Archiv der Mathematik

, Volume 95, Issue 6, pp 575–581

On some results of A. E. Livingston and coefficient problems for concave univalent functions



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.

Mathematics Subject Classification (2000)

Primary 30C45 Secondary 30C50 


Concave univalent functions Taylor and Laurent coefficients 


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Copyright information

© Springer Basel AG 2010

Authors and Affiliations

  1. 1.Department of MathematicsIndian Institute of ScienceBangaloreIndia

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