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

Abstract

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