Support points of the set of univalent functions

Abstract

Let S be the usual set of analytic, univalent, normalized functions on the unit disk A. Let f E S. Then f is a support point of S, if there exists a continuous linear functional J on the space of analytic functions on A, J nonconstant on S, such that Re J(f)=max{ReJ(g):g e S}. THEOREM. Let f be a support point of S. Then f is analytic in the closed unit disk except for a pole of order two at one point of the unit circle. The complement off(A) is a single arc r, regular and analytic everywhere, tending to o in such a way that the angle between r and the radial direction is always less than 7T4. Near co, r can be described in the form :,=-, dnt(O < t <6, d I 0O). In particular, r is asymptotic to a line d-1t-1 +do. Let A be the open unit disk of the complex plane C, and let J be a continuous linear functional on the space of functions analytic on A. (The continuity of J means that if g.-*g uniformly on compact subsets of A, then J(gn)-J(g) in C.) We assume further that J does not have the trivial form J(g)= ocg(0)+ fg'(0). Let S be the usual set of analytic univalent functions g on A satisfying g(O)=0 and g'(0)= 1. If f E S we shall say thatf is a support point of S provided there exists a functional J as described above such that Re J(f) = max{Re J(g):g e S}. In geometrical language S has a nontrivial supporting hyperplane passing throughf. We shall establish the following theorem. THEOREM. Let f be a support point of S. Then the following statements hold. I. f is analytic in the closed unit disk A except for a pole of order two at one point of the unit circle MA. Received by the editors January 29, 1973 and, in revised form, May 21, 1973. AMS (MOS) subject classifications (1970). Primary 30A36, 30A38; Secondary 30A40. 1 Research partially supported by National Science Foundation Grant PO 19709000. 2 Research partially supported by National Science Foundation Grant PO 12020000, and a State University of New York Fellowship. ? American Mathematical Society 1974