Convex hulls and extreme points of families of starlike and close-to-convex mappings

THEOREM 1. Let X = {x:\x\ = 1 and Imx ^ 0}, a ^ 1, denote the set of probability measures on X. Given v e 0* there exists a μ e & so that

THEOREM 2. Let where X and & are as in Theorem 1. Ifp>0 and q > 0 then Fp'FgCZ Fq+q where Fp Fq = [/:/ = gh and g e Fp, he Fg).

Theorem 1 by the same arguments used to prove Theorem 1 in [3]. Now suppose p + q ^ 1. Consider the linear operator L defined by p 1 — z 2 1 — z 2 It is easily verified that L is a linear map from Fp onto Fp+1 since L applied to 1/[(1 - xz)(l - xz)f yields 1/[(1 - xz)(l - xz)] p+1. Let w [(1 - χz)(l - xz)]> [(1 - yz)(l - yx)Y A computation shows that p + q 1 — z

THEOREM 3. Let X and & be as in Theorem 1. Then given ^9 3ve^ such that - xz)(l - xz)dμ(x)} = j^[(l ~ xz)(l - xz)]-'dv(x) . exp I j ^ -

THEOREM 4. Let X and & be as in Theorem 1, a < 1, k be any positive integer and J^ be the set of functions fμ on Δ defined by then &~ = βί?StR{a9 k) and VMTSt/μ, k)^{{1_χzΎ-J{χ_WzΎ-.β -1*1 = 1. fin* 2: 0} .

LEMMA 1. Suppose Rep(z) > 0, p(0) = 1, p(z) is even, and p(z) is real on ( — 1,1). Then = \ l—f-—dμ(x) Jx (1 — xz 2)(l — xz 2) where X = {x: \x\ — 1 and \xax ^ 0} and μ is a probability measure on X.

THEOREM 5. Let X = {x: \x\ = 1, Im# ^ 0}, & be the set of probability measures on X, and J^ be the set of functions fμ on Δ defined by i

THEOREM 6. Let {X = x: \x\ = 1, Im x ^ 0}, & he the set of pro- bability measures on X, and J^~ be the class of functions fμ on Δ defined by — xz) J (1 — JχL(l — xz)(l — xz) J (1 — xz)(l — xz) Then &~ = £$fK!R(β), where β ^ l and K'R{β).$

THEOREM 7. Suppose F(z) e K{β) where β ^ 1. If f(z) < F{z) then f{z) < fk(z) where

THEOREM 8. Let X 2 = {(x, y): \x\ = \y\ = 1}, ^ δe ίfte se£ 0/ probability measures on X 2, and ^ be the class of functions fμ on Δ defined by J — yz k) then ^ = έ%fC f k, where C[ is the set of derivatives of functions in