Subordination theorems for some classes of starlike fumctions

THEOREM 1. Let α, d, and M be fixed nonnegative numbers satis- fying 0 < α < ° o 9 θ ^ d < l , and 1 < M ^ oo. Then there exists a func- tion F = F( ,a,d,M)E S(α, d, M) with the following properties: (A) The function g(z) = log F(z)lz, z G K,(g(0) = 0) w univalent and convex in the direction of the imaginary axis, (B) /// G S(α, d, M), ί/ien log/(z)/z, z G K, w subordinate to g. In order to describe F we first make the following definition. DEFINITION 1. Let a be given, 0 < a < oo. Then γ is sai

COROLLARY 1. Let α, d, M, and F be as in Theorem 1. Let Φ be a given nonconstant entire function. If f E S(a, d, M), then (A) For given z E K - {0} (B) For given r, 0 < r < 1, and λ > 0, P |/(re ίθ)| λd<9 ^ Γ" |F Jo Jo (C) For a given positive integer N ^ = z + ΣI=2 α*z k and F(z) = z + Σl=2Λkz k

Theorem 2 implies, as in the discussion after Theorem 1, the following corollary.

COROLLARY 2. Let d and M be as in Corollary 1. Replace F by F* and 5(α,d,M) by S*{d,M) in Corollary 1. Then Corollary 1 is valid for F*. We remark that Theorem 2 and Corollary 2 are well known in the simple case d = 1, M = oo (see Goluzin [4, Thm. 1, p. 531]). Moreover, Suffridge [16] proved (C) of Corollary 2 for S*(d, oo) and N = 2. Barnard [2] proved (A) of Corollary 2 for S*(d,M) and Φ(H>) = ± w. With these exceptions, Theorem 2 and Corollary 2 are new results for starlike functions. For giv

LEMMA 1. Let D be a bounded domain containing w = 0 with the property that each ray through w = 0 intersects 3D in exactly one point. Let a be a fixed positive number and suppose there exists a sufficiently small η>0 such that whenever A, BE 3D and 0< I Arg (ΛB) I < η < Trα, then either β C D U {A, B} or β C 3D. Then there is a function g &S(a) and a number t >0 such that tg(K) = D.

LEMMA 2. Let /ES(α,0,M) for some M<oo and put D = f(K). Then each ray through w = 0 intersects 3D in exactly one point. If A, β, A 7^ B, are in 3D*and if β is an a arc with endpoints A and B, then (a) either β C3D or β CD U {A,B}, (b) // Ω denotes the component of D - β containing w = 0, then there exists a g E S(a) and t >0 such that tg(K) = Ω.

Lemma 2. Choose a sector V containing β in its interior and of angle opening φ, 0 < φ < πa. This choice is possible by (2.1a). Let p be an analytic 1/α power of w on V. Then (2.3a) implies that p(V ΠDr) is

LEMMA 3. Let f and D be as in Lemma 2 and put Γ = 3D. Then Γ has a unique right (left) hand tangent at each wGΓ. Consequently, there exists exactly one a curve γ which is tangent to Γ at w from the right (left). IfβCγ is an a arc with one endpoint w, then β ΠD - {φ}.

Lemma 3 follows easily from (2.3b) and geometric proper- ties of convex domains. We omit the details. 3. Applications of Lemmas 1-3. We now determine δ(M,a) (see §1) for fixed M and a satisfying 1< M < oo and 0 < a < oo. To do this we let /, D, and Γ be as in Lemma 3 and put d(f) = min {| w |: w G Γ}. We shall use the following remark which also will be used in §4 and §8.

Lemma 3 and the fact that the γ, corresponding to e2 is tangent to Γ2(€,) for 0<€i<6 2ge 0. (4.6) then follows from (4.5), the definition of D2(e), and the fact that lim^. Qn = A (see III). To define III when B satisfies II (b) we choose a point Ao E Γ D dKd near A, A0^A, and let A, be a point on the arc of dKd ΠT with endpoints Ao, A. With this notation γ, γ0, and yx are defined as in HI (a). Let γ* be the a curve tangent to Γ at B which does not contain β. Let J5j be the point nearest B in V w

LEMMA 4. If Ω E sέ, then there exists a sequence of domains {Ωn} with Ωn E sdn such that Ωn -» Ω in the sense of kernel convergence.

Lemma 4 and A Theorem of Caratheodory imply that if f(K) = Ω, /n(K) = Ωn, where /,/„ ES(a,d,M), then uniformly on compact subsets of K.

LEMMA 5. Let H} and Ω^ = H}(K) be as above for j g n0. Then all but at most two of the a sides of d Ω, are either a chords of dKM or are tangent to dKd.

LEMMA 6. Let dx and Mx be fixed positive numbers satisfying d <d]<Mι<M. Let Ω, be as in Lemma 5 for j ^ n0. Then there exists, independently of j, a maximum number N of a sides of dΩ} that intersect the closed annulus L{dx,Mx).

LEMMA 7. Let dx and Mx be as in Lemma 6. Let Ω = H(K). Then dfiΠ{w: dλ < \ w | < M,} consists of a finite number of a arcs.

LEMMA 8. For some real θ, Ω = e iθF(K), where F = F( , α, d, M) is as in (i)-(iv).

Theorem 2 of Jenkins [6] we see that (8.3) \F(re iθ>)\ > \F(re iθή\, 0 ^ 0, < θ2^ π, whenever 0 < r < l . From (8.3) and Theorem 3 of Kaplan [8], we deduce that the function g(z) = log F(z)/z, z E X, is univalent and convex in the direction of the imaginary axis. Suppose now for some f £S(a, d, M) that the function h(z) = log/(z)/z, z E K, is not subordi- nate to g(z). Then for some z0EK-{0} we would have wo = h(zo)gig(K). It would then follow from Runge's Theorem (see Rudin