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arXiv · 1705.03660

On some results for meromorphic univalent functions having quasiconformal extension

Bappaditya Bhowmik·Goutam Satpati
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Abstract

We consider the class $Σ(p)$ of univalent meromorphic functions $f$ on $\ID$ having simple pole at $z=p\in[0,1)$ with residue 1. Let $Σ_k(p)$ be the class of functions in $Σ(p)$ which have $k$-quasiconformal extension to the extended complex plane $\sphere$ %with $q=\frac{1+k}{1-k}$ where $0\leq k < 1$. We first give a representation formula for functions in this class and using this formula we derive an asymptotic estimate of the Laurent coefficients for the functions in the class $Σ_k(p)$. Thereafter we give a sufficient condition for functions in $Σ(p)$ to belong in the class $Σ_k(p).$ Finally we obtain a sharp distortion result for functions in $Σ(p)$ and as a consequence, we get a distortion estimate for functions in $Σ_k(p).$

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Bappaditya Bhowmik, Goutam Satpati. 2017-05-10. On some results for meromorphic univalent functions having quasiconformal extension. https://arxiv.org/abs/1705.03660

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