Abstract
We consider the class \(\Sigma (p)\) of univalent meromorphic functions f on \({\mathbb D}\) having a simple pole at \(z=p\in [0,1)\) with residue 1. Let \(\Sigma _k(p)\) be the class of functions in \(\Sigma (p)\) which have k-quasiconformal extension to the extended complex plane \({\hat{\mathbb C}}\), where \(0\le k < 1\). 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 \(\Sigma _k(p)\). Thereafter, we give a sufficient condition for functions in \(\Sigma (p)\) to belong to the class \(\Sigma _k(p).\) Finally, we obtain a sharp distortion result for functions in \(\Sigma (p)\) and as a consequence, we obtain a distortion estimate for functions in \(\Sigma _k(p).\)
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The authors would like to thank Toshiyuki Sugawa for his suggestions and careful reading of the manuscript.
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Communicating Editor: Kaushal Verma
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Bhowmik, B., Satpati, G. On some results for a class of meromorphic functions having quasiconformal extension. Proc Math Sci 128, 61 (2018). https://doi.org/10.1007/s12044-018-0442-z
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DOI: https://doi.org/10.1007/s12044-018-0442-z