J. Math. Computer Sci., 29 (2023), 40–51

Lemma 2.1 ([7]). For a function p (z) ∈P of the form (1.2), the sharp inequality |pn| ⩽2 holds for each n ⩾1. Equality holds for the function p (z) = 1+z 1−z.

Lemma 2.2 ([8]). Let p (z) ∈P of the form (1.2) and µ ∈C. Then |pn −µpkpn−k| ⩽2max {1, |2µ −1|} , 1 ⩽k ⩽n −1. If |2µ −1| ⩾1, then the inequality is sharp for the function p (z) = 1+z 1−z or its rotations. If |2µ −1| < 1, then the inequality is sharp for the function p (z) = 1+zn 1−zn or its rotations.

Theorem 3.1. If f (z) ∈A and of the form (1.1) belongs to SC∗(α, δ, A, B) , then |a2| ⩽T, |a3| ⩽T 2 |−ξ + Υ −1| , |a4| ⩽T 6  |−3ξ + 4Υ −2| + ξ2 −3Υξ + 2Υ2 , |a5| ⩽T 24  2 |−4ξ + 6Υ −3| + −6ξ2 + ξ (−3 + 22Υ) + 6Υ (1 −3Υ)

Corollary 3.2. For special values of the parameters α, δ, A and B, we have following. (a) If f (z) ∈A given by (1.1) belongs to SC∗(0, 0, 1, −1) ≡SC∗, then |a2| ⩽2, |a3| ⩽3, |a4| ⩽4, |a5| ⩽5, |a6| ⩽6 and |a7| ⩽7. (b) If f (z) ∈A given by (1.1) belongs to SC∗(0, δ, 1, −1) ≡SC∗(δ) , then |a2| ⩽2Φ, |a3| ⩽Φ (2Φ + 1) , |a4| ⩽Φ 3  2 1 + 3Φ + 2Φ2
 , |a5| ⩽Φ 6 

Theorem 3.3. If f (z) ∈A and of the form (1.1) belongs to SC∗(α, δ, A, B), then a22 −a3 ⩽T |ξ + Υ −1| 2 , where ξ = Te−iα, T = Ψtαδ, Ψ = A −B, tαδ = cos α −δ and Υ = 1 + B.

Corollary 3.4. For special values of the parameters α, δ, A, and B, we obtain following. (a) Let f (z) ∈SC∗(0, 0, 1, −1) ≡SC∗. Then a22 −a3 ⩽1. (b) Let f (z) ∈SC∗(0, δ, 1, −1) ≡SC∗(δ) . Then a22 −a3 ⩽Φ |2Φ −1| , where Φ = 1 −δ. (c) Let SC∗(0, 0, A, B) ≡SC∗(A, B) . Then a22 −a3 ⩽AΨ 2 . (d) Let f (z) ∈SC∗(α, δ, 1, −1) ≡SC∗(α, δ) . Then a22 −a3 ⩽tαδ

Theorem 3.5. If f (z) ∈A and of the form (1.1) belongs to SC∗(α, δ, A, B), then a32 −a5 ⩽T 24 −6ξ2 + ξ (−3 −10Υ) + 6Υ (−1 + 3Υ) + 2 |−4ξ + 6Υ −3| + 5ξ3 −6Υξ2 −5Υ2ξ + 6Υ3 , where ξ = Te−iα, T = Ψtαδ, Ψ = A −B, tαδ = cos α −δ, and Υ = 1 + B.

Corollary 3.6. For special values of the parameters involved, we obtain following. (a) Let f (z) ∈SC∗(0, 0, 1, −1) ≡SC∗. Then a32 −a5 ⩽23 3 . (b) Let f (z) ∈SC∗(0, δ, 1, −1) ≡SC∗(δ) . Then a32 −a5 ⩽Φ 6  3Φ |4Φ + 1| + |8Φ + 3| + 20Φ3 , where Φ = 1 −δ. (c) Let f (z) ∈SC∗(0, 0, A, B) ≡SC∗(A, B) . Then a32 −a5

Theorem 3.7. If f (z) ∈A and of the form (1.1) belongs to SC∗(α, δ, A, B) , then a42 −a7 ⩽ T 720  24 |−6ξ + 10Υ −5| + 6 15ξ2 + 3ξ (5 −21Υ) + 20Υ (−2 + 3Υ)

Corollary 3.8. For special values of the parameters α, δ, A, and B, we obtain following. (a) Let f (z) ∈SC∗(0, 0, 1, −1) ≡SC∗. Then a42 −a7 ⩽73 5 . (b) Let f (z) ∈SC∗(0, δ, 1, −1) ≡SC∗(δ) . Then a42 −a7 ⩽Φ 90 [6 |12Φ + 5| + 45Φ |2Φ + 1| + 30Φ2 |9Φ + 4| + 20Φ3 |21Φ + 4| + 20Φ +15Φ2 + 152Φ5 , where Φ = 1 −δ. (c) Let SC∗(0, 0, A, B) ≡SC∗(A, B) . Then a42 −a7 ⩽Ψ 720

Theorem 3.1, it is natural to devote further investigation to other properties for functions in the class SC∗(α, δ, A, B) such as the Fekete-Szeg¨o functional, Hankel and Toeplitz determinants, Krushkal inequal- ity and logarithmic coefficients. One may also refer to, for example, [2, 13, 14, 24, 26, 29] for some ideas on these properties for newly defined subclasses of analytic and univalent functions. Meanwhile, the results obtained in Theorems 3.3 and 3.5 could lead to an alternative method dif