Radius of convexity of partial sums of functions in the close-to-convex family

Abstract

Let F denote the class of all normalized analytic functions f that are locally univalent in the unit disk |z|<1 satisfying the condition Re(1+z f″(z)f′(z))>-12 for |z|<1. Functions in F are known to be close-to-convex (univalent) in the unit disk. This class plays a crucial role in the discussion on certain extremal problems for the class of complex-valued and sense-preserving harmonic convex functions and some other related problems in determining univalence criteria for sense-preserving harmonic mappings. In this article, we show that every section of a function in the class F is convex in the disk |z|<1/6. The radius 1/6 is best possible. We conjecture that every section of functions in the family F is univalent and close-to-convex in the disk |z|<1/3. © 2013 Elsevier Ltd. All rights reserved.