A Method for Investigating Geometric Properties of Support Points and Applications

Johnny E. Brown
1985 Transactions of the American Mathematical Society  
A normalized univalent function / is a support point of S if there exists a continuous linear functional L (which is nonconstant on S) for which / maximizes ReL(g), g £ S. For such functions it is known that T = C -/(£/) is a single analytic arc that is part of a trajectory of a certain quadratic differential Q(w) dw2. A method is developed which is used to study geometric properties of support points. This method depends on consideration of \m{w2Q(w)} rather than the usual Re{w2Q(w)}.
more » ... e{w2Q(w)}. Qualitative, as well as quantitative, applications are obtained. Results related to the Bieberbach conjecture when the extremal functions have initial real coefficients are also obtained.
doi:10.2307/2000411 fatcat:2uplkvjqdnbihm2iwekfv2mo6q