Int. J. Anal. Appl. (2024), 22:90

Lemma 2.1. ( [8]) If p ∈P, then |Pk| ≤2 (2, 3, 4, · · · )

Lemma 2.2. ( [6]) Let g be a Sigmoid function defined in (1.2) and ϕ(z) = 2g(z) = 1 + ∞ X m=1 (−1)m 2m   ∞ X n=1 (−1)m n! zn

Lemma 2.3. ( [6]) Let g be a Sigmoid function defined in (1.1) and ϕn,m(z) = 1 + ∞ X m=1 (−1)m 2m   ∞ X n=1 (−1)m n! zn

Lemma 2.4. ( [6]) Let ϕ(z) ∈P and be starlike, then f is a normalized univalent function of the form (1.1). Setting m = 1, Fadipe et al. [6] remarked that ϕ(z) = 1 + ∞ X n=1 cnzn (2.3) where cn = (−1)n+1 2n! , then |cn| ≤2 for n = 2, 3, 4, · · · and the result is sharp for each n. 3. Some coefficient estimates for the class of Mλ,(∗) (η, ϕn,m) In this section, we will find the estimates on the coefficients ap+1bp+1, ap+2bp+2 and ap+3bp+3 for functions in the class Mλ,(∗) (η, ϕn,m).

Theorem 3.1. Let ϕn,m(z) = 1 + ∞ X m=1 (−1)m 2m   ∞ X n=1 (−1)m n! zn

Corollary 3.1. For coefficient ap+1bp+1, ap+1bp+1 = (1 −λ + λp)|η| 2p(1 + λp) is written and since ϕ(λ) = (1−λ+λp) (1+λp) , ϕ ′(λ) < 0 in the interval 0 ≤λ ≤1 and ϕ(λ) is decreasing, it will be |η| 2(p + 1) ≤|ap+1bp+1| ≤|η| 2p (3.10) for 1 2 ≤(1−λ+λp) (1+λp)

Theorem 4.1. If F(z) ∈AP given by (1.1) belongs to the class Mλ,(∗) (η, ϕn,m) then, |ap+2bp+2 −µ  ap+1bp+1 2 | = |η|2 4p(p + 1)

Theorem 4.2. If F(z) ∈AP given by (1.1) belongs to the class Mλ,(∗) (η, ϕn,m) then, |(ap+1bp+1)(ap+3bp+3) −  ap+2bp+2 2 | ≤ |η|2 48p2(p + 1)2(p + 2)  (p + 1)|3η2 −p(p + 1)| + 3(p + 2)|η|2 (4.2)