On extreme points and support points of the family of starlike functions of order

LEMMA 1.1. Let U denote the unit circle { z e C| \z\ = 1} and let μ and v be two probability measures on U. If p and q are two non-negative real numbers with p + q > 1, then there exists a probability measure λ on U X U such that = ίuχu (1 -yz

THEOREM 1.2. Let Ube the unit circle {z e C| \z\ = 1} and a < 1/2. A Iso let ϊF consist of the functions where λ varies over the probability measures on U X U. Then coS(St(α)) = 3? and

COROLLARY 1.3. Let f(z) e S(St(α)) and f(z) = Σ^xanz n. If a < 1/2, then ( 2 - 2 « ) ( 3 - 2 « ) ( π - 2 α ) { and the inequality is sharp.

LEMMA 2.1. (D. Cantor, R. R. Phelps [5].) Let av...,an be complex numbers with \ak\ = 1 (k = 1,2,..., n) and bl9...,bn be distinct complex numbers with \bk\ = 1 (k = 1,2,..., w). Then there exists a finite Blaschke product B(z) such that B{bk) = ak (k = 1,2,..., n).

THEOREM 2.2. %= ( B(y)z u (l-yz)' My) B is a finite Blaschke product and v is a probability measure on U\

LEMMA 3.1. Let φ(z) be a finite Blaschke product with φ(0) = 0 and let c be a complex number with \c\ = 1. // a < 1/2 and φ(z)/(l - cφ(z)) 2(l~ a) is a support point of S(St(a)) then φ(z) = xz for some \x\ = 1. . We first note that a result in [6, p. 83] gives , v ± i

THEOREM 3.2. Let a < 1/2 and J be a continuous linear functional on s? not of the form J(f) = af(0) + bf\ϋ) (a,b<EC and f e J / ) . ///O W fl support point of S(St(a)) associated with J, then fo(z) = xz/(l —