A variational method for functions of bounded boundary rotation

Abstract

Let f be a function analytic in the unit disc, properly normalized, with bounded boundary rotation. There exists a Stieltjes integral representation for 1 + zf(z)/f'(z). From this representation, and in view of a known variational formula for functions of positive real part, a variational formula is derived for functions of the form q(z) =1 + zf(z)/f'(z). This formula is for functions of arbitrary boundary rotation and does not assume the functions to be univalent. A new proof for the radius of convexity for functions of bounded boundary rotation is given. The extremal function for Re {F(f'(z))} is derived. Examples of univalent functions with arbitrary boundary rotation are given and estimates for the radius in which Re {f'(z)} > 0 are computed. The coefficient problem is solved for a4 for all values of the boundary rotation and without the assumption of univalency.