Radius of Convexity of Partial Sums of Odd Functions in the Close-to-Convex Family

We consider the class of all analytic and locally univalent functionsfof the form f ( z ) = z + ∑ n = 2 ∞ a 2 n − 1 z 2 n − 1 , | z | 1 , satisfying the condition Re ( 1 + z f ″ ( z ) f ′ ( z ) ) > − 1 2 . We show that every section s 2 n − 1 ( z ) = z + ∑ k = 2 n a 2 k − 1 z 2 k − 1 , off, is convex in the disk | z | 2 / 3 . We also prove that the radius 2 / 3 is best possible, i.e. the number 2 / 3 cannot be replaced by a larger one.