uSIGMOID FUNCTION IN THE SPACE OF UNIVALENT lambda-PSEUDO STARLIKE FUNCTIONS

Lemma 1.1. (see [9]). If a function p ∈P is given by p(z) = 1 + p1z + p2z2 + · · · (z ∈U),

Lemma 1.3. If z is a complex number having positive real part, then for any real number t such that t ∈[0, 1], we have ℜzt ≥(ℜz)t.

Lemma 2.1. [4]Let h be a sigmoid function and Φ(z) = 2h(z) = 1 + ∞ X m=1 (−1)m 2m ∞ X n=1 (−1)n n! zn !m

Lemma 2.2. [4] Let Φn,m(z) = 1 + ∞ X m=1 (−1)m 2m ∞ X n=1 (−1)n n! zn !m

Lemma 2.3. [4] If Φ(z) ∈P and it is starlike, then f is a normalized univalent function of the form(1). Taking m = 1, Joseph et al [4] remarked the following:

Theorem 2.5. If f ∈A and of the form (1) is belonging to Lβ λ(Φ) (λ ≥ 1 ∈R, ) then |a2| ≤ 1 −β 2(2λ −1) (7) |a3| ≤ (1 −β)2(4λ −2λ2 −1) 4(2λ −1)2(3λ −1) (8) |a4|

Corollary 2.6. If f(z) ∈A given by (1) belongs to Lβ 1(Φ) ≡S∗(β, Φ), then |a2| ≤1 −β 2 ; |a3| ≤(1 −β)2 8 and |a4| ≤1 −β 72 + (1 −β)3 48

Corollary 2.7. If f(z) ∈A given by (1) belongs to Lβ 2(Φ) ≡G(β, Φ), then |a2| ≤1 −β 6 ; |a3| ≤(1 −β)2 180 and |a4| ≤1 −β 168 + 27(1 −β)3 22680 . By taking β = 0 in Corollary 2.6 and 2.7we get

Corollary 2.8. If f(z) ∈A given by (1) belongs to L0 1(Φ) ≡S∗(Φ), then |a2| ≤1 2; |a3| ≤1 8 and |a4| ≤ 1 144.

Corollary 2.9. If f(z) ∈A given by (1) belongs to L0 2(Φ(z)) ≡G(Φ), then |a2| ≤1 6; |a3| ≤ 1 180 and |a4| ≤ 1 140. 3. The Fekete-Szeg¨o Inequality Recently there has been interest to obtain the Fekete-Szeg¨o inequality for the

Theorem 3.1. If f(z) ∈A given by (1) be in the class Lβ λ(Φ) and µ ∈R. Then |a3 −µa2 2| ≤ (1 −β)2 4(2λ −1)2

Theorem 3.3. If f(z) ∈A given by (1) be in the class Lβ λ(Φ), then |a2a4 −a2 3| ≤ (1 −β)2 48(2λ −1)(4λ −1) 1 −λ(1 −β)2[24λ4 −60λ3 + 44λ2 −12λ + 1] (2λ −1)3(3λ −1)2 . (21)