Subordination and extreme-point theory

THEOREM 1. Let K denote the subset of S of convex mappings

THEOREM 2. Let Si* = {f:f<g for some g in St}, let U = {x: I a? I = 1} ami let X — U x U. Then ξ>St* consists of the functions represented by where μ varies over the probability measures on X. Also,

THEOREM 3. Let C denote the subset of S of close-to-convex func- tions and let C* = {/:/ -< g for some g in C}. Let X = U x U x U where U — {z: | z \ = 1}. Then §C* consists of all functions = \ , y, w) (1 + WXZf where μ varies over the probability measures on X. Also,

THEOREM 4. Let R{a) denote the set of analytic functions f so that /(0) = 0, /'(0) = 1 and Re Vf(zjjz > a(z e J)(0 ^ a < 1). Let X denote the unit circle, P the set of probability measures on X, and fμiz) = \ Jl — xz Then = {/„: μeP} and @%R(a) 1 — xz I

THEOREM 5. Let j ^ ~ = {f:f<F} where F(z) = (1 - z)~ p. If ^y p ^ l and f(z) = Σ«M , then \ an \ £ P(v + 1) • • {p + n - 1) {n =

THEOREM 6. Let F be analytic in A and let J^ be the family of functions subordinate to F in A. Then the functions f(z) = F(xz), \x\ = 1, belong to

Theorem 6 can be proved by appealing to other unique extremal properties of the functions F{xz). For example, say that ^'(0) Φ 0, 30 that for all sufficiently small r, F maps | z \ < r one-to-one onto a convex domain. Lindelof's theorem [13, p. 22] asserts that if Ar = {z:\z\ < r} then f(Aτ)cF(Ar) and the boundary of f(Ar) meets the boundary of F(Ar) only if f(z) = F(xz), \x\ = 1. The convexity of jP(Jr) thereby implies that Ree ία/(z) is uniquely maximized for each z,\z\ — r, over ^ 7 With vary

Theorem 6 may be thought of as prescribing the minimal pos- sibility for the set ©φ ^ 7 This minimal situation is achieved for several examples discussed earlier. Our next theorem gives informa- tion about when the set @^>^ can be much more varied for certain functions F. We need to recall some results about subordination and its rela- tion to H p spaces. (For a general discussion of the theory of H p spaces see [3] or [9].) For a function / analytic in A we set (p > 0, 0 < r < 1). In [11] J. E.

THEOREM 7. Let F be analytic in A and let J? be the family of functions subordinate to F in Δ. If F e H p where 1 < p < oo and if φ is an inner function so that φ{0) — 0, then f = F(φ) e 6fξ> ^.