The radius of close-to-convexity of functions of bounded boundary rotation

Abstract:

An analytic function whose boundary rotation is bounded by $k\pi (k \geqq 2)$ is shown to map a disc of radius ${r_k}$ onto a close-to-convex domain, where ${r_k}$ is the solution of a transcendental equation when $k > 4$ and ${r_k} = 1$ when $2 \leqq k \leqq 4$. The above value of ${r_k}$ is shown to be the best possible for each k and an asymptotic expression for ${r_k}$ is obtained.