Close-to-convex functions and linear-invariant families

Abstract

correct. We shall modify this result for linear-invariant families. Families of closeto-convex functions and of functions of bounded boundary rotation will be showed to be linear-invariant. Because of the coefficient estimate for close-to-convex functions and functions of bounded boundary rotation derived by Aharonov and Friedland [1], it is possible to get the distortion theorem for the n-th. derivative for all n, but here we obtain the same conclusion more elementarily (and without using the linear-invariance), just because the coefficient estimate is given for all n. All functions / considered here are analytic functions on the unit disk with normalization /(0)=0,//(0)=l, and they are locally schlicht, i.e., {z|/'(z)=O}=0. Let N be the class of such functions. Pommerenke defined a linear-invariant family in [9] and showed some properties of such families. A subset F of N is called linear-invariant if it is closed under the re-normalized composition with a schlicht automorphism of the unit disk. If the modulus of the second Taylor coefficient is bounded in F, we define the order a of the linear-invariant family to be