathbb D$.
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