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

Let FF denote the class of all normalized analytic functions ff that are locally univalent in the unit disk |z|<1|z|<1 satisfying the condition Re(1+zf″(z)f′(z))>−12 for |z|<1|z|<1. Functions in FF 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 FF is convex in the disk |z|<1/6|z|<1/6. The radius 1/61/6 is best possible. We conjecture that every section of functions in the family FF is univalent and close-to-convex in the disk |z|<1/3|z|<1/3.