J. Indones. Math. Soc.

Lemma 1.1. Let f ∈A. Then (i) J0,α,β λ,0 f(z) = f(z), (ii) J1,0,β 1,0 f(z) = Z z 0 f(t) t dt = f(z) ∗(−log(1 −z)) = z + ∞ X n=2 anzn ∈A.

Lemma 1.2.[14] Let p ∈P, that is, p be analytic in U, be given by p(z) = 1 + P∞ n=1 pnzn and ℜ{p(z)} > 0 for z ∈U. Then |p2 −p2 1 2 | ≤2 −|p1|2 2 and |pn| ≤2 for all n ∈N. 2. General Properties of Jk,α,β λ,δ In this section we study the characterization properties and distortion theo- rems for the function f(z) ∈A to belong to the classes Sk,α,β λ,δ (µ) and Ck,α,β λ,δ

Theorem 2.1. Let f(z) ∈A. If for α ≥0, β ≥1, δ ≥0 and λ ≥0 ∞ X n=2 (n −µ)|an| [(βn)α + (n −1)(βn)αλ]kC(δ, n) ≤1 −µ, 0 ≤µ < 1, (4) then f(z) ∈Sk,α,β λ,δ (µ). The result (4) is sharp.

Theorem 2.2. Let f(z) ∈A. If for α ≥0, β ≥1, δ ≥0 and λ ≥0 ∞ X n=2 n(n −µ)|an| [(βn)α + (n −1)(βn)αλ]kC(δ, n) ≤1 −µ, 0 ≤µ < 1, (5) then f(z) ∈Ck,α,β λ,δ (µ). The result (8) is sharp.

Theorem 2.3. Let the hypotheses of Theorem 2.1 be satisfy. Then for z ∈U and 0 ≤µ < 1 |Jk,α,β λ,δ f(z)| ≥|z| −1 −µ 2 −µ and |Jk,α,β λ,δ f(z)| ≤|z| + 1 −µ 2 −µ.

Theorem 2.4.Let the hypotheses of Theorem 2.21 be satisfy. Then for z ∈U and 0 ≤µ < 1 |Jk,α,β λ,δ f(z)| ≥|z| −(1 −µ) 2(2 −µ)|z|2 and |Jk,α,β λ,δ f(z)| ≤|z| + (1 −µ) 2(2 −µ)|z|2.

Theorem 2.5. Let the hypotheses of Theorem 2.1 be satisfy. Then (n −µ) [(βn)α + (n −1)(βn)αλ]kC(δ, n) ≥1, ∀n ≥2 and 0 ≤µ < 1 implies |f(z)| ≥|z| −(1 −µ)|z|2 and |f(z)| ≤|z| + (1 −µ)|z|2.

Theorem 2.6. Let the hypotheses of Theorem 2.2 be satisfy. Then (n −µ) [(βn)α + (n −1)(βn)αλ]kC(δ, n) ≥1, ∀n ≥2 and 0 ≤µ < 1 poses |f(z)| ≥|z| −(1 −µ) 2 |z|2 and |f(z)| ≤|z| + (1 −µ) 2 |z|2. 3. Fekete-Szeg¨o for the Classes Sk,α,β λ,δ (µ) and Ck,α,β λ,δ

Theorem 3.1. Let the hypotheses of Theorem 2.1 be satisfy. Then |a2| ≤2(1 −µ)[(β2)α(1 + λ)]kC(δ, 2) 1 + µ and for all ν ∈C the following bound is sharp |a3 −νa2 2| ≤2(1 −µ)[(3β)α(1 + 2λ)]kC(δ, 3) max n 1, |1 + 2(1 −µ) (1 + µ)[(3β)α(1 + 2λ)]kC(δ, 3) −2ν 1 −µ (1 + µ)2 {[(2β)α(1 + λ)]kC(δ, 2)}2 [(3β)α(1 + 2λ)]kC(δ, 3) |

Corollary 3.1. Let the assumptions of Theorem 3.1 hold. Then for µ = 0 |a2| ≤2[(2β)α(1 + λ)]kC(δ, 2) and |a3 −νa2 2| ≤2[(3β)α(1 + 2λ)]kC(δ, 3) max n 1, |1 + 2 [(3β)α(1 + 2λ)]kC(δ, 3) −2ν {[(2β)α(1 + λ)]kC(δ, 2)}2 [(3β)α(1 + 2λ)]kC(δ, 3) | o . In the similar manner we can prove the following result.

Theorem 3.2. Let the hypotheses of Theorem 2.2 be satisfy. then |a2| ≤2(1 −µ)[(2β)α(1 + λ)]kC(δ, 2) 3 + µ

Corollary 3.2. Let the assumptions of Theorem 3.2 hold. Then for µ = 0 |a2| ≤2 3[(2β)α(1 + λ)]kC(δ, 2) and |a3−νa2 2| ≤1 3[(3β)α(1+2λ)]kC(δ, 3)max n 1, |1 2 + 2 3 −ν 3{[(2β)α(1+λ)]kC(δ, 2)}2| o . Acknowledgement: The work here is fully supported by MOHE: UKM-ST-06-