Turk. J. Math. Comput. Sci.

Lemma 2.1 ( [7,23,26]). Let d ∈P be of the form (2.1). (i) Then, for n ≥1 |dn| ≤2. (2.2) The inequality holds for all n ≥1 if and only if d(z) = 1 + λz 1 −λz, |λ| = 1. (ii) Also, if µ ≥0 then |dn+k −µdndk| ≤2 max {1; |2µ −1|} =  2, if 0 ≤µ ≤1, 2|2µ −1|, otherwise. (2.3)

Lemma 2.2 ( [4], Lemma 2.2.). If d ∈P has the form (2.1), then αd3 1 −βd1d2 + γd3 ≤2 (|α| + |β −2α| + |α −β + γ|) . (2.4)

Lemma 2.3 ( [32], Proposition 1). Let d ∈P be given by (2.1). Let B1, B2 and B3 be numbers such that B1 ≥0, B2 ∈C and B3 ∈R. Define ψ+(d1, d2) and ψ−(d1, d2) by ψ+(d1, d2) = |B2d2 1 + B3d2| −|B1d1|, and ψ−(d1, d2) = −ψ+(d1, d2). Then, ψ+(d1, d2) ≤  |4B2 + 2B3| −2B1 when |2B2 + B3| ≥|B3| + B1, 2|B3| otherwise, (2.5) and ψ−(d1, d2) ≤

Lemma 2.4 ( [17]). Let p ∈P be given by (2.1). Then, 2d2 =d2 1 + tξ, 4d3 =d3 1 + 2d1tξ −d1tξ2 + 2t(1 −|ξ|2)η, 8d4 =d4 1 + 3d2 1tξ + (4 −3d2 1)tξ2 + d2 1tξ3 + 4t(1 −|ξ|2)(1 −|η|2)γ (2.7) + 4t(1 −|ξ|2)(d1η −d1ξη −¯ξη2), for some ξ, η, γ ∈¯D and t = (4 −d2 1).

Theorem 3.1. Let the function f ∈A given by (1.1) be a member of the class S ∗∗ s,e. Then, |a2| ≤1 2, |a3| ≤1 2, |a4| ≤5 16, |a5| ≤78226 196476 = 0.3981453205. The first two coefficient estimates are sharp.

Theorem 3.2. If f ∈S ∗∗ s,e has of the form (1.1), then for any complex number µ, we have |a3 −µa2 2| ≤1 2 max ( 1,

Corollary 3.3. If the function f ∈A given by (1.1) belongs to the function class S ∗∗ s,e, then |a3 −a2 2| ≤1 2. The estimate is sharp and the extremal function is given in (1.3). In following theorem, we investigate the upper bound of Hankel determinant of order two for the function that belongs to the class S ∗∗ s,e.

Theorem 3.4. If the function f ∈A given by (1.1) belongs to the class S ∗∗ s,e, then |a2a4 −a2 3| ≤1 4. This bound is sharp.

Theorem 4.1. Let the function f ∈A given in the form (1.1) be the member of the class S ∗∗ s,e. Then, |Γ1| ≤1 4, |Γ2| ≤1 4, |Γ3| ≤1 6, |Γ4| ≤7 16. The first three estimates are sharp.

Theorem 4.2. Let the function f ∈A given in the form (1.1) be the member of the function class S ∗∗ s,e. Then, |H2(1)(F f −1/2)| ≤1 16. This bound is sharp.

Theorem 4.3. Let the function f ∈A given by (1.1) be a member of the class S ∗∗ s,e. Then, |T2(1)(F f −1/2)| ≤1 16. This bound is sharp.

Theorem 4.4. Let the function f ∈A given by (1.1) be a member of the class S ∗∗ s,e. Then, −1 2 ≤T2,1(F f −1/2) ≤15 256.

Theorem 5.1. Let the function f ∈S ∗∗ s,e. Then, − 1 2 √ 5 ≤(|Γ2| −|Γ1|) ≤1 4.

Theorem 6.1. Let f ∈S ∗∗ s,e be of the form (1.1) and its logarithmic inverse coefficients F f −1/2 is given by (1.8). Then, J2,2(F f −1/2) = |Γ3 −Γ2 2| ≤2754681 14204928 = 0.1939243198.

Theorem 6.2. Let the function f ∈S ∗∗ s,e. Then, |Γ4 −Γ2Γ3| ≤ 917 1536.

Lemma 2.1 and (2.4) of Lemma 2.2, we have |Γ4 −Γ2Γ3| ≤4 ( 13 6144 +

Theorem 7.1. Assume that f ∈S ∗∗ s,e. Then, |Γ4 −Γ23| ≤ 495 1024.

Definition 1.1. A function f ∈S given by (1.1) is said to be the member of the class S ∗∗ s,e if the following subordination condition holds: 2zf ′(z) f(z) −f(−z) ≺1 + zez = p(z) (z ∈D), where the function p maps the unit disk D onto a cardioid domain in the right half plane.