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Abstract. A function /(z) = z+2°=2 onzn regular in the open unit disk A = {z : |z| 0 for z in A; in this case we write f(z)e&a.A fundamental result of this paper shows that the transformation f (z\ = azf(iz + a)lil+âz))efines a function in &a whenever/(z) is in &a and a is in A. If g(z) is regular in A, g(0) = 0 and g'(0) = 1, then g(z) is in 0O if and only if zg'(z) is in &". The main result of the paper is the derivation of the sharp radius of close-toconvexity for each class ^"; it is given as the solution of an equation in r which is dependent only on a. (Approximate solutions of this equation were made by computer and these suggest that the radius of close-to-convexity of the class <& = \Ja &a is approximately .99097+ .) Additional results are also obtained such as the radius of convexity of 0O, a range of a for which g(z) in ^" is always univalent is given, etc. These conclusions all depend heavily on the transformation cited above and its analogue for @a.

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Abstract.An analytic function whose boundarv rotation is bounded by k-n (k^.2) is shown to map a disc of radius rt onto a close-to-convex domain, where rk is the solution of a transcendental equation when k>4 and rk = \ when 2^/t^4.The above value of rk is shown to be the best possible for each k and an asymptotic expression for rk is obtained.Let Vk (k^.2) denote the class of functions/(z) which are analytic in the unit disc £={z:|z|4, then functions in Vk need not be close-to-convex or even univalent. In this paper we determine the radius of close-to-convexity of Vk for each k, i.e. the radius of the largest disc centered at the origin which is mapped onto a close-to-convex domain by all/in Vk. The techniques used are similar to those used by Krzyz in determining the radius of closeto-convexity of the class of univalent functions [2]. Some related problems were posed by M. O. Reade [5].

A class of functions si is defined which contains the spirallike and close-to-convex functions. By decomposing si into subclasses in a natural way, some basic properties of si and these subclasses are determined, including solutions to extremal problems; distortion theorems; coefficient inequalities; and the radii of convexity and close-to-convexity.