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26 Nov 2010
On some results of A. E. Livingston and coefficient problems for concave univalent functions
 Bappaditya Bhowmik
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We consider functions that map the open unit disc conformally onto the complement of a bounded convex set. We call these functions concave univalent functions. In 1994, Livingston presented a characterization for these functions. In this paper, we observe that there is a minor flaw with this characterization. We obtain certain sharp estimates and the exact set of variability involving Laurent and Taylor coefficients for concave functions. We also present the exact set of variability of the linear combination of certain successive Taylor coefficients of concave functions.
This work of the author was done during the period of stay as a Postdoctoral fellow under UGCDr. D.S. Kothari postdoctoral fellowship at the Indian Institute of Science, Bangalore, India.
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 Title
 On some results of A. E. Livingston and coefficient problems for concave univalent functions
 Journal

Archiv der Mathematik
Volume 95, Issue 6 , pp 575581
 Cover Date
 20101201
 DOI
 10.1007/s0001301002034
 Print ISSN
 0003889X
 Online ISSN
 14208938
 Publisher
 SP BirkhĂ¤user Verlag Basel
 Additional Links
 Topics
 Keywords

 Primary 30C45
 Secondary 30C50
 Concave univalent functions
 Taylor and Laurent coefficients
 Authors

 Bappaditya Bhowmik ^{(1)}
 Author Affiliations

 1. Department of Mathematics, Indian Institute of Science, Bangalore, 560012, India