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Quasiconformal extension of meromorphic functions with nonzero pole


Authors: B. Bhowmik, G. Satpati and T. Sugawa
Journal: Proc. Amer. Math. Soc. 144 (2016), 2593-2601
MSC (2010): Primary 30C62, 30C55
Published electronically: October 22, 2015
MathSciNet review: 3477076
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Abstract: In this note, we consider meromorphic univalent functions $ f(z)$ in the unit disc with a simple pole at $ z=p\in (0,1)$ which have a $ k$-quasiconformal extension to the extended complex plane $ {\widehat {\mathbb{C}}},$ where $ 0\leq k < 1$. We denote the class of such functions by $ \Sigma _k(p)$. We first prove an area theorem for functions in this class. Next, we derive a sufficient condition for meromorphic functions in the unit disc with a simple pole at $ z=p\in (0,1)$ to belong to the class $ \Sigma _k(p)$. Finally, we give a convolution property for functions in the class $ \Sigma _k(p)$.


References [Enhancements On Off] (What's this?)

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Additional Information

B. Bhowmik
Affiliation: Department of Mathematics, Indian Institute of Technology Kharagpur, Kharagpur - 721302, India
Email: bappaditya@maths.iitkgp.ernet.in

G. Satpati
Affiliation: Department of Mathematics, Indian Institute of Technology Kharagpur, Kharagpur - 721302, India
Email: g.satpati@iitkgp.ac.in

T. Sugawa
Affiliation: Graduate School of Information Sciences, Tohoku University, Aoba-ku, Sendai 980-8579, Japan
Email: sugawa@math.is.tohoku.ac.jp

DOI: https://doi.org/10.1090/proc/12933
Keywords: Quasiconformal map, convolution
Received by editor(s): February 9, 2015
Received by editor(s) in revised form: August 3, 2015
Published electronically: October 22, 2015
Additional Notes: The first author would like to thank NBHM, DAE, India (Ref.No.- 2/48(20)/2012/ NBHM(R.P.)/R&D II/14916) for its financial support
The third author would like to thank JSPS Grant-in-Aid for Scientific Research (B) 22340025 for its partial financial support
Communicated by: Jeremy Tyson
Article copyright: © Copyright 2015 American Mathematical Society