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

Abstract Let U={z:| z |<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, (0Duren and Schober gave some results on extreme and support points of S 0 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 w 0(w 0≠0) to ∞, which is intersecting each ellipse, with foci 0 and a, exactly once.