Coefficients of symmetric functions of bounded boundary rotation

The well-known inclusion relation between functions with bounded boundary rotation and close-to-convex functions of some order is extended to m m -fold symmetric functions. This leads solving the corresponding result for close-to-convex functions to the sharp coefficient bounds for m m -fold symmetric functions of bounded boundary rotation at most k π kπ when k ≥ 2 m k ≥ 2m . Moreover it shows that an m m -fold symmetric function of bounded boundary rotation at most ( 2 m + 2 ) π (2m + 2)π is close-to-convex and thus univalent.