TAMKANG JOURNAL OF MATHEMATICS

Lemma 1. Let f = h + ¯g ∈VH be given by (1.2) and P∞ k=2 k|ak| + P∞ k=1 k|bk| ≤1 −α, (0 ≤α < 1) then f ∈VHP(α). 2. Main results

Theorem 2.1. A function f of the form (1.2) is in RH(β) if and only if ∞ X k=2 k|ak| + ∞ X k=1 k|bk| ≤β −1. (2.1)

Theorem 2.2. If f ∈RH(β), then |f(z)| ≤(1 + |b1|)r + 1 2(β −1 −|b1|)r2, |z| = r < 1 and |f(z)| ≥(1 −|b1|)r −1 2(β −1 −|b1|)r2, |z| = r < 1. The bounds are sharp for the functions f(z) = z + |b1|¯z + 1 2(β −1 −|b1|)¯z2 and f(z) = z + |b1|¯z + 1 2(β −1 −|b1|)z2 for |b1| ≤β −1.

Corollary 2.1. If f ∈RH(β), then {ω : |ω| < 1 2(3 −β −|b1|)} ⊂f(U). (2.2) Next we determine the extreme points of the closed convex hulls of RH(β), denoted by clco RH(β).

Theorem 2.3. f ∈clco RH(β), if and only if f(z) = ∞ X k=1 (λkhk + γkgk) (2.3) where h1(z) = z, hk(z) = z + β−1 k zk (k = 2, 3, 4, . . .), gk(z) = z + β−1 k ¯zk (k = 1, 2, 3, . . .) and P∞ k=1(λk + γk) = 1, λk ≥0 and γk ≥0. In particular the extreme points of RH(β) are {hk} and {gk}.

Theorem 2.4. If f ∈RH(β) then f ∈VHP(2 −β).

Theorem 2.5. RH(β) ⊆S∗ H, where 1 < β ≤2.

Theorem 2.6. Each function in the class RH(β) maps a disks Ur where r < infk n 1 k(β−1−|b1|) o 1 k+1 onto convex domains for β > 1 + |b1|.

Theorem 2.7. For 1 < β ≤α ≤2 let f ∈RH(α) and F ∈RH(β). Then f ∗F ∈ RH(β) ⊆RH(α).

Theorem 2.1. For F(z) ∈RH(α) we note that |Ak| ≤1 and |Bk| ≤1. Now, for the convolution function f ∗F, we have ∞ X k=2 k β −1|akAk| + ∞ X k=1 k β −1|bkBk| ≤ ∞ X

Theorem 2.8. The class RH(β) is closed under convex combination.

Theorem 2.9. Let f ∈RH(β) and δ ≤2−β. If F ∈Nδ(f), then F is harmonic starlike function.

Theorem 3.1. Let f(z) = h(z) + g(z) ∈SH be given by (1.2) and f(z) ∈RH(β) then F(z) be defined by (3.1) also belong to RH(β).