Area distortion under meromorphic mappings with nonzero pole having quasiconformal extension

2017/11/23 by Bhowmik, Bappaditya, Satpati, Goutam
#Complex Variables (math.CV) #FOS: Mathematics

paper · doi:10.48550/arxiv.1711.08684

Abstract

Let Σ_k(p) be the class of univalent meromorphic functions defined on \mathbbD with k-quasiconformal extension to the extended complex plane \widehatℂ, where 0≤ k < 1. Let Σ_k⁰(p) be the class of functions f ∈ Σ_k(p) having expansion of the form f(z)= 1/(z-p) + ∑_n=1^∞b_n zⁿ on \mathbbD. In this article, we obtain sharp area distortion and weighted area distortion inequalities for functions in Σ_k⁰(p). As a consequence of the obtained results, we present a sharp estimate for the bounds of the Hilbert transform.

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