TAMKANG JOURNAL OF MATHEMATICS

Lemma 1.1 ([15]). Let q(z) = P∞ n=1 Bnzn be analytic and convex univalent in ∆. If p(z) = P∞ n=1 Anzn is analytic in ∆and satisfies the subordination p(z) ≺q(z), then |An| ≤|B1| (n = 1, 2, . . . ) .

Lemma 1.2. [6, p.254] If the function ω ∈B given by (1.3). Then w2 = ξ 1 −w2 1
, w3 = 1 −w2 1
1 −|ξ|2
ζ −w1 1 −w2 1
ξ2,

Lemma 1.3. [7, p.10] If the function ω ∈B given by (1.3), then w2 −µw2 1 ≤max {1, |µ|} . Let us denote by Q the class of functions f that are analytic and injective on ∆\ E(f), where E(f) =  ζ : ζ ∈∂∆ and lim z→ζ f(z) = ∞  , and are such that

Lemma 1.4. [10, p.24] Let q ∈Q with q(0) = 1 and let p(z) = 1 + p1z + · · · be analytic in ∆ with p(z) ̸= 1. If p ̸≺q in ∆then there exits points z0 ∈∆and ζ ∈∂∆\ E(q) and there exits a real number m ≥1 for which p(|z| < |z0|) ⊂q(∆), p(z0) = q(ζ), z0p′(z0) = mζq′(ζ). The purpose of this work is to define a new subfamily of P related to a domain bounded by LB(λ) =  ρeiϕ : ρ =  2 cos ϕ λ λ

Lemma 2.1. The functions qλ(z) are convex univalent in ∆for each 0 < λ < 1. Moreover gλ(z) = (qλ(z) −1)/λ ∈CV((1 + λ)/2). Also, if |z| = r < 1, then min |z|=r |qλ(z)| = qλ(−r) and max |z|=r |qλ(z)| = qλ(r).

Theorem 2.1. Let p(z) ∈H with p(0) = 1. If p(z) ≺qλ(z), (z ∈∆) , then |Arg {p(z)}| < λπ 2 , 0 < ℜ{p(z)} < 2λ, (z ∈∆) , (2.3) and p 1 λ (z) −1 < 1, (z ∈∆) .

Theorem 2.2. If a function f belongs to the class G(λ), then f ′ ≺qλ in ∆. Also, f is univalent function in ∆.

Corollary 2.3. Let f ∈G(λ) for 0 < λ < 1. Then the function g(z) = z exp Z z 0 f ′(t) −1 t dt  belongs to S∗ L(λ). Example 1. The function f(z) = z exp(−Az) belongs in class S∗ L(λ) if |A| ≤ λ 2+λ.

Theorem 2.4. If f ∈S∗ L(λ) and |z| = r < 1, then −Fλ(−r) ≤|f(z)| ≤Fλ(r), F ′ λ(−r) ≤|f ′(z)| ≤F ′ λ(r), |Arg {f(z)/z}| ≤max |z|=r Arg {Fλ(z)/z} . Equality holds for some z ̸= 0 if and only if f is a rotation of Fλ. Also, If f ∈S∗ L(λ), then either f is a rotation of Fλ or {w ∈C: |w| ≤−Fλ(−1)} ⊂f(∆). Here −Fλ(−1) is understood to be the limit of −Fλ(−r) as r tends to 1. For the special case λ = 1/2, results for functions belonging to the class S∗

Theorem 3.1. Let f ∈S∗ L(λ). Then the logarithmic coefficients of f satisfy |γn| ≤λ 2n (n ≥1) . All the inequalities are sharp.

Theorem 4.1. let f ∈S∗ L(λ) given by (1.1). Then a2a4 −a2 3 ≤λ2 4 . The inequalities are sharp.

Theorem 4.2. let f ∈S∗ L(λ) given by (1.1). Then we have sharp inequalities a3 −δa2 2 ≤        −λ2δ + 1−3λ 4λ

Theorem 4.3. Let f ∈S∗ L(λ) and F(z) = z/f(z) given by (1.1) and (4.3), respectively. Then we have sharp inequalities b2 −δb2 1 ≤        −λ2δ −λ+1 4λ

Theorem 4.4. let f ∈S∗ L(λ) and f −1(z) given by (1.1) and (4.6), respectively. Then we have sharp inequalities A3 −δA2 2 ≤        −λ2δ −5λ+1 4λ