KYUNGPOOK Math. J. 53(2013), 467-478

Theorem 2.1. Let the function f = h + g be given by (1.1). If (2.1) ∞ X k=2 k −β β −1 |ak| + ∞ X k=1 k + β β −1 |bk| ≤1, where 1 < β ≤4 3, then f ∈MH(β).

Theorem 2.2. Let the function f = h + g ∈SH be given by (1.1). If (2.3) ∞ X k=2 k(k −β) β −1 |ak| + ∞ X k=1 k(k + β) β −1 |bk| ≤1, where 1 < β ≤3

Theorem 2.3. A function f of the form (1.4) is in VH(β), if and only if (2.4) ∞ X k=2 (k −β) |ak| + ∞ X k=1 (k + β) |bk| ≤β −1, where 1 < β ≤4 3.

Theorem 2.4. A function f of the form (1.5) is in UH(β). If and only if (2.6) ∞ X k=2 k(k −β) |ak| + ∞ X k=1 k(k + β) |bk| ≤β −1, where 1 < β ≤3 2.

Theorem 2.5. f ∈clco VH(β), if and only if (2.7) f(z) = ∞ X k=1 {xkhk(z) + ykgk(z)} , where h1(z) = z, hk(z) = z + β−1 k−β zk, (k = 2, 3, ...) and gk(z) = z + β−1 k+β zk, (k = 1, 2, ...), P∞ k=1(xk + yk) = 1, xk ≥0, yk ≥0. In particular, the extreme points of VH(β) are {hk} and {gk}.

Theorem 2.6. f ∈clco UH(β), if and only if (2.8) f(z) = ∞ X k=1 {xkhk(z) + ykgk(z)} , where h1(z) = z, hk(z) = z + β−1 k(k−β)zk, (k = 2, 3, ...) and gk(z) = z + β−1 k(k+β)zk, (k = 1, 2, ...), P∞ k=1(xk + yk) = 1, xk ≥0, yk ≥0. In particular, the extreme points of UH(β) are {hk} and {gk}.

Theorem 2.7. Let f ∈VH(β). Then for |z| = r < 1, we have |f(z)| ≤(1 + |b1|)r + β −1 2 −β −β + 1 2 −β |b1|  r2 and |f(z)| ≥(1 −|b1|)r − β −1 2 −β −β + 1 2 −β |b1|  r2.

Theorem 2.8. Let f ∈UH(β). Then for |z| = r < 1, we have |f(z)| ≤(1 + |b1|)r + β −1 2 −β −β + 1 2 −β |b1|  r2 2 and |f(z)| ≥(1 −|b1|)r − β −1 2 −β −β + 1 2 −β |b1|  r2 2 .

Corollary 2.9. Let f of the form (1.4) be so that f ∈VH(β). Then  ω : |ω| < 3 −2β 2 −β + 2β −1 2 −β |b1|  ⊂f(U).

Corollary 2.10. Let of the form (1.5) be so that f ∈UH(β) . Then  ω : |ω| < 5 −2β 2(2 −β) + 3(β −1) 2(2 −β)|b1|  ⊂f(U). For our next theorem, we need to define the convolution of two harmonic func- tions. For harmonic functions of the form f(z) = z + ∞ X k=2 |ak|zk − ∞

Theorem 2.11. For 1 < α ≤β ≤ 4 3, let f ∈VH(α) and F ∈VH(β). Then f ∗F ∈VH(α) ⊆VH(β).

Theorem 2.12. For 1 < α ≤β ≤ 3 2 let f ∈UH(α) and F ∈UH(β). Then f ∗F ∈UH(α) ⊆UH(β).

Theorem 2.13. The class VH(β) is closed under convex combination.

Theorem 2.14. The class UH(β) is closed under convex combination. 3. Inclusion Relations To prove our next theorem, we shall require the following lemma due to Jahangiri [8].

Lemma 3.1. Let f = h + g ∈VH be given by (1.4) and if ∞ X k=2 k −α 1 −α |ak| + ∞ X k=1 k + α 1 −α |bk| ≤1, (0 ≤α < 1). Then f ∈V ∗ H(α).

Theorem 3.1. If f ∈VH(β) then f ∈V ∗ H  4−3β 3−2β  .

Theorem 3.2. If f ∈UH(β) then f ∈Vk  4−3β 3−2β  . Following corollaries are an easy consequences of Theorem 3.1 and 3.2.

Corollary 3.3. VH(β) ⊂VH 4 3
⊂V ∗ H.

Corollary 3.4. UH(β) ⊂UH 3 2
⊂Vk.

Corollary 3.5. UH 4 3
⊂VH 6 5
.

Corollary 3.6. UH 4 3
⊂V ∗ H 2 3
. 4. A Family of Class Preserving Integral Operator Let f(z) = h(z) + g(z) be defined by (1.1). Let us define F(z) by the relation (4.1) F(z) = c + 1 zc

Theorem 4.1. Let f(z) = h(z) + g(z) ∈SH be given by (1.4) and f ∈VH(β), where 1 < β ≤4 3. Then F(z) defined by (4.1) is also in the class VH (β) .

Theorem 4.2. Let f(z) = h(z) + g(z) ∈SH be given by (1.5) and f ∈UH(β), where 1 < β ≤3 2. Then F(z) defined by (4.1) is also in the class UH (β) . Acknowledgements. The authors are thankful to the referee for his valuable com- ments and suggestions. References [1] O. P. Ahuja, Planar harmonic univalent and related mappings, J. Inequal. Pure Appl. Math., 6(4)(2005), Art. 122, 1-18. [2] Y. Avci and E. Zlotkiewicz, On harmonic univalent mappings, Ann. Univ. Mariae Curie-Sklodowska Sect., A44(1990),