Extremal properties of extreme and support points of univalent functions with montel normalization

Abstract

Let U = {z : vertical bar z vertical bar < 1} be the unit disk. The Montel class, M(a), is the class of functions f (z) = a(1)z + a(2)z(2) +..., which are analytic and univalent in the unit disk and satisfy the conditions f (0) = 0 and f (a) = a, (0 < a < 1). Duren and Schober gave some results on extreme and support points of So in [P.L. Duren and G. Schober, Nonvanishing univalent functions, Math. Z. 170 (1980), pp. 195-216.] and [P.L. Duren and G. Scober, Nonvanishing univalent functions III, Ann. Acad. Sci. Fennicae Series A, I, 10 (1985), pp. 139-147]. The purpose of this paper is to obtain analog results for the corresponding class M(a). Using the well-known Brickman representation for univalent functions, it is shown that the extreme and support points of the class M (a) of the univalent functions with Montel normalization map the unit disk onto the component of a continuous arc extending from omega(0)(omega(0) not equal 0) to infinity, which is intersecting each ellipse, with foci 0 and a, exactly once.