Integral representations of bounded starlike functions

Theorem 2.1. Let Ωα = {w : |w −α| ≤Re w + α}. Define Pα(z) to be the analytic and univalent function with the properties Pα(0) = 1, P ′ α(0) > 0 and Pα(U) = Ωα. Then (2.1) Pα(z) = α  1 + 4 π2  log 1 + p wα(z) 1 − p

Theorem 2.2. Let Pα(z) be as in (2.1). Then for 0 < α < ∞, Pα ∈H2. P r o o f. Let Qα(z) be the function in (2.2) and define Ak such that Qα(z) = α + 4α π2 ∞ X k=1 Akzk. Then, from [10], we know that Ak = 4 k k X m=1 1

Theorem 3.1. If f ∈SP(α) then there is a constant K(α) such that |f(z)| < |z|K(α), |z| < 1. P r o o f. If α1 < α2 then SP(α1) ⊂SP(α2), so it is enough to prove the theorem for α > 1. Let kα be the function in SP(α) with the property

Theorem 3.2. Let f ∈SP(α), 0 < α < ∞. Then f ′ ∈H2. P r o o f. It is enough to prove that zf ′ ∈H2, and from Theorem 3.1 it follows that |zf ′(z)|2 < K(α)2

Corollary 3.3. Every function in SP(α) belongs to F0. R e m a r k. It is natural to compare the classes SP(α) to the classes of strongly starlike functions, SS(α), studied e.g. in [1]. A function f ∈ SS(α) if and only if |arg(zf ′(z)/f(z))| < πα/2, so in this case we have an angular domain instead of a parabola. According to results in [1] the functions in SS(α) share many properties of the functions in SP(α), e.g. that they are bounded and that f ′ ∈H1. However, we do not get f ′ ∈H2 as in SP(α

Corollary 3.4. Every function in SP(α) maps |z| = 1 onto a rectifiable Jordan curve. If f(z) = z + P∞ k=2 akzk is a function in SP(α) then there is a µ ∈ M such that f(z) = R Γ log(1/(1 −xz)) dµ(x), which means that an = (1/n) R Γ xn dµ(x) and therefore we have

Corollary 3.5. The order of growth of the coefficients in SP(α) is O(1/n). R e m a r k. Ma and Minda [8] proved that the order of growth of the coefficients for functions in SP 1 2, 1 2
is O(1/n). Note that by Corollary 3.5 this order of growth holds in all the classes SP(α, β). 4. Some special cases. We now go back to the more general classes SP(α, β). Because of the inclusion SP(α, β) ⊂SP(α, 0), 0<β < 1, the re- sults about boundedness and membership in F0 will also hold for SP(α, β). As menti

Theorem 4.1. Let f ∈SP 1−γ 2 , 1+γ 2
and let Kγ be as in (4.1), −1 ≤ γ < 1. Then f(z) maps |z| = 1 onto a rectifiable curve of length at most 2πKγIγ where Iγ = ∞R 0 p (1 + γ)2 + 2(3 −4γ + γ2)v2 + (1 −γ)2v4eπv/2 1 + eπv dv.

Theorem 4.2. Let f ∈SP 1−γ 2 , 1+γ 2
. Then M2(r, f ′) ≤Kγ p 3 −4γ + 2γ2, where Kγ is as in (4.1). P r o o f. As in the proof of Theorem 4.1 we have 2πR 0 |f ′(eiθ)|2 dθ ≤K2 γ