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
Bhowmik B., Ponnusamy S., Wirths K.-J.: Domains of variability of the Laurent coefficients and the convex hull of the concave univalent functions. Kodai Math. J. 30, 385–393 (2007)MATHCrossRefMathSciNetGoogle Scholar
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