Mathematics & Statistics

Lemma 1.2. [9] Consider the function ep (A, B; z) defined by (1.6) . Then i. The function ep (A, B; z) is univalent in the disk |z| < τ 2, where τ = 1− √ 5 2 . ii. If p (z) ≺ep (A, B; z) , then Rep (z) > α, where α = 2 (A + B −2) τ + 2 (2AB −A −B) τ 3 + 16 (A + B) τ 2η 4 (B −1) (τ + Bτ 3) + 32Bτ 2η , (1.8) where η = 4+τ 2−B2τ 2−4B2τ 4−(1−Bτ 2)√ 5(2Bτ 2−(B−1)τ+2)(2Bτ 2+(B−1)τ+2)

Lemma 1.4. [14] Let p ∈P such that p (z) = 1 + ∞ P n=1 cnzn. Then |cn| ≤2, n ≥1. (1.11) c2 −v 2c2 1 ≤max {2, 2 |v −1|} =  2, 0 ≤v ≤2, 2 |v −1| ,

Theorem 2.1. Let p (z) = 1 + p1z + p2z2 + . . . be in the class CP [A, B] . Then |p1| ≤ (A −B) |τ| 2 , (2.1) |p2| ≤ (A −B) (5 −B) |τ|2 22 . (2.2) Results are sharp.

Theorem 2.2. Let f ∈CS∗[A, B] , −1 ≤B < A ≤1 and of the form (1.1) . Then |a2| ≤ 1 2 |τ| (A −B) , (2.7) |a3| ≤ |τ|2 8 (A −B) (A −2B + 5) . (2.8) These results are sharp.

Theorem 2.3. Let f ∈CS∗[A, B] and of the form (1.4) . Then a3 −µa2 2 ≤(A −B) |τ| 8 max {2, |τ (−(A −2B + 5) + 2 (A −B) µ)|} . (2.16) This result is sharp.

Theorem 2.4. Let f ∈CS∗[A, B] and f−1 have the coefficients of the form (1.3) . Then for τ = 1− √ 5 2 , |A2| ≤ |τ| 2 (A −B) , |A3| ≤ |τ| 8 (A −B) max {2, |τ (3A −2B −5)|} . These results are sharp.

Theorem 2.5. Let f ∈CS∗[A, B] and having inverse coefficients of the form (1.3) . Then for µ a complex number and for |z| < τ 2, where τ = 1− √ 5 2 , A3 −µA2 2 ≤(A −B) |τ| 8 max {2, |τ (3A −2B −5 −2µ (A −B))|} . This result is sharp.

Theorem 2.6. Let f ∈CS∗[A, B] and the coefficients of log f(z) z be given by (1.4) . Then |γ1| ≤ |τ| 4 (A −B) , |γ2| = τ 2 16 (A −B) (5 −B). These results are sharp.

Theorem 2.7. Let f ∈CS∗[A, B] and the coefficients of log f(z) z be given by (1.4) . Then for µ, a complex number, we have γ2 −µγ2 1 ≤(A −B) |τ| 16 max {2, |τ (B −5 + µ (A −B))|} .