J. Math. Computer Sci., 26 (2022), 379–394

Lemma 2.1 ([17, 21]). Let P be the class of all analytic functions h(z) of the following form h(z) = 1 + ∞ X n=1 cnzn, (z ∈U) satisfying ℜh(z) > 0 and h(0) = 1. Then the sharp estimates |cn| ⩽2(n ∈N) are true. In Particular, the equality holds for all n about the next function h(z) = 1 + z 1 −z = 1 + ∞ X n=1 2zn. For the sake of brevity we assume that Λ2 and Λ3 are assumed as in (1.3), α and β are positive and

Theorem 2.2. Let f(z) be given by (1.1). If f ∈QΣγ,k α,β(ϑ, ρ; φ), then |a2| ⩽min  E1 ϑ + ρ, s 2(|E2 −E1| + E1) (ρ + 1)(2ϑ + ρ)|B0|, E1 √2E1 p |Π(ϑ, ρ, B0, E1, E2)|  |B0|

Corollary 2.3. Let f(z) be given by (1.1). If f ∈BΣγ,k α,β(ρ; φ), then |a2| ⩽min  E1 ρ + 1, s 2(|E2 −E1| + E1) (ρ + 1)(ρ + 2)|B0|, E1 √2E1 p |Π(ρ, B0, E1, E2)|  |B0|

Corollary 2.4. Let f(z) be given by (1.1) and ϑ > 0, and if f ∈SΣγ,k α,β(ϑ; φ), then |a2| ⩽min    E1 ϑ , s |E2 −E1| + E1 ϑ|B0| , E1 √E1 q

Theorem 2.6. Let f(z) be given by (1.1). If f ∈QΣγ,k α,β(ϑ, ρ; φ) and δ ∈R, then |a3 −δa2 2| ⩽    (|B0|+|B1|)E1 (2ϑ+ρ)|Λ3| , if 2(2ϑ + ρ)|B0|E2 1|Λ2 2 −δΛ3| ⩽|ΠΛ2 2|, |B1|E1 (2ϑ+ρ)|Λ3| + 2|B0|2E3

Corollary 2.7. Let f(z) be given by (1.1). If f ∈QΣγ,k α,β(ϑ, ρ; φ), then |a3| ⩽  (|B0|+|B1|)E1 (2ϑ+ρ)|Λ3| , if 2(2ϑ + ρ)|B0|E2 1 ⩽|Π|, |B1|E1 (2ϑ+ρ)|Λ3| + 2|B0|2E3 1 |ΠΛ3| , if 2(2ϑ + ρ)|B0|E2 1 ⩾|Π|, where Π = Π(ϑ, ρ, B0, E1, E2) is given in (2.3). 3. Functional estimates for f ∈MΣγ,k

Theorem 3.1. If f(z) is assumed as in (1.1) and f ∈MΣγ,k α,β(τ, ϑ, ρ; φ), then |a2| ⩽min{F1, F2, F3}

Corollary 3.2. If f(z) is as assumed in (1.1) and f ∈MΣγ,k α,β(τ, ρ; φ), then |a2| ⩽min    |B0τ|E1 (ρ + 2)|Λ2|, s |B0τ|(|E2 −E1| + E1) |Ω(ρ)Λ2 2| , |B0τ|E1 √E1

Corollary 3.3. If f(z) is as assumed in (1.1) and f ∈MΣγ,k α,β(τ, ϑ; φ), then |a2| ⩽min  |B0τ|E1 (ϑ + 1)|Λ2|, s |B0τ|(|E2 −E1| + E1) |Ω(ϑ)Λ2 2| , F  for F = |B0τ|E1

Theorem 3.5. Let f(z) be fixed as in (1.1) and f ∈MΣγ,k α,β(τ, ϑ, ρ; φ) and δ ∈R. Then |a3 −δa2 2| ⩽ |B1τ|E1 2|6ϑρ + 2ϑ −2ρ + 1||Λ3| + |B2 0τ2|E3 1|Λ2 2 −δΛ3| |B0E2 1τΩ(ϑ, ρ) + (E1 −E2)(2ϑρ + ϑ −ρ + 1)2||Λ2|2|Λ3| if 2|B0τ|E2 1|(6ϑρ + 2ϑ −2ρ + 1)(Λ2

Corollary 3.6. Let f(z) given by (1.1) belongs to the class MΣγ,k α,β(τ, ϑ, ρ; φ). Then |a3| ⩽ |B1τ|E1 2|6ϑρ + 2ϑ −2ρ + 1||Λ3| + |B2 0τ2|E3 1 |B0E2 1τΩ(ϑ, ρ) + (E1 −E2)(2ϑρ + ϑ −ρ + 1)2||Λ3| if 2|B0τ|E2 1|6ϑρ + 2ϑ −2ρ + 1| ⩾|B0E2 1τΩ(ϑ, ρ) + (E1 −E2)(2ϑρ + ϑ −ρ + 1)2|, or