Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
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AREA DISTORTION UNDER MEROMORPHIC MAPPINGS WITH NONZERO POLE HAVING QUASICONFORMAL EXTENSION

  • Bhowmik, Bappaditya (Department of Mathematics Indian Institute of Technology Kharagpur) ;
  • Satpati, Goutam (Department of Mathematics Indian Institute of Technology Kharagpur)
  • Received : 2018.03.30
  • Accepted : 2018.09.19
  • Published : 2019.03.01
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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 ${\hat{\mathbb{C}}}$, 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 ${\mathbb{D}}$.

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References

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