TAMKANG JOURNAL OF MATHEMATICS

Lemma 2.1 ([3]). Let P(z) = 1+Cw(z) 1+Dw(z) = 1+P∞ n=1 pnzn, then |pn| ≤(C −D). Result is sharp for the functions Pn(z) = 1+Cδzn 1+Dδzn , |δ| = 1 and n ≥1.

Lemma 2.2 ([4]). Let g ∈S∗(A,B), then, for A −(n −1)B ≥(n −2), (n ≥3), |bn| ≤ 1 (n −1)! nY k=2 (A −(k −1)B). Equality holds for the function g0(z) defined by g0(z) = z(1+Bδz)(A−B)/B, |δ| = 1. Since g(z) ∈C(A,B) implies that zg ′(z) ∈S∗(A,B), we have the following

Lemma 2.3. Let g ∈C(A,B), then, for A −(n −1)B ≥(n −2), (n ≥3), |bn| ≤1 n! nY k=2 (A −(k −1)B). Result is sharp for the function g1(z) defined by g ′ 1(z) = (1+Bδz)(A−B)/B, |δ| = 1.

Lemma 2.4 ([5]). Let g ∈S∗(A,B), then, for |s| ≤1, |t| ≤1, (s ̸= t) tg(sz) sg(tz) ≺    ³1+Bsz 1+Btz ´(A−B)/B , B ̸= 0; exp A(s −t)z, B = 0.

Lemma 2.5. If g ∈S∗(A,B), then, for |z| = r < 1, r(1−Br)(A−B)/B ≤|g(z)| ≤r(1+Br)(A−B)/B, B ̸= 0; (2.1) r exp(−Ar) ≤|g(z)| ≤r exp(Ar), B = 0; (2.2) ¯¯¯arg g(z) z ¯¯¯ ≤(A −B) B sin−1(Br), B ̸= 0; (2.3)

Lemma 2.6. If g ∈C(A,B), then, for |z| = r < 1, 1 A {1−(1−Br)A/B} ≤|g(z)| ≤1 A {(1+Br)A/B −1}, B ̸= 0; 1 A {1−exp(−Ar)} ≤|g(z)| ≤1 A {exp(Ar)−1}, B = 0; ¯¯¯arg g(z) z ¯¯¯ ≤A B sin−1(Br), B ̸= 0; ¯¯¯arg g(z)

Lemma 2.7 ([2]). Let f and g are analytic functions and h be convex univalent function in E such that f ≺h and g ≺h. Then (1−λ)f +λg ≺h, (0 ≤λ ≤1). 3. Coefficient estimates

Theorem 3.1. Let f ∈K (A,B;C,D). Then, for A −(n −1)B ≥(n −2), (n ≥3), |an| ≤1 n! nX k=2 {A −(k −1)B}+ (C −D) n ³ 1+ n−1 X k=2 1 (k −1)! kY

Theorem 3.2. Let f ∈K1(A,B;C,D). Then, for A −(n −1)B ≥(n −2), (n ≥3), |an| ≤1 n h 1 n! nY k=2 (A −(k −1)B)+(C −D) ³ 1+ n−1 X k=2 1 k!

Theorem 4.1. Let f ∈K (A,B;C,D), then ³ 1−Cr 1−Dr ´ (1−Br)(A−B)/B ≤|f ′(z)| ≤ ³ 1+Cr 1+Dr ´ (1+Br)(A−B)/B, B ̸= 0; (4.1) ³ 1−Cr 1−Dr ´ exp(−Ar) ≤|f ′(z)| ≤

Theorem 4.2. Let f ∈K1(A,B;C,D), then 1 Ar ³ 1−Cr 1−Dr ´ {1−(1−Br)A/B} ≤|f ′(z)| ≤1 Ar ³ 1+Cr 1+Dr ´ {(1+Br)A/B −1}, B ̸= 0; 1 Ar

Theorem 5.1. Let f ∈K (A,B;C,D), then |arg f ′(z)| ≤(A −B) B sin−1(Br)+sin−1n (C −D)r (1−CDr 2) o , B ̸= 0; (5.1) |arg f ′(z)| ≤Ar +sin−1n (C −D)r (1−CDr 2) o , B = 0. (5.2)

Theorem 5.2. Let f ∈K1(A,B;C,D), then |arg f ′(z)| ≤A B sin−1(Br)+sin−1n (C −D)r (1−CDr 2) o , B ̸= 0; |arg f ′(z)| ≤Ar +sin−1n (C −D)r (1−CDr 2) o , B ̸= 0. The results are sharp for the function f3(z) defined in (4.7) where δ1 and δ2 are given by (5.4) and (5.5), respectively.

Theorem 6.1. If f and h ∈K (A,B;C,D), then (1−λ)f +λh ∈K (A,B;C,D), (0 ≤λ ≤1).