, Volume 95, Issue 6, pp 575-581
Date: 26 Nov 2010

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

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Abstract

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 Post-doctoral fellow under UGC-Dr. D.S. Kothari postdoctoral fellowship at the Indian Institute of Science, Bangalore, India.